![NOTATIONAL CONVENTIONS xv
infinite dimensional analogs of Gelfand pairs. That elegant area is extremely active
but its level of technicality takes it out of the scope of this book.
Acknowledgment s
Much of the material in Parts 1, 2 and 3 was the subject of courses I taught at
the University of California, Berkeley, over a period of years. Questions, comments
and suggestions from participants in those courses certainly improved the exposi-
tion. Some of the material in Part 3 relies on earlier treatments of J. Dieudonne [Di]
and J. Faraut [Fa], and much of the material in Part 4 depends on O. Yakimova's
doctoral dissertation [Y3]. In addition, a number of mathematicians looked at early
versions of this book and made useful suggestions. These include D. Akhiezer (com-
munications concerning his work with E. B. Vinberg on weakly symmetric spaces),
D. Bao (discussions on Finsler manifolds), R. Goodman (advice on how to organize
a book), I. A. Latypov and V. M. Gichev (communications concerning their work
on invariant function algebras), J. Lauret, H. Nguyen and G. Olafsson (for going
over the manuscript), G. Ratcliff and C. Benson (communications concerning their
work with J. Jenkins on spherical functions for commutative Heisenberg nilmani-
folds), and the three mathematicians who refereed this volume (for some very useful
remarks).
I especially want to thank O. Yakimova for a number of helpful conversations
concerning her work and E. B. Vinberg's work on classification of smooth commu-
tative spaces.
Notational Conventions
M, C, M and O denote the real, complex, quaternionic and octonionic number
systems. If F is one of them, then x H-> X* denotes the conjugation of F over R,
F m x n
denotes the space o f m x n matrices over F, and if x G F m x n
then x* e F n X m
is its conjugate transpose. We write R e F n x n
for the hermitian (x = x*) elements of
F n x n
and ReFp Xn
for those of trace 0, and we write ImFn X n
for the skew-hermitian
(x + #* = 0) elements of Fn X n
; that corresponds to the case n = 1.
In general we use upper case roman letters for groups, and when possible we
use the corresponding lower case letters for their elements. If G is a Lie group then
g denotes its Lie algebra. If I) is a Lie subalgebra of g then (unless it is defined
differently) H is the corresponding analytic subgroup of G.](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-17-320.jpg)
![4 1. BASIC TOPOLOGICALGROUP THEORY)
PROOF. Let F C X be a closedsubset and let g E G with g 302242
F. By applying
69.1 We can assume g = 1. Now W =
GF is a neighborhood of 1. As G>< G 342200224>
G is
continuous We have an openneighborhood U of 1 with U >4 U C W. So U = UI"IU3422002301
is an open neighborhood of 1 with U = U3422002341
and U2 C W.)
The closure U C W. For if h E U then every neighborhood of h meets U, so
hU meetsU, say hb = a where a,b E U. Now h = ab3422002341
6 UU3422002301
= U2 C W.)
Set V =
GU. Then U and V are disjoint, g E U, and F C V. El)
When G be a topological group and H is a subgroup, H carries the subspace
topology unless weexplicitlyspecifyto the contrary. So the open sets in H are the
U357254202
H where U is open in G. It is easy to checkthat H is a topological group.)
COROLLARY 1.1.6. Let H be a closed subgroup of a topologicalgroup G. If G
is locally compactthen H is locally compact. If G is compact then H is compact.)
PROOF. A closed subspace of a compact Hausdorfftopologicalspaceis com-
pact.GisHausdorff (Tychonofl separation condition T2) by Lemma 1.1.5. D)
1.2. Subgroups, Quotient Groups, and Quotient Spaces)
The cosetspaceG/H carries the quotient topology: if 7r : G 342200224342200224>
G / H denotes
the projection,7r(g)
=
gH, then a set U C G/H isopen if and only if 7r3422002341(U)
is
open in G. Also, if U is openin G so is the union UH =
Uheh Uh of open sets,
and then 7r(U) is open in G/H because UH = 7r3422002301(7r(U)). So the map 7r is both
continuous and open. The opensubsets of G/H are the sets 7r(U)
=
{gH ] g E U}
where U runs over the opensubsets of G.)
If H is a normal subgroup of G then G/ H inherits a group structure from G,
and the map (gH,g342200231H)
+342200224>
(gH)(g342200231H)3422002311
=
gg342200231_1H
is continuous. So then G/H is
a topological
group if and only if its points are closed.In this connection, note that
G / H is T1if and only if H is closed in G. SoWe have)
LEMMA 1.2.1. Let H be a closednormal subgroup of G. Then G/H is a topo-
logical group, the projection 17 : G 342200224>
G /H is a continuous homomorphism,and H
is the kernel of 7r. (Note that 7r : G 342200224>
G/H is an open map.))
The converseis the topological
version of the standard isomorphism theorem:)
PROPOSITION
1.2.2. Let f : G 342200224>
L be a continuous homomorphism between
topological groups. Let H C G be the kernel of f and let M = f (G)C L, the image
of f. Then f factors through a continuous injectiue homomorphism 7 ofG/H onto
M.)
In general one cannot expect 7 to beahomeomorphism. For example, let L be
the multiplicative group of all 2 X 2 diagonal matrices diag{e342200230/:T0, e342200230/:i342200230l342200231}
with 0,11)
real. Let G be the additive group of real numbers. Choose an irrational number
357254202
and de357254201ne
f : G 342200224342200224>
L by f (t) = diag{e342200230/342200230_1342200230,e342200230/342200230_15342200231}.
Since ,6342200231
is irrational, the
image of f can342200231t
be a closed curve, but its closuremust be a torus, so that closure
is all of L. Thus the image M = f (G) is densein L. Evidently f is one to one.)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-21-320.jpg)
![253)
,s.a,302273vP)
1.4. COVERING GROUPS 7)
image is connected because G is connected,so the image is the single point {h}
because H is totally disconnected. D)
1.4. Covering Groups)
Let G be a topological
spacethat is connected, locally arcwise connected, and
semi342200224locally simply connected in the large3. Fix a basepoint g E G. Every sub-
group I342200230
C 711 (G, g) of the fundamental group correspondsto a covering space
fr : (G,_?j) 342200224>
(G, g) such that fp maps 7r1(G,g)isomorphically onto I342200230.
In this
section we will prove)
THEOREM 1.4.1. Let G be a connected, locally arcwise connected topological
group that is semi342200224loca.lly simply connected in the large. Then the fundamental
group 7r1(G, 1) is commutative. Suppose that I342200230
C 7r1(G,1) is a subgroup and let
fr : (G, T) 342200224+
(G, 1) denote the corresponding coveringspace.Then G has a unique
structure of topological group such that T is the identity element and fr is a group
homomorphism.)
PROOF. We will write 1g rather than 1 for the identity element ofG,in order to
avoid confusion. First recallone of the standard constructions of the coveringspace
fp : (G, 342200224342200224>
(G,1g). The elements of G are the homotopy classes (endpoints
357254201xed)
[0] of continuous arcs 0,)
0 : [0,1]342200224>
G such that 0(0) = lg)
modulo the equivalence relation)
[0] 2 [T] <:>
{0(1)
= 7(1)and)
I342200230
contains the homotopy class (endpoints 357254201xed)
of 'r3422002301
- 0.)
Here, as usual, r3422002351(t)
=
r(1342200224t)
and p-0(t)
= 0(2t) for 0 302247
t g %, p-0(t) = p(2t3422002241)
for
% 302247
t 302247
1. We write [[0]] for the equivalenceclassof
[0]. The base point 1}}
is the class[[01]]
of the trivial arc, 01 (t) = 1g for 0 302247
t 302247
1, and the projection is
fr([l0ll) = 0(1)-
We
first suppose that F = {I}, so we are dealing with the universal covering
space f : (G, 342200224>
(G, lg).)
De357254201ne
inverse in G by [[0]]3422002301
= where n(t) = 0(t)3422002301.
If [[0]]
= [[0342200231]]
then
we have a homotopy H :0 2 0342200231
with endpoints 357254201xed.
In other words, if yo = 0(1)
then H: [0,1]><[0,1]
342200224>
G is continuous, H(s,0) = lg, H(s, 1)= go, H(0,t)
=
0(t),
and H(1,t) = 0342200231(t)
. Now K 2 7; 2 n342200231
where n(t)
= 0(t)3422002311,
n342200231(t)
=
0342200231(t)3422002301,
and
K (s,t) = H (5,t)3422002301.
Thus inverse is well de357254201ned,
and the de357254201nition includes the
fact that f([[Ull3422002301)= f(l[Ull)3422002351~)
De357254201ne
composition in G by = where n(t) = 0(t)r(t). If [[0]]
=
[[0342200231]]
and =
[[r342200231]],
so we have homotopies H : 0 2 0342200231
and K : r 2 7" with
endpoints 357254201xed,
then J : r] 2 77342200231
where r](t)
= 0(t)r(t), n342200231
(t)
= 0342200231
(t)r342200231
(t) and)
3This last condition means: if g 6 G there is an arcwise connected open neighborhood U of
g such that, if 'y is a closed curve in U based at g then 1 is nullhomotopic (endpoints 357254201xed)
in G.)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-24-320.jpg)
![1.6. THE LOCALLY COMPACT CASE 11)
Now we can start to settle someproblems raised in Proposition 1.2.2 and
Lemma 1.5.4:)
PROPOSITION 1.6.5. Let G be a topologicalgroup, L a closed subgroup, and K
a compact subgroup. Let 71' 2 G 342200224>
G/K denote the projection onto the quotient
space, let 7rL : L 342200224342200224>
7r(L) denote 7rlL, and let 7;: L/LFWK
342200224>
7r(L) be the pushdown
of 7r]L to L/L H K. Then 7rL is both open and closed, and 357254201
is a homeomorphism.)
PROOF. We 357254201rst
prove that 7rL is a closed map. LetT bea closed
subset of L.
Then T is closed in G. By Lemma 1.6.3TK isclosed
in G. Now G TK is open
in G. The map 7r is open so 7r(G TK) is open in G/K. Thus 7r(T) is closed in
G / K, soit also is closed in 7r(L). This provesthat 7rL is closed.)
Let u : L 342200224>
L/L O K denote the projection. If Y is closedin L/L H K then
,u.3422002301Y
is closed in L, so 357254201(Y)
=
7rL(u3422002301Y) is closed in 7r(L). Thus if is a closed
map. But 357254201
is continuous and bijective, so now it is a homeomorphism. It follows
that in; is open. El)
COROLLARY 1.6.6. If K is a compact normal subgroup of G and L is a closed
subgroup of G, then 7r : G 342200224>
G/K induces a topological group isomorphism4 of
L/L 0 K onto LK/K closed in 7r(L).)
We can also handle a question of separateversus joint continuity:)
THEOREM 1.6.7. Let G be a locally compact and o~compact5 topologicalgroup.
LetX be a locally compact Hausdor357254201342200230
space. Suppose that CI) : G><X 342200224>
X satis357254201es
all
the conditions for a topologicaltransformation group such that X is a homogeneous
space of G, except that <I> is only assumed to be separately continuous. Then {D is
jointly continuous. Let me E X and let H denote the isotropy subgroup of G at 930.
Then g s342200224>
T9 (:30) de357254201nes
an open map 7 : G 342200224>
X , and 7 induces a homeomorphism
of G / H onto X.)
PROOF.We need only check that if U is a neighborhood
of 1 in G then <P(U,9:0)
is a neighborhood
of mg in X. For then 7 is open.Since we already know that 7
is continuous and surjective, it will follow that 7 induces a homeomorphism 7 of
G / H onto X. Thenjoint continuity of <I> will follow from joint continuity of the
map G X G / H 342200224342200224>
G / H induced by multiplication in G.)
Let U be a neighborhood of 1 and choose an open neighborhood
V = V3422002341
of
1 with V2 C U. As G is locally compact, V contains a compact neighborhood F
of 1. As G is o~compact it contains a sequence {gn},n = 1,2,3,
..., such that
G = U gnF. Since7 iscontinuous and surjective, the 7(gnF) are compact and they
cover X. Now some 7(gnF) contains an opensubset Z C X. If f E F such that
7(gn f) E Z then also belongs to the open set <I>((gn f )3422002311,
Z), which is contained in
Q(f_1F,l'0) C 302242(l/2,1130) C .719) D)
COROLLARY 1.6.8. Let f : G 342200224>
G342200231
be a continuous surjectioe homomorphism
of locally compact topological groups. If G is a342200224compact then f is open.)
4By topological group isomorphism we mean an isomorphism in the category of topological
groups, that is, a. group isomorphism which is a homeomorphism.
5A topological space is called o342200224compact
if it is a countable union of compact subsets.))](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-28-320.jpg)
![12 1. BASIC TOPOLOGICAL GROUP THEORY)
Finally we note something that we can342200231t
do without:)
PROPOSITION 1.6.9. Let H be a subgroup of a topological group G. If H and
G / H are locally compact, then G is locally compact. If H and G/H arecompact,
then G is compact.)
PROOF. Let 7r : G 342200224>
G / H be the projection,U a closedneighborhood of 1 in
G such that H (1U3422002301U
is compact, and U1 a closed neighborhood of 1 such that
Uf1U1 C U. As G /H is locally compact there is a compact neighborhood 0 of
7r(1) such that C342200231
C 7r(U1). Set V = U1 0 7r3422002301(C),
closed neighborhood of 1 in G.
We are going to prove that V is compact.)
Let B be any family of subsets of V that is maximal for the property that
357254201nite
intersections are non-empty. We will prove compactness of V by showing
that HBEBB # (ll. As 7r(V)
= C, which is compact, 357254202BEB7r(B) 79 (7), say with)
v E V such that 1r(v) E 7r(B) for all B E B. H 0 v3422002301U
is compact because it is
closedin the compact set H 0 U3422002301U.
Now UH O U is compact.)
If HBB were empty, to every g E UH (1U we could associatea neighborhood
V9 of 1 and some By E B such that
Vg2g
(1 B9
= (0. As UH Fl U is compact it is
covered by a 357254201nite
union Vglgl U U Vgngn. Set B = By,
F1 (1 B9342200235.
Then
B E B by maximality of the latter. W = U1 0 V91
0 302253
- - (1 V9" is a neighborhood
of 1 in G such that W(vH F] U) F1 B = 0. But 7r(Wv) meets 7r(B)becauseit is a
neighborhood of 7r(v). Sothere existin 6 VV, b E B, h E H such that wvh = b. Now
ch = w3422002311b
E Uf1U1 C U, so b E W(vH (1 U) H B = (ll. This contradiction proves
357254202BB
75 9}. Thus V is compact, and it follows that G is locally compact.)
If H and G/H are compact, wejust take C =
G/H so U = G and V = G is
compact. El)
1.7. Product Groups)
In this Section wediscussdirectproducts and semidirect products of topological
groups.)
DEFINITION 1.7.1. Let {G1-}i342202254A
be a family of topological groups indexedby
some set A. The direct product G = H,-EAG, of topological groups is direct
product G = H,-GAG, as abstract group with the direct producttopology. Thus the
group composition in G is)
(1-7-2) (gt-)(y302253E)
=
(9.-gl) and <90342200235
=
(95 )
and a basis for open sets in G is given by the products 1'I,EAU,- where U, = G,-)
except for a 357254201nite
number of indices 1342200231
E A, and for those indices U, is an open
subset of G,-. O)
LEMMA 1.7.3. G = H,-EAG, is a topological group, and each G, is a closed
normal subgroup.)
PROOF. Let (g,-), E G and let W =
ILEAW, be an open subset of G that
contains (g,-)(h,~)3422002301.
For each 2' with VV, 75 G, choose open neighborhoods U, of g,
and V, of h, in G, such that
U,-V171 C I/V,-. If W, = G, let U, = V;
= G,-. Then))](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-29-320.jpg)
![16 1. BASIC TOPOLOGICALGROUP THEORY)
Then p(g,h) = p(h, g) 3 O and p(g,h) 2 f(g,h) by construction, and it follows
that p(g, h) = 0 <=> g
= h. Right invariance comesfrom)
ply342200234:
=
1U-b0L342200230EG|f(.gu7x)
_
=
1ubzeG|f(9,wu'1)
342200224
f(h,93u'1)|
1342200230-1b97342202254Glf(gv37)
342200224 : p(ga h)302273
and the triangle inequality comes from
p(g:k)
= h1bacEGlf(gam)
342200224
+ 342200224
302247
p(g9h))
Thus p is a metric on G.)
To see that the topology induced by p is the original topology on G, we need
only show that they have the same neighborhoods of 1 E G, because both are
right342200224invariant. De357254201ne
BT to be the metric ball {gE G 1 p(g, 1) < 7"}. Unwinding
the de357254201nition of p one sees that V1/2n
C
B1/2n
. Thus the open metric ballsB1/2342200235
are neighborhoods of 1 in the original topology. And if 0 is a neighborhood of 1
in the original topology, then some Ok C 0, soB1/2:c+1
C 0 because g E B1/2k+1
implies p(g,1) < 1/2'342200234+1,
which implies f(g, 1) < 1/2k342200234,which in turn implies
g E
l/12/2k+1
C
V1/2:: C Ok C 0. That completes the proof. Cl)
DEFINITION 1.8.2. Let (G, p) be a topologicalgroup with a right342200224invariant
metric. A sequence {gn} C G is Cauchyif, given 6 > 0, there is an no > 0 such
that p(gm,gn) < 6 for m,n > no. A sequence {gm} C G converges to g E G if,
given 6 > 0, there is an no > 0 such that p(gn,g) < e for n > no. One then says
that g is the limit of {gm},written g = lim gn and also {gn} 342200224>
g. This forces {gm}
to be Cauchy. (G,p) is complete if every Cauchy sequencein G converges in G.
(G,p) is the completion of (H, pH) if H is a subgroup of G, (ii) pH = p|HXH,
and (iii) every g E G is the limit of a Cauchy sequence from (H, pH). 0)
We will now construct the completionof (G, p) and show that it is a topological
group. In effect we follow the Cauchy sequence construction that yields the real
number 357254201eld
as the completion of the rationals.)
Recall that Cauchy sequences {gn}, in a metric space (G,p) are called
equivalent if {p(gn,
342200224>
0 in the real numbers.)
EEMMA
1.8.3. Let (G, p) be a topologicalgroup with a right342200224in1Ja7iant metric.
get
G consist of the equivalenceclasses of Cauchy sequences {gn} in G. Then
G is a group with well de357254201ned
composition
= and inverse)
l{.9n}l_1= [{9;1}]-)
PROOF. De357254201ne
F : G X G 342200224+
G by F(g, h) = gh3422002301.
If {gn} C G is Cauchy,
so is {g;1}.To see this, note that the Cauchy condition for {gn} just says: if U
is a neighborhood
of 1 E G then F3422002301(U)
contains {(gm, gn) | m,n > no} for some
no > 0. Since g 1->
g3422002301
is a homeomorphism of G, the sameholds for the map
H (g, h) = g3422002301h,
not necessarily with the same no. Thus {gg1}isCauchy.)
If {gn}, C G are equivalent Cauchy sequences,then {gg
1}and {g357254201fl}
(which we just saw to be Cauchy) are equivalent. For the equivalence condition on
{gm} and just says: if U is a neighborhood
of 1 E G then F3422002301(U)contains)
r1;3](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-33-320.jpg)
![00224302242~
DQ_,)
1.8. INVARIANT METRICS ON TOPOLOGICAL GROUPS 17)
342200230l(9m
9:1) I n > no} for some no > 0. Now the same holds for H3422002311(U),for some no.
Thus {gg 1} and {g357254201fl}
are equivalent.)
If {gm}, C G are equivalent Cauchy sequences, and {h,,}, C G are
equivalent Cauchy sequences, then {gnhn} and {g;,h;,}are Cauchy sequences and
are equivalent. To see this, let U be an open neighborhood of 1 in G. We must
show that there is a number n1 such that
m,n 2 n1 implies(gmhm)(g,,h,,)3422002301,(gmhm)(g;,h;,)'1
E U.
First, let E = E3422002311
be an open neighborhood of 1 such that E3 C U and let no > 0
such that, if E 2 no then gg E Egno. Now W =
g;01Eg,,0is an openneighborhood
of 1 such that, if E 2 no then ggWg[1 C Egno Wg;O1E
= E3 C U. At this point we
choose an open neighborhood
V = V3422002341
of 1 in G such that V3 C W, and we let
n1 2 no such that)
m,n 2 n1 implies hmh;1,hmh302247,'1 E V and a,b 2 n1 impliesg302247,_1ga,g;1ga
E V.
If a,b, m, n 2 n1 now)
(9ahm)(9bhn)_1
: gm
'
(91l119a)(hmhr_z1)(9b34220022419n1) 342200230.9111
C gm Wrlil C E3 C U
and
(9ahm)(gbh342200231,n)_1
2
gni
'
(g'r:11ga)(hmhl1,_1)(gb3422002301gn1)
'
97:11 C gni W977: C E3 C U342200230)
That proves our assertions.)
We have now proved that
the rules of composition, describedabove, specify a
well342200224de357254201ned
group structure on G. [3)
LEMMA 1.8.4. Let (G, p) be a topological group with a right342200224invariant metric.
Construct the group G as in Lemma 1.8.3. De357254201ne
o5 : G 342200224>
G by 302242(g)
=
where gn = g for all n. Then (1) is an lnjectiue homomorphism of G into G. De357254201ne
o : G X G 342200224>
R by 357254201([{g,,}],
=
limg(gn,hn).
Then o is a well342200224de357254201ned)
right342200224invariant metric on the abstract group G, and 342200230p342200230(q$(g),
=
p(g, h) for all
g, h E G.)
PROOF. It is clear from the de357254201nition that (Z) : G 342200224>
G is an injective homo-
morphism.)
We check that 357254201([{g,,}],
=
limp(gn,hn) is well342200224de357254201ned
on G. Let
{gn}, C G be equivalent Cauchy sequences, and let {hn}, C G be equiv-
alent Cauchy sequences; we must prove that limn_,oo p(g,,, hn) exists and is equal
to limnnoo p(g;,,h302247,).
For the existence, let 6 > 0 and chooseno > 0 such that
E, m 2 no implies
p(-9579771) < 6/2302273
and <
Then if 342202254,m
2 no we have p(gg,hg)342200224
p(gm,hm) < 6, so limp(g,,,h,,) exists. Now
choose n1 > 0 such that m,n 2 n1implies p(gm,g302247,) < 6/2 and ,o(hm,h;,) < 6/2.
Then m, n 2 n1implies p(gm, hm)
342200224
p(g;,, < 6. So the two limits are the same,
and 5 is well de357254201ned.)
By construction, if g, h E G then o(q5(g), gb(h))
=
p(g, h).)
Now we check that ]6 is a right342200224invariant metric on G. The metric properties
p(gn7hn)
: /7(hn3022739n)
and p(gn:k342200231rL)
302247
p(9nvhn) + p(h342200231n=kn)
Survive to the limit342200230)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-34-320.jpg)
![18 1. BASIC TOPOLOGICAL GROUP THEORY)
Note : 1imp(gnagn)= 0- If : 03 then {.971} is
equivalent to {hn}, i.e. = Th_us p is a metric on the underlying set)
of G. It is right342200224invariant because [{an}] E G gives us
357254201(l{9n}ll{an}l, l{hn}ll{an}l)
=
1imp(9nan,gnan)
= 1imp(9m on) = /_7(l{9n}l302273 l{hn}l)-
Thus p is a right342200224invariant metric as asserted. Cl)
THEOREM 1.8.5. Let (G, p) be a topological group with a right342200224inuariant metric.
Construct the group G as in Lemma 1.8.3,the injection 302242
: G 342200224+
G as in Lemma
1.8.4, and the metric p on G as in Lemma 1.8.4.Then the underlying topology from
p givesG the structure of topological group, (p becomes an injectiue homomorphism,
and (p is a homeomorphism of G onto 302242>(G). Thus we can view G as a subgroup of
G, p as p|gXG, and (G,p) as the completion of (G, p).)
PROOF. We use Lemmas 1.8.3 and 1.8.4 to identify g e G with 302242(g)
e 6, thus
viewing G as a metric subgroupof G. Here p = plgxg and the p342200224-metric
balls
in G are just the intersections
of G with the p342200224metricballs in G. So the original
topology of G is its subspace topology in G. This completes the proof that (Gj)
is the completion of (G, p). D)
1.8.6. Different invariant metrics can give different completions. For example,
considerthe metricsonthe additive group of the rational number 357254201eld
Q given by
po302260(.1:,
y)
=
[ac
342200224
y] (usual absolute value) and pp(:17, y)
= 342200224342200224
jg/Hp (p342200224adic
norm, p
prime). The corresponding completions are the additive group of the real number
357254201eld
R and the additive groups of the p342200224adic
number 357254201elds
Qp.)
1.8.7. Everything is essentially the samefor_left342200224invariant metrics, and the
homeomorphism g n342200224342200224>
g'1 carries the completion (G,p) for a right342200224invariant metric)
p over to the completionfor the corresponding left342200224invariant metric.)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-35-320.jpg)
![CHAPTER 2)
Some Examples)
Lie groupsand linear algebraic groups are the most interesting and useful topo-
logical groups. Without going into any structure theory here, we summarize some
basicfacts on linear Lie groups and then turn to a number of important exam-
ples.Thoseexamplesinclude general linear groups G'L(n; IF), special linear groups
SL(n;IF), unitary groups U (p, q; IF) of various signatures over the real, complex
and
quaternion 357254201elds,
real and complex symplectic groups S'p(n;IF), p342200224adic
completions
of linear algebraic groups de357254201ned
over the rationals, and various types of Heisen-
berg groups. Other examples are concernedwith application to harmonic analysis
on spheres, projective spacesand Grassmann manifolds. All of these exampleswill
be used later in the book.)
2.1. General and SpecialLinear Groups)
Let IF be the real number 357254201eld
IR, the complex number 357254201eld
(C, or the quaternion
division algebra ll-II. Let V be a 357254201nite
dimensional vector space over IF. Since we342200231ll
be concentrating on groups of linear transformations of V it will be convenient to
have linear transformations act from the left, so that the correspondence between
matrices and linear transformations does not reverse order of products. Thus we
want scalars to act from the right, that is, we take V to be a right vector space over
IF. Now a choice ,6 = {e1,...,en}
of basis of V gives an isomorphismof V with the
right vector space IF" of n X 1 (column) vectors with entries from IF, and here every
lF342200224linear
transformation of V corresponds to an n X n matrix with entries from IF
acting on IF342200235
by g : v 302273342200224+
gv. Now we have the general linear group)
(21.1) GL(V)
E
GL(n; IF) : invertible IF342200224linear
transformations of V E IF".)
Here we think of G'L(V) as a group of linear transformations and GL(n; IF) as a
multiplicative group of matrices.)
Recall the notation IF""" for the space of n X n matrices over IF. If IF is IR or
(C then IF342200235X"
is a vector space of dimensionn2 over IF. If IF = H then it is a vector
spaceof dimension 477,2 over IF.)
We can View V as a vector space of dimension m342200230
over R where 1" =
dim], IF.
Then GL(V) consists of all lR342200224l'1near
transformations g : V ~> V with det,R (g) a303251
0
that also are lF~linear, i.e. that commute with the (right) scalar action of IF. This
exhibits GL(n; IF) as an opensubset of the real vector space IF
"X"
ER R342200231"342200231342200230342200231
In the
subspace topology the matrix group GL(n;IF) now is a locally compact topological
group, and GL(V)acquires the same structure from the isomorphismof (2.1.1).)
19))](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-36-320.jpg)
![20 2. SOME EXAMPLES)
We can also view GL(V) as a closedsubmanifold of lF2"X2", where
(2.1.2) on E GL(V) corresponds to (3 (L91)6 lF2""2".)
When IE342200230
is R or C, so that one has the determinant, it is more usual to use)
GL0/)
3'
3 (det(3z)3422002311
) i
3422002301
E
Fnxn})
with the group isomorphismgiven by a I342200224>
(3 det(;_1
The general linear group has someobvious and useful subgroups, such as
(2.1.3) GL'(V)%
GL'(n;lF) : all elements that preserve Lebesgue measure on V.
So
GL'(n;lR)
= {g E GL(n;lR) | det(g) = :l:1},
GL342200231(n;<C)
=
{g E GL(n;C) | |det(g)| = 1}= {gE GL(n; (C) |detRg = 1},and
GL'(n;H) = {.9
E GLWH) I detm(9)
=
1}-
A slightly smaller but more famous subgroup is the special linear group
(2.14) SL(V)
2 SL(n;lF) : the derived group of GL(V)
302247
GL(n;)
All these are closedsubgroupsof general linear groups, hence also are locally com-
pact topological
groups.)
These groups have an interesting involutive automorphism, the Cartan invo-
lution given on the matrix level by)
(2.1.5) 0 : g v342200224->
tg3422002301
where the bar designates conjugation of IF over R. It has 357254201xed
point set
(2-1-5) {9 E GLWJF) I 9(9)
=
9} = U(n;F),)
which will be de357254201ned
below and which turns out to be a maximal compact subgroup.)
2.2. Linear Lie Groups)
For our purposes, a linear Lie group is a group G of linear transformations
of a (real, complex or quaternionic) vector space V de357254201ned
by some polynomial
equations (in the matrix entries for a choice of basis of V). Thus G is a linear
algebraicgroup by de357254201nition, and we will see just how G is a Lie group. By
de357254201nition, a Lie group is a C342200234
(real analytic) differentiable manifold with a group
structure such that the group operationsare C342200234.)
First, we may always regard V as a real vector space, perhaps at the cost of
doubling or quadrupling its dimension,sowe may View G as a real linear Lie group
perhaps at the additional costof increasing the number of de357254201ning
equations. That
done, 71 = dim V, and a basis chosenfor V, now G may be viewed as the group
of all n X n real matrices (gm) that satisfy a collection of polynomial equations
f;,(g171, . . . ,g,,,,,) 2 0. The implicit function theorem now ensures that generically
and locally G is a C342200234
submanifold of the vector space ]R"X" of n X n real matrices.
Applying group translations G is globally a C342200235
submanifold of Rm342200235.
The group
operations are polynomial in lR""", thus C342200235
on the C"" submanifold G. Thus, by
de357254201nition, G is a Lie group.)
iaw](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-37-320.jpg)
![2.3. GROUPS DEFINED BY BILINEAR FORMS 21)
Equation (2.1.2) shows that the general linear groups GL(n;lF) are linear Lie
groups. It follows that the GL342200231
(n;lF) and the special linear groups SL(n;lF)are
linearLiegroups because we obtain them by just enlargingthe setof lR342200224polynomial
equations used to define GL(n;)
We will want the Lie algebra g of a linear Lie group G. By de357254201nition (only
for linear Lie groups!) g consists of all linear transformations E of V such that
exp(t302247) E G for all real t. Hereexp(t302247) means the exponential series 2,20 %302247",
and
it is easy to seethat this series converges for all real t. Onecanprove thz357254201;
g is closed
under linear combination and alsounder the composition [5, 77]
=
517
342200224
775. Thus g
is a vector spaceof linear transformations of V with the Lie algebracomposition)
[30224377]-)
DEFINITION 2.2.1. The automorphism 0 : GL(n;F) 342200224+
GL(n; lF), given by
0(g) = E? as in (2.1.5), is a Cartan involution of GL(n;lF). If a linear Lie group
G C GL(n;lF)is stableunder 0 (or one of its conjugates in the group of automor-
phisms of GL(n; then 6|G(orthe restrictionof the conjugate that stabilizes G)
is a Cartan involutionof G. If I357254202
is a Cartan involution of G then dzp : g 342200224>
g is
called a Cartan involution of the Lie algebra g, and the 357254201xed
point set K = G342200230(342200231
is
a maximal compact subgroup of G. O)
2.3. Groups De357254201ned by Bilinear Forms)
Let b be a bilinear form on V. That means b: V X V 342200224+
]F is an lF342200224bilinearmap.
In general an lF~bilinea.r map T : U x V 342200224>
W factors through an lF342200224linear
map
U (8) V 342200224+
W, so U 302256
V must be de357254201ned
and must be a (right, in our case) vector
space over IF. This requires F to be commutative. Sowe only consider bilinear
forms in the real and complex cases.)
Fix a bilinear form b on V. Then a choice,5342200231
=
{e1, . . . , en} of basis of V gives
a matrix B = (b(e,-,e,-))such that, identifying a vector v = Ev,-6," E V with the
column vector 342200230(vb
. . . , 11,") we have
(2.3.1) b(u,1J) =
Z u,-b,-7]-vj
= t'u,B'u.
1302247i,j302247n
7
The bilinear form now de357254201nes
a group
O(V, b) = {g E GL(V)|b(gu,gv)= b(u,v) Vu,v E V}
-%
{9 E GL('n;TF) l *9Bg
=
B}-
Each of the groups(23.2)is a subgroup of the general linear group de357254201ned
by
a system of quadratic equations in the matrix entries, thus a closed subgroupof
the general linear group, and thus a locally compact topological
group.Thegroup)
(2.3.2) is called the orthogonal group of b when b is symmetric and nondegener-
ate,calledthesymplecticgroupof b when b is antisymmetric and nondegenerate.)
(2.32))
Suppose that b is symmetric and nondegenerate.In the complexcasethere
is a basis in which the matrix B = I, the identity matrix, and the corresponding
group is the complexorthogonal
group)
(23.3) O(n;C)
= {g E GL(n;<C) I tgg
=
I}.)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-38-320.jpg)
![22 2. SOME EXAMPLES)
In the real case there is a basis in which B is the matrix Ipjq
=
(15 jg ),
0 302247
13 302247
n
and p + q = n, and the corresponding
groups are the real orthogonal groups)
(23.4) 0(p,q) = {gE GL(n;lR) | tgIp,qg
=
Ipyq} Where Ip,q =
(15 _0Iq))
of 342200234signature342200235
(p, q). They are called inde357254201nite orthogonal groups when pq a303251
0.
If pq = 0, then O(p,q) is the ordinary orthogonal group O(n) = O(0,n)
=
O(n, 0),
and the condition of (2.3.4) is the standard condition tgg = I.)
Taking determinant on the de357254201ningconditions of (2.33) and (2.34) we see
that it forces det(g)2 = 1. The additional condition det(g)= 1de357254201nes
the special
orthogonal groups SO(n,C) C O((n,(C)and SO(p,q) C O(p,q). In all cases the
specialorthogonal group is a subgroup of index 2 in the orthogonal group. S O(n, (C)
and SO(n) are connected,
but S'O(p, q) has 2 components when pq 75 0.)
Now suppose that b is antisymmetric and nondegenerate. Then 71 = dimp V is
even, n = 2m, and there is a basis in which B is the matrix J = (_?m 15"
The
corresponding groups are the complex symplectic group)
(2.3.5) Sp(m;(C)
=
{g E GL(n;<C) | tgJg = J} where J =
(_(}m 15342200235).
and the real symplectic group
(23.6) Sp(m;R)= {ge GL(n;R) 5 guy
=
J} where J = (_9m 15302273).)
The groups Sp(m, ]F) are connected. Note that S'p(m; R) simply consists of the real
matricesin Sp(m;
C).)
In all cases, we have the Cartan involution 6 of G given by the restriction of
the Cartan involution (2.1.5) of the general linear group. It has 357254201xed
point sets)
(2-3-7) {9 E 0(n;<C) I 9(9)
=
9} = 0(n; C) (7 GL(n;R)
=
0(9),
(2-3-8) {9 6 0(p.q) I 0(9)
=
9} = 0(1), 9) 0 0(9) g 0(9)X
0(9),
(2-3-9) {9 6 Sp(m;<C) I 9(9)
=
9} = Sp(m; C) 0 U(2m) 342200231302243
Sp(m),)
(2-3-10) {9342202254Sp(m;R)|0(9)=9} = S;v(m;1R)3572542020(2m)
%
U(m),)
respective maximal compact subgroupsof O(n; (C), O(p, q), Sp(m; (C) and S'p(m; R).
Here Sp(m) will be de357254201ned
later.)
2.4. Groups De357254201ned by Hermitian Forms)
Let h be an hermitian form on V. This will mean that h : V X V 342200224342200224>
F is
linear in the 357254201rst
variable, conjugate3422002241inear in the second variable, and satis357254201es
h(v, u)
= h(u, 11).This isthe usual (for mathematicians 342200224
not for physicists) role of
the 357254201rst
and second variables. It is slightly inconvenient in the quaternionic case,
where conjugation reverses the order of products, but we meet that problem with
the formulation (2.4.1) for the matrix expressions
of hermitian forms.)
Now, as for bilinear forms,a choice,6
=
{e1, . . . , en} of basis of V gives a matrix
H = (hm) = (h(ej,e302242))
such that, identifying a Vector v =
261% E V with the
column vector 342200230(v1,.
, . ,vn), we have)
(2.4.1) h(u,v) = Z v_ih,-,3-uj
=
v*Hu, 11* = 342200230E.
1302247i,j302247n)
6](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-39-320.jpg)
![2.4. GROUPS DEFINED BY HERMITIAN FORMS 23)
The hermitian form now de357254201nes
a group
(2.42) U(V, h) = {gE GL(V) 1 h(gu,gv)
=
h(u,v) 1u,v E V}
% {9 E GL(n;1F) lg*Hy = H}, 9*
=
342200230@-)
As mentioned above, there is a delicatepoint here in the quaternionic case, where
it usesthe calculation)
h(9ua!]U)
=
Z
91312112)
hm"
!Jj,mUm>
I m)
id342200231)
:
Z(g*)342202254,'ihi,jgj,m
um =
If m) i,j l,m)
Each of the groups (2.4.2)isa subgroup of the general linear group de357254201ned
by a
system of quadratic equations in the real components of the matrix entries,thus is
a closed subgroup of the generallinear group, soit is a locally compact topological
group and a linear Lie group.)
V has a basis that is orthonormal in the sense that H = Ipvq
=
(if _[}q ),)
0 302247
p 302247
n and p + q = m, the matrix we met while describing the real orthogonal
groups. The corresponding
groups are the (inde357254201nite) unitary groups over F,)
(2-4-3) U(p, q;F) = {9E GL(n;F) I 9* my
=
Im}, 0 E p,q and :0 + q = 7302273-
When h is de357254201nite,
i.e. when pq = 0, these are the usual unitary groupsover F,
U(n;lF)
= U(n,0;lF) = U(0,n;
This notation is standard, but unfortunately there are several standards.Another
standard notation is to write
(2-44) 0(1),q) for U (p,q;1R), U (P, 4) for U(:0,q;C), and 317(1), q) for U(:D,q;1HD,)
reflecting the respective names orthogonal, unitary, and symplectic unitary,
of 342200234signature342200235
(p, q), for these groups. In the de357254201nite
case one has the standard
notation)
O(n) for U(n;lR), U(n) for U(n;(C), and S'p(n)for U('n;lHl)
and one usually refers to thesegroups as the orthogonal, unitary and symplectic
unitary groups.
Thecorresponding (inde357254201nite) special unitary groups are the
(2.4.5) SU(p,q; IF)
=
U(p, q;lF) 357254202
SL(n;lF) where 0 302247
1), q and p -1-q = n.
In the real and complexcasesS'U(p,q;lF)
=
{g E U(p,q;lF) I det(g)
=
1}. There is
no distinction in the quaternion case: SU(p, q; H) = U(p,q; llil). Except in the case
where F = R and n = 2, SU (19, q; IF) is the derived group of U(p, q;lF As before,
when h is de357254201nite
these are the usual special unitary groups over F,
SU(n; F) = SU(n, 0;lF) = S'U(0,
n;F),
and one has the other standard notation
(2-46) 30(1),
c1)
=
5U(;v,q;R), 571(1), <1)
=
3U(p,q;<C), Sp(p,q) = 3U(P,q;IH1))](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-40-320.jpg)
![24 2. SOME EXAMPLES)
re357254202ectingthe respective names special orthogonal, special unitary, and sym-
plectic unitary, of 342200234signature342200235
(p, q), for these groups. And of course in the
de357254201nite
case we have the notation)
SO(n) for SU(n;]R), SU(n) for .S'U(n; (C), and Sp(n) for SU(n;lHI),
calledthe special orthogonal,specialunitary and symplectic unitary groups.)
Again the Cartan involutions of our groups are of the form (21.5), that is,
9(g) = (g*)'1. It has 357254201xed
point set)
(24-7) {9 E U(p,q;1F) I 9(9)
=
9} = U(p,q;1F) F7 U(n;1F)
% U(p;F) ><
U(9;IF))
in the full unitary case, but is slightly more complicated in the special unitary case.
Of course nothing new happens in the quaternionic setting, but)
(2-4-8) {9 E SU(p,9;1F) 19(9)
=
9} = 5U(p,9;1F) F) U(n;1F)
ES(U(p;lF) ><
U(q;lF)), l6342200230=R,(C.)
The 357254201xed
point set of 9 is a maximal compact subgroup in all cases.)
One can alsoconsidergroups de357254201ned
by skew342200224hermitian forms s on V. So
s:V x V 342200224+
F is linear in the second Variable, conjugateslinear in the 357254201rst
variable,
and satis357254201es
s(u, v) + s(v, u) = 0. Inthe real case, skew342200224hermitian is the same as
antisymmetric, so We get the groups S'p(n;lR). In the complex case,a matrix S is
skew342200224hermitian if and only if it is of the form 5' ==
/3422002243422002243422002241H
where H is hermitian.
In that case, g*Sg = S if and only if g*Hg = H, sowe get the groups U(p, q) and
SU(p,q). But in the quaternionic case we get something new,)
(24.9) SO*(V, s) 2 {g E GL(V) | s(gu,gv)= s(u,v) for all 14,11 6 V}.
We can choosea basis of V in which s has matrix U342200235,
and then this group is written
(24.10) S'O*(2n)
= {gE GL(n;lHI) | g.*Sg = S} where S = H342200235.
We embed GL(n; H) 342200230342200224>
GL(4n; R) by the usual map)
91 -92 -93 -94
92 91 -94 93
93 94 91 -92
94 -93 92 91)
(24-11) 342200230P
I 91 + 9'92 +993 + 1994
302273342200224>)
That is induced from)
U1
V
U2
U3
U4)
ll-ll342200224>lR4
by ul +ugi+u3j+u4k r342200224>342200231
Similarly GL(2n; (C)
=>
GL(4n;lR) by 342200230I>
: a + ib 302273342200224>
(_342200234b
One can check that)
(2.4.12) II(S'0*(2n))= II(GL(n; 11-11))(W <I>(U(n, n)),)
which is another standard way of describing SO*(2n).)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-41-320.jpg)
![2.5. DEGENERATE FORMS 25
2.5. Degenerate Forms
Let b be a symmetric bilinear form on V, not necessarily nondegenerate. Thus
the subspace
(2.5.1) U =
Ker(b)
==
{1} E V I b(v, V)
= 0} C V)
can be nonzero. In any case b inducesa nondegenerate
symmetric bilinear form
b on the quotient spaceV/U. LetW be any vector space complement to U in V,
so V : U {B W. Then b is nondegenerate on W. Let O(V, b) denote the group of
(2.32) and choose a basis H
=
{u1, . . . ,u,.;'w1, . . .,ws}of V that starts with a basis
of U and 357254201nishes
with a basis of W. Relative to 357254202
the matrix group O(V, b) is)
(2.52) O(V, b) =
(9
=)
Let X = ]F342200231342200234"3
E
Hom]p(l/V, U A quick computation with (2.5.2) shows that
O(V,b) = X >4 (GL(U) ><
O(l/V,b|WXW)), semidirect product for
the action (3 2) ::cI342200224>
aaslfl of GL(U) X
0(W,b|WXW) on X)
(1 E and b E .)
(2.53))
Next, let b be an antisymmetric bilinear form on V, not necessarily nondegen-
erate. Again, the subspace)
(2.5.4) U = Ker(b) = {U E V I b('v, V)
= 0} C V)
can be nonzero, and b induces a nondegenerate
antisymmetric bilinear form b on
the quotient spaceV/U . As before, let W be any vector space complement to U in
V, so V = U EB W. Then b is nondegenerate on W. Let Sp(V, b) denote the group
of (2.32) and choose a basis 3 = {u1,
...,u,;w1,
...,w3}
of V that starts with a
basis of U and 357254201nishes
with a basis of W. Relative to ,8 the matrix group O(V, b)
is)
(26.5)
Sp<V,b>=(g= (3 if))
Let X = IFTX342200231
E
Hom[(.~(I/V, U As before, computation with (255) shows that
Sp(V, b) = X >4 (GL(U) ><
Sp(VV, b[WXW)), semidirect product for
the action (3 E)
: a: 302273342200224>
axlfl of GL(U) ><
Sp(W,b|WXW) on X)
a E GL(U) and b E
Sp(VV,b|WXW)}.)
(25.6))
Now let h be an lF342200224hermitian form on V, not necessarily nondegenerate. Yet
again, the subspace)
(25.7) U = Ker(h) = {U E V | h(v, V) = 0} C V
can be nonzero, and h inducesa nondegenerate
lF342200224hermitian form H on the quotient
space V/U. As before, let W be any vector space complement to U in V, so
V = U 69 W. Then h is nondegenerate on W. Let U (V, h) denote the group of
(2.42) and choose a basis, 357254202
=
{u1, . . . ,u,.;'w1, . . . , ws} of V that starts with a basis
of U and 357254201nishes
with a basis of W. Relative to B the matrix group U(V, h) is)
(2.58) U(V, h) =
{g
=
(3
aeGL(U) and b e U(W,
h|WxW)}.)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-42-320.jpg)
![26 2.SOME EXAMPLES)
Let X = IFTXS 302247
Hom11.~(I/V, U Once again, computation with (2.5.8) shows that)
U(V, h) = X )4
(GL(U) ><
U(l/V, h|WXw)), semidirect product for)
2.5.9
( )
the action (3 2) ::cH axlfl of
GL(U) ><
U(W,h|WxW) on X
Finally, let s be a skew342200224hermitian form on a vector space V over lHl, not neces-
sarily nondegenerate. Yet again, the subspace)
(2.5.10) U= Ker(s)
=
{v E V I s(v, V)
= 0} C V)
can be nonzero, and s induces a nondegenerateskew342200224hermitian form 302247
on the
quotient space V/ U. As before, let W be any vector space complement to U in V,
so V = U 69 W. Then s is nondegenerate on W. Let SO*(V, s) denote the group of
(2.4.9) and choose a basis 357254202
=
{u1, . . . ,uT;w1, . . .,ws}of V that starts with a basis
of U and 357254201nishes
with a basis of W. Relative to (3 the matrix group SO*(V,s) is)
(2.511)
SO*(V,s)={g=<3)
a E and b E
S|WXW)}
.)
Let X = ll-ll342200231""3
%
Hom]F(W, U). Once again, computation with (25.11) shows that)
S0*(V, s) = X ><1
(GL(U) ><
SO*(W', s|wXW)), semidirect product for)
(2.5.12)
th
_ Q 0 _ _, ,
e action (0 b) . x 1342200224>
axb of GL(U) X 5'0 (W,s|W,(W) on X)
2.6. Automorphism Groups of Algebras)
Let A be a 357254201nite
dimensional algebra over R or C, not necessarily associative.
In other Words, the multiplication on A c011ld be any bilinear map A X A 342200224->
A.
Then one has the automorphism group of A,)
(2-6-1) Aut(A) = {9 6 G342200231L(3022534)
l9(:vy)
=
g(rv)g(y) V :0. y 6 A}-)
It is a closedsubgroup of GL(A) defined by some quadratic equations,so it is a
topological group.)
For example, if A is the associativel algebra Mn of n X n matrices over 11''342200230,
then every automorphism is inner, so Aut(A) is the projective general linear group
PGL(A) = GL(A)/lF*. If A is a semisimple Lie algebrathen Aut(A) has identity
component that is the adjoint group of A.)
Let (0) denote the algebra of octonions (Cayley division algebra) over R. It
can be de357254201ned
as the algebra with basis {(30,. ..,e7} over R with multiplication
de357254201ned
by (a) eoej
= ej = ejeo for 1 302247
j 302247
7, (b) e?
= 342200224e0
for 1 302247
j 302247
7,
(c) ejek +ekej = 0 for 1 302247
j,k 302247
7 with j a303251
k, (d) 6162 = e3, 6365 = e5,
e5e7= 61,(3164
=
65, e3e4 = 342200224342200224e7,
6564 = eg and e2e5 = e7, and (e)eachequation
in (cl) remains true when the subscripts involved in it are cyclically permuted. This)
1Thealgebra A is called associative if its multiplication is associative, i.e. if (ab)c
= a.(bc)
in A.))](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-43-320.jpg)
![28 2. SOME EXAMPLES)
Second, J could be the real simpleJordan algebraV (B R, where V is a real
vector space of dimension n < oo with a positive de357254201nite
symmetric bilinear form
f. .7 has composition(u,a)
0 (1), b)
= (av + bu, ab + f(u,v). liere Str0(._7) is
the orthogonal group of signature (n, 1) associated to the form f ((u, a), (11,b)) =
f (u,v)
342200224
ab and Aut(J) is the orthogonal group of (V, f). It is not obvious, but in
this case J is special.)
Finally, third, we have the exceptional simplereal JordanalgebraJ consisting
of 3 x 3 hermitian matrices over the octonion division algebra (D). It has composition
:v o y
=
%(acy + Then Str0(J) is the 78~dimensional simple Lie group E673,
Whosemaximal compact subgroupisthe 52342200224dimensional compact simple Lie group
F4, and that F4 is Aut(._7 We will see these groups later in connection with
projective planes. (One can try this construction for the n X n hermitian matrices
over (0),but that does not result in a Jordan algebrawhen n > 3.))
Summarizing, the simple formally real Jordan algebras and their automorphism
and structuregroups are)
.7 Str(..7) Aut(.7)
Sym lR""" G'L(n; R) SO(n)
Re (C"342200231342200230342200235
GL(n; CC) SU(n)
Re lHI"X" G-'L(n; ll-ll) .S'p(n)
IR342200235
GB R O(n, 1) O(n)
Re 3022513X3
E67F4
X RX F4)
where the hermitian and skew342200224hermitian parts of a square matrix m are given by)
(2.64) Rem := %(m+m*) and Imm := %(m
342200224
m*).)
(2.6.3))
(These are real and imaginary parts only in a rather generalizedsense.)See[F-K]
for complete proofs.)
2.7. Spheres, Projective Spaces and Grassmannians)
The underlying set of the projective space P(V) of a vector space V over
IE342200230
is the set of all 1342200224dimensional linear subspaces of V. Thus it can be described
in terms of compact groupsas a homogeneous space of a unitary group, or in
terms of noncompact groups as a homogeneousspaceof a general linear group. If
n = dirny V then
(2.7.1) P(V) %
U(n;lF)/U(1;lF) X U(n
342200224
1;lF),
which can also be described as
G'L(n;E4342200230)/{(3%)
Ia e GL(1,]F), b e GL(n 342200224
1;]F) and .1342200230
e
rm"-1)}.)
For U(n;lF) is transitive on P(V) and the stabilizer of e1lF is U(1;lF) X
U(n
342200224
1;lF),
While GL(n; F) is of coursealsotransitive and the isotropy subgroup of GL(n;lF)
at e1lF is the subgroup (call it described just above. Thus we have real,
complexand quaternion n342200224space
given by)
(2.72) P"(]R) = O(n + 1)/0(1)><
O(n)
=
GL(n + 1;R)/Q,,+1(]R)
(2.73) P"(<C)
=
U(n + 1)/11(1) x U(n) GL(n + 1;C)/Q,,+1(<C)
(27-4) P"(H)
= Sp(n + 1)/317(1)X 52001)
=
GL(n + 1;1H1)/Qn+1(1H1)))](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-45-320.jpg)
![2.7.SPHERES, PROJECTIVE SPACES AND GRASSMANNIANS 29)
For n > 1 the universal cover of P" (R) is the sphere S"with the covering given by
.S'O(n + 1)/S'O(n) E342200231
0(n + 1)/O(n) 342200224>
O(n + 1)/0(1) ><)
The complex and quaternion projective spaces are simply connected.Thereis an-
other locally compact projective space in the background,
the octonion(orCayley)
projective plane,
(27.5) P2(o) = F4/Sp2'n(9) = Em,/Q3(o).
HereE534 and F4 are the groups mentioned above in connection with the excep-
tional simpleJordanalgebraJ, which is used to construct2 P2 ((0)), and Q3((D)) and
Sp2'n(9) are certain subgroups.P2(302251) is simply connected. The projective lines in
P" (R) are circles S1, in P"(C) are Riemann spheres S2, in P" (H) are 4342200224spheres
S4, and in P2((U>)are 8342200224spheres S8.)
More generally, the Grassmann manifold of k342200224dimensional linear subspaces
of V is given by)
Gk(V)
E U(n;lF)/U(k;lF) >< U(n
342200224342200224
k;lF))
2.7.6
( )
E GL(n;lF)/ {(3 3;)| a e GL(k,lF342200230),
b e GL(n
342200224
k;lF) and :10 e
r342200231342200235<"342200224342200
For again U (n; IF ) is transitive on P(V) and the stabilizerof e1lF EB e2lF EB
- - -
EB ek]F'
is U (k; IF342200230
) X U (n
342200224
k; F), while GL(n; F) is of course also transitive and the isotropy
subgroupof GL(n;lF) at e1lF EB e2lF G3 EB ekl357254201342200230
is the subgroup (call it
Qk,,,(lF))
described just above. Thus we have real, complex and quaternion Grassmann
manifoldsgiven by)
(27-7) Gk,n(R)
= 001)/0(k) X 001
~
1?) = GM342200235
+ 1; R)/Qn+1(1R)
(3-7-8) Gk,n(C) = U(n)/U (k) X U (71
- k) = GL(n + 1; C)/Qn+1(342200230C)
(2-7-9) Gk,n(H)
= Sp(n)/510(k) X 31001
-
k) = GL0?302273
+ 1;1H1)/Qn+1(1H1))
For k(n
342200224
k) > 1 the universal cover of
Gk7n(lR) is the 2342200224sheeted cover by the)
grassmannian C~302245k7n(lR)
of oriented kaiimensional linear subspaces of an oriented
R", and that covering is given by)
SO(n)/S'0(k) >< S'O(n
342200224
k)
342200224>
S'O(n)/{g E O(k) ><
O(n
342200224
k) I det(g)
=
1}.
The complex and quaternion Grassmann manifolds are simply connected.)
There is an important natural generalization of the real, complex and quater-
nion Grassman manifolds, the compact riemannian symmetric spaces.SeeSection
11.2 for a quick description and Cartan342200231sclassi357254201cation. In differential geometry,
in a certain sense 342200234most342200235
of the compact riemannian symmetric spaces are grass-
mannians. By this I mean that if a theorem can be veri357254201ed,
or a phenomenon
can be understood, for the Grassmann manifolds mentioned above, then this often
is enough of an indication to carry it through for compact riemannian symmetric
spaces.)
2Here is a uniform construction of all four types of projective spaces P"3422002311(lF
Let .7 denote
the Jordan algebra of n X 17, hermitian matrices over F, with n = 3 in the case 17 = (0). An element
e E J is idempotent if e2 = :3. An idempotent e E .7 is primitive if e 75 0 and 6 cannot be
expressed in the form e1 + (:2 where the e,- are nonzero idempotents that are orthogonal in the
sense e1e2 = 0. Then the set I(J) of primitive idempotents is a compact submanifold of .7 that
is Stro(.7)342200224equivariantly diffeomorphjc to P"3422002301 See [F-K, Exercises 4 & 5, Chapter IV] and
[F-K, Exercise 5, Chapter V].))](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-46-320.jpg)
![2. SOME EXAMPLES)
For noncompact symmetric spacesthere is something similar. Let h be the)
hermjtian form on F" with matrix
(I3 In0_k)
Then U(k,n 342200224
k;lF) is its unitary)
group. The orbitsof U(k,n
342200224
k;lF) on Gk,n(lF) are the various subsetson which
h has equivalent restriction. The possibilities for that restriction are given by
triples (p,q,u) where p is the number of +342200231s,
q is the number of -342200231s,
and u is the
dimension ofthe kernel,of the restriction of h to our k-dimensionalsubspace.Thus
p 302247
k,q 302247
71 342200224
k,u 3 min{k,n
342200224
kt}, andp+ q + u = k. The most important of
these orbits, for riemannian geometry,is)
(27.10) U(k,n
342200224
k; lF)([e1 / ~ - - / ek])
% U(k,n
342200224
k';lF)/U(k;lF) >< U(n
342200224-
k;lF),)
where [e1 / - - - / ek]
=
Span1g{e1, e2, . . .,ck}.The manifold (2.7.10) can be realized
as the boundeddomain of k X (n
342200224342200224
k) matrices Z over IF such that I 342200224
ZZ* >> 0.
The other orbits, at leastin the caseIF =
C, are useful in complex function theory
and in the unitary representation theory of the group U(k,n 342200224
lc;)
2.8. Complexi357254201cation of Real Groups)
Certain of our groups are obtained from others by a processthat could be called
complexi357254201cation. For example, GL(n;C) is the complexi357254201cation of GL(n;lR) and
O(n; C) is the complexi357254201cation of O(n; R). This notion, however,goeswell beyond
those obvious cases, and it has a number of geometric and analytic consequences.
We will formalize it from the viewpoint of linear groups.)
DEFINITION 2.8.1. Let G be a topological
group.We say that another topolog-
ical group GC
is a complexi357254201cation of G if there exist n > 0 and homomorphisms
45: G 342200224+
GL(n;lR) and Lb : GC 342200224->
GI/(n; (C) such that)
o ()5 is a homeomorphism of G onto 302242>(G)
and 1/} is a homeomorphism of GC
onto 1,b(GC),
o
qb(G)
= {g E RM" 1 F(g)
= 0 for all F E I} for some set I of real
polynomial functions on Rnx", and
0 1/J(GC)= {gE C"342200231342200230342200235
| F(g)
= 0 for all F E I} when we view the elements
of I as polynomial functions on <C"X".)
If GC is a complexi357254201cation of G then we also say that G is a real form of GC. 0)
From now on, when we write GCand G, it is understood that Gc is a complex-
i357254201cation
of G.)
We now make sure that the de357254201nition of complexi357254201cation is what we want, by
verifying
(2.82) GL(n;R)c
= GL(n; (C)and S342200231L(n;lR)C
=
SL(n; (C).)
For this, let 302242>(g)
=
1,b(g)
=
<31/dgt(g)).
Then q5(GL(n;lR))consistsof all real
(71 + 1) X (n + 1) matrices m that satisfy the equations mn+1,j = Ofor 1 g j 302247
n,
ml-,n+1
= 0 for 1 302247
7} 302247
n, and det(m) = 1. Note that 1/1(GL(n;C)) consists of all
complex (n + 1)X
(n + 1) matrices that satisfy the samesystem of equations. That
proves (2.8.2) for GL. The sameargument, with q5(g)
= 1/;(g) = g, provesit for SL.)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-47-320.jpg)
![2.8. COMPLEXIFICATION OF REAL GROUPS 31)
Now let342200231s
run through some cases that we met earlier.
(2.8.3) GL(n;ll-ll)C
=
GL(2n;(C) and SL(n;lHl)C = SL(2n; (C).
When we view H342200235
as a 4n342200224dimensional real vector space R4342200235,
we obtain a map
qb : GL(n;lHI) 342200224+
GL(4n;lR), and the scalar multiplications by the pure imaginary
unit quaternions i, j and k are transformed to anticommuting linear transformations
I,J,K E GL(4n;IR)of square 342200224I.
Note that)
302242>(GL(n;1H1))
=
{9 E GL(4n;R) I 91
= Ig,gJ = J9 and 9K = K9}-)
When we View C2342200235
as R4" we obtain a map 1/; : GL(2n;(C) 342200224>
GL(4n;]R), such
that scalar multiplication by i is transformed to the samelineartransformation
I E GL(4n;]R), and 1/1(GL(2n;(C))= {gE GL(4n;R) l gI
=
Ig}. Now we view the
spaceof 4n x 471. real matrices that commute with I as a complex vector space V,
with complex conjugation given by an 302273342200224342200224>
J;2:J3422002351.
So U = {cc 6 V 1 :rJ =
Jae} is
a real vector space with complexi357254201cation V. Making use of the map that realizes
(2.82), we obtain the 357254201rst
statement of (2.8.3). The second statement follows.)
(2-84) 0(p, q)c
= 0(1) + q;(C) and 30(1). q).;
= 50(1) + <1; C)-)
For the full orthogonal groups we use the system of equations on the matrix entries
given by 342200230gI,,,qg
342200224
Ipyq
= 0, and for the special orthogonal groups we use the
additional equations that come from det(g)
342200224
1 = 0. Then (2.8.4) is immediate.)
(2.8.5) S'p(n;lR)C
=
S'p(n; C).
The proof of (2.8.5)is the sameas the proof of (2.84), with J in place of IN}.)
(2-8-5) U(p,q)c = GL(:v+ M3) and
5U(p,q)c
=
SL(p+ q; C)-
Write 2 E U(p, q) as 2 = :1:
+ iy with a:,y real, and de357254201ne
302242(z)
=
(
Z
Then)
my Z
<z5(U(p,q))
=
{9 6 0(2p, 291) I N
= J9} Where J = (P15)~ Thus 302242(U(1>,q)) is
the set of all 2n >< 2n rea.l matrices, n = p + q, that satisfy certain real polynomial
equations. As affine variety or as differentiable manifold it has real dimension n2,
so its complexi357254201cation has real dimension 2722. That complexi357254201cation still consists
of matrices that commute with J, in other words of matrices in q$(GL(n;
(C)),which)
has real dimension 2n2. That proves (2.8.6).)
(2-8-7) 511(1).
q)c
=
Sp(p + q; C).
The proof of (2.8.7) is the same as the proof of (2.8.6).)
(2.8.8) S'0*(2n)C
= S'O(2n;(C).
The proof of
(2.8.8) is the same as the proof of (2.8.6).)
(2.8.9) GL(n;<C)c = GL(n;(C) X
GL(n;(C) and S'L(n;C)C
= SL(n;(C) >< SL(n;(C).)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-48-320.jpg)
![34 2. SOME EXAMPLES)
and its unitary group U(p,q; In the octonion case one has to be careful:
U(p,q;302251)
=
SO(p, q) >< Spin(7). We now have a group)
Hp,q;]F : real vector space Im]F + lF342200235342200230q
with group composition)
(2101)
(2, w)(z', w342200231)
=
(2 + 2' + Im h(w, w342200231),
w + 11)342200231).)
Groups Hp7q;]1:
% Hp/,q/;F/ if and only if F = IF342200231
and (ii) p -1-q = p342200231
+ q342200231.
Finally,
g(z,w)
= (2, de357254201nes
an action of the unitary group U(p, q; F) by automor-
phisms on Hp1q;]5342200230.
The semidirect product group G,,7q;1;
=
Hpg357254202p
><1 U(p,q;lF) has
group composition)
(2.102) (2,w,g)(z342200231,
111', g342200231)
=
(2 + 2' -1-Im h(w, g(w342200231)),
w + g(w342200231),
gg342200231).
We will mostly be concerned with the case q = 0, leading to the groups
(2.103) Hn;};.342200230
'.= Hn,0;][r and Gm357254201w
2: Gn;0;]p.)
The usual Heisenberg group is H := Hmc. Sothe groups H,,,q;;p, and even the
groups Hn;}F,
form a slight generalization. Of coursethe HWER
are just a real vector
groups. The othersare 2342200224step
nilpotent.)
Occasionally one wants more of a generalization of the usual Heisenberg group.)
Given an integer 3 g 1 we decompose IFSXS = Im IFSXS + Re F5342200234,
direct sum of
real vector spaces, where the projectionsare given by)
1)
Imz =
302247(z
342200224
2*) and Rez 2 + z*) where 2* is conjugate transpose.)
Given integers t, u 3 O, we have a hermitian map
H I F342200235(342200230+342200234)
>< F342200235(t+342200234)
342200224>
F3342200234
given by 7'l((13422002301,rv2),($11,312))
=
$131?
- $22/3
where a:1,y1 E IFS342200235
and .7cg,y2 E FSX342200234.
We write lE3422002305>342200230("342200234)
for lF3422002303"(342200234*'342200234)
with the
hermitian map H. Putting these together we have a (very) generalized Heisenberg
group
Hs,t7u;]302247342200230
: real vector space Im IFS342200234
+ lF5X("342200234)
with group composition)
(2.104)
(2, w)(z', w342200231)
=
(2 + z342200231
+ Im'H(w, w342200231),
w + 10').)
Since the (t,u) pertainsto rows in F5"(t*342200234)
the automorphism action of U(t,u;lF)
on H3yt,u;]14342200230
is g(z, w) = (z, wg*). The semidirect
product group G_;342200231t,302273u_;]]4342200230
=
H57t342200231u;]p342200230
>4
U (t, n; F) has group composition)
(2.105) (2, w, g)(z342200231,
w',g')
2
(2 + 2' + ImH(w,w342200231g*),
w + u/g*,gg342200231).)
In particular Glsytyu357254202p342200230
has center Im]F3"5, H1yt,u;];.~
342200234=342200231
Ht342200231u;]F
and G3422002311,tyu;][302253342200230
%
Gt,u;]F,
and Hsytwp is commutativejust when either 3 =: 1 with F = R, or t = u = 0.)
The groups H satyudlr
and Crlsvtj-u;]F appear in the study of maximal parabolic
subgroupsof unitary groups U(p, q; See [W9], [W10] and [W11]. We will
meet a few of them in Chapter 13 when we look at the classi357254201cation of commutative
nilmanifolds, and in Chapter 14 we will see that harmonic analysison thosegroups,
based on a theory of square integrable representations, is particularly elegant.)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-51-320.jpg)
![273
73m302273..
273..
273.,g..~)
CHAPTER 3)
Integration and Convolution)
Let G be a locally compact topological group. In this chapter weconstruct the
basictoolfor analysis on G: a left translation invariant Radon measurel known as
left Haar measure MG on G. As always, there is a certain emphasisonexamples.
We then study properties of the associated left Haar integral fa f(g) dp,G
These
include structure and properties of the LebesguespacesLP(G),the convolution
product, the modular function, the group algebraL1(G),
and the measure algebra
M eas(G). Finally, we use the notion of modulus of an automorphism to construct
adele groups of linear algebraic groups.)
3.1. De357254201nition and Examples)
The invariant integral on G will be left invariant, in other words invariant
under left translations. That means fa f(a:g)dug (g)
=
fa f(g) d,uG
In other
words, the associated measure is left invariant in the sense that ac (xA) = as (A)
whenever an E G and A C G is a Borelset. Formally,)
DEFINITION 3.1.1. A left Haar measure on G is a left invariant Radon mea-
sure on G that is not identically zero. A left Haar integral is the left invariant
integral de357254201ned
by a left Haar measure. <))
Before going on we look at some examples.)
EXAMPLE 3.1.2. Let G be the additive group R342200235.
Then Lebesgue measure is a
left Haar measureon G, and the Lebesgue integral is a left Haar integral on G. 0)
EXAMPLE 3.1.3. The circle group G = {zE (C I
=
1} has left Haar measure
357254201d357254202
where z = 62"342200234).
It has total mass 1 and the corresponding
invariant integral)
is f I342200224>
.217 02" f(e342200234342200231)d6. <>
EXAMPLE 3.1.4. Let G be a discretegroup. Thencounting measure is a left
Haar measure on G and f 302273342200224+
Zgea f (g) is a left Haar integral. 0)
EXAMPLE 3.1.5. Let G = GL(n;]R) and let dx denote the volume element of
Lebesgue measure on lR"X342200234.
The left action g : av r342200224>
gm scales Lebesgue measure
on each column of :1: by {det(g)|, so d(ga:) = |det(g)|"d:r. Thus ldet(x)|""d:c
is a
left Haar measure on GL(n;lR). This same argument shows that |det(x)|3422002302"d;zt
is
a left Haar measure on GL(n;(C).)
1Let 8 denote the a342200224a.lgebra
of Borel sets in G. It is the cr342200224algebra
generated by the open
sets. A R/adon measure on G is a positive inner342200224regular Borel measure ,u : B 342200224>
[0, 00] that is
357254201nite
on compact sets. Inner regularity means that ,u(A) = sup{p(F) I F C A and F compact}
for all A E B.)
35)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-52-320.jpg)
=
h(g) implies f 302247
Z?:1(|[f|lm/c)Zx, We have exhibited)
(3.26) n > 0 integer,'y,- 2 0, {$1, . . . ,x,,} C G such that f g Zn:*y,342202254,,,(h).
i=1
De357254201ne
a ratio
(32.7) (f 2 h)
= inf | f satis357254201es
(32.6) for some {$1, . . .,ac,,}C
G}.
2'21
Notice that f g ||h|[oO(f : h) and that (f : h) > 0 iff a303251
0.)
Ifk7303251Other1(f:k)302247(f:h)(h:k). Iff7E0then(h:k)302247(h:f)(f:k).
Combine these: if f 72$ 0 then 1/(h : 302247 : :
k) 302247
(f 2 h). In other words,)
extended the result to arbitrary locally Compact groups. After that, Kakutani noted that Haar342200231s)
construction is valid for arbitrary locally compact groups. The proof here is due to H. Cartan. It
has the feature that uniqueness comes along with the existence.)](https://image.slidesharecdn.com/2474791-250619060351-6533796a/85/Harmonic-Analysis-On-Commutative-Spaces-Joseph-A-Wolf-54-320.jpg)
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Harmonic Analysis On Commutative Spaces Joseph A Wolf
- 1. Harmonic Analysis On Commutative Spaces Joseph A Wolf download https://ebookbell.com/product/harmonic-analysis-on-commutative- spaces-joseph-a-wolf-4949582 Explore and download more ebooks at ebookbell.com
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- 3. Harmonic Analysis On Reductive Padic Groups Robert S Doran https://ebookbell.com/product/harmonic-analysis-on-reductive-padic- groups-robert-s-doran-4765218 Harmonic Analysis On Exponential Solvable Lie Groups Hidenori Fujiwara https://ebookbell.com/product/harmonic-analysis-on-exponential- solvable-lie-groups-hidenori-fujiwara-4976236 Harmonic Analysis On Symmetric Spaceshigher Rank Spaces Positive Definite Matrix Space And Generalizations 2nd Edition Audrey Terras Auth https://ebookbell.com/product/harmonic-analysis-on-symmetric- spaceshigher-rank-spaces-positive-definite-matrix-space-and- generalizations-2nd-edition-audrey-terras-auth-5482440 Harmonic Analysis On Finite Groups Representation Theory Gelfand Pairs And Markov Chains 1st Edition Tullio Ceccherinisilberstein https://ebookbell.com/product/harmonic-analysis-on-finite-groups- representation-theory-gelfand-pairs-and-markov-chains-1st-edition- tullio-ceccherinisilberstein-1007430 Harmonic Analysis On Semigroups Theory Of Positive Definite And Related Functions Berg https://ebookbell.com/product/harmonic-analysis-on-semigroups-theory- of-positive-definite-and-related-functions-berg-11344302
- 6. Harmonic Analysis on Commutative Spaces http://dx.doi.org/10.1090/surv/142
- 7. Mathematical Surveys and Monographs Volume 142 Harmonic Analysis on Commutative Spaces Joseph A. Wolf American Mathematical Society
- 8. E D I T O R I A L C O M M I T T E E Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 20G20, 22D10, 22Exx, 53C30, 53C35. For additional information and updates on this book, visit www.ams.org/bookpages/surv-142 Library of Congress Cataloging-in-Publication Data Wolf, Joseph Albert, 1936- Harmonic analysis on commutative spaces / Joseph A. Wolf. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 142) Includes bibliographical references and indexes. ISBN 978-0-8218-4289-8 (alk. paper) 1. Harmonic analysis. 2. Topological groups. 3. Abelian groups. 4. Algebraic spaces. 5. Geometry, Differential. I. Title. QA403.W648 2007 515'.2433—dc22 2007060807 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission@ams.org. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07
- 9. To Lois
- 10. Contents Introduction xiii Acknowledgments xv Notational Conventions xv Part 1. GENERAL THEORY OF TOPOLOGICAL GROUPS Chapter 1. Basic Topological Group Theory 3 1.1. Definition and Separation Properties 3 1.2. Subgroups, Quotient Groups, and Quotient Spaces 4 1.3. Connectedness 5 1.4. Covering Groups 7 1.5. Transformation Groups and Homogeneous Spaces 8 1.6. The Locally Compact Case 9 1.7. Product Groups 12 1.8. Invariant Metrics on Topological Groups 15 Chapter 2. Some Examples 19 2.1. General and Special Linear Groups 19 2.2. Linear Lie Groups 20 2.3. Groups Defined by Bilinear Forms 21 2.4. Groups Defined by Hermitian Forms 22 2.5. Degenerate Forms 25 2.6. Automorphism Groups of Algebras 26 2.7. Spheres, Projective Spaces and Grassmannians 28 2.8. Complexification of Real Groups 30 2.9. p-adic Groups 32 2.10. Heisenberg Groups 33 Chapter 3. Integration and Convolution 35 3.1. Definition and Examples 35 3.2. Existence and Uniqueness of Haar Measure 36 3.3. The Modular Function 41 3.4. Integration on Homogeneous Spaces 44 3.5. Convolution and the Lebesgue Spaces 45
- 11. viii CONTENTS 3.6. The Group Algebra 48 3.7. The Measure Algebra 50 3.8. Adele Groups 51 Part 2. REPRESENTATION THEORY AND COMPACT GROUPS Chapter 4. Basic Representation Theory 55 4.1. Definitions and Examples 56 4.2. Subrepresentations and Quotient Representations 59 4.3. Operations on Representations 64 4.3A. Dual Space 64 4.3B. Direct Sum 64 4.3C. Tensor Product of Spaces 65 4.3D. Horn 67 4.3E. Bilinear Forms 67 4.3F. Tensor Products of Algebras 68 4.3G. Relation with the Commuting Algebra 69 4.4. Multiplicities and the Commuting Algebra 70 4.5. Completely Continuous Representations 72 4.6. Continuous Direct Sums of Representations 75 4.7. Induced Representations 77 4.8. Vector Bundle Interpretation 81 4.9. Mackey's Little-Group Theorem 82 4.9A. The Normal Subgroup Case 82 4.9B. Cohomology and Projective Representations 84 4.9C. Cocycle Representations and Extensions 85 4.10. Mackey Theory and the Heisenberg Group 87 93 93 96 97 99 101 104 107 107 107 110 111 111 112 113 115 Chapter 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5. Representations of Compact Groups Finite Dimensionality Orthogonality Relations Characters and Projections The Peter-Weyl Theorem The Plancherel Formula Decomposition into Irreducibles Some Basic Examples 5.7A. The Group 17(1) 5.7B. The Group SU(2) 5.7C. The Group SO(3) 5.7D. The Group 50(4) 5.7E. The Sphere S2 5.7F. The Sphere S3 Real, Complex and Quaternion Representations The Frobenius Reciprocity Theorem
- 12. CONTENTS ix Chapter 6. Compact Lie Groups and Homogeneous Spaces 119 6.1. Some Generalities on Lie Groups 119 6.2. Reductive Lie Groups and Lie Algebras 122 6.3. Cartan's Highest Weight Theory 127 6.4. The Peter-Weyl Theorem and the Plancherel Formula 131 6.5. Complex Flag Manifolds and Holomorphic Vector Bundles 133 6.6. Invariant Function Algebras 136 Chapter 7. Discrete Co-Compact Subgroups 141 7.1. Basic Properties of Discrete Subgroups 141 7.2. Regular Representations on Compact Quotients 146 7.3. The First Trace Formula for Compact Quotients 147 7.4. The Lie Group Case 148 Part 3. INTRODUCTION TO COMMUTATIVE SPACES Chapter 8. Basic Theory of Commutative Spaces 153 8.1. Preliminaries 153 8.2. Spherical Measures and Spherical Functions 156 8.3. Alternate Formulation in the Differentiable Setting 160 8.4. Positive Definite Functions 165 8.5. Induced Spherical Functions 168 8.6. Example: Spherical Principal Series Representations 170 8.7. Example: Double Transitivity and Homogeneous Trees 174 8.7A. Doubly Transitive Groups 174 8.7B. Homogeneous Trees 175 8.7C. A Special Case 176 Chapter 9. Spherical Transforms and Plancherel Formulae 179 9.1. Commutative Banach Algebras 179 9.2. The Spherical Transform 184 9.3. Bochner's Theorem 187 9.4. The Inverse Spherical Transform 191 9.5. The Plancherel Formula for KG/K 192 9.6. The Plancherel Formula for G/K 194 9.7. The Multiplicity Free Criterion 197 9.8. Characterizations of Commutative Spaces 198 9.9. The Uncertainty Principle 199 9.9A. Operator Norm Inequalities for KG/K 199 9.9B. The Uncertainty Principle for KG/K 202 9.9C. Operator Norm Inequalities for G/K 203 9.9D. The Uncertainty Principle for G/K 204 9.10. The Compact Case 204
- 13. x CONTENTS Chapter 10. Special Case: Commutative Groups 207 10.1. The Character Group 207 10.2. The Fourier Transform and Fourier Inversion Theorems 212 10.3. Pontrjagin Duality 214 10.4. Almost Periodic Functions 216 10.5. Spectral Theorems 218 10.6. The Lie Group Case 219 Part 4. STRUCTURE AND ANALYSIS FOR COMMUTATIVE SPACES Chapter 11. Riemannian Symmetric Spaces 225 11.1. A Fast Tour of Symmetric Space Theory 225 11.1A. Riemannian Basics 225 11.IB. Lie Theoretic Basics 226 11.1C. Complex and Quaternionic Structures 229 11.2. Classifications of Symmetric Spaces 231 11.3. Euclidean Space 236 11.3A. Construction of Spherical Functions 236 11.3B. General Spherical Functions on Euclidean Space 238 11.3C. Positive Definite Spherical Functions on Euclidean Space 240 11.3D. The Transitive Case 242 11.4. Symmetric Spaces of Compact Type 245 11.4A. Restricted Root Systems 245 11.4B. The Cartan-Helgason Theorem 246 11.4C. Example: Group Manifolds 249 11.4D. Examples: Spheres and Projective Spaces 250 11.5. Symmetric Spaces of Noncompact Type 252 11.5A. Restricted Root Systems 253 11.5B. Harish-Chandra's Parameterization 254 11.5C. Hyperbolic Spaces 255 11.5D. The c-Function and Plancherel Measure 257 11.5E. Example: Groups with Only One Conjugacy Class of Cartan Subgroups 258 11.6. Appendix: Finsler Symmetric Spaces 260 Chapter 12. Weakly Symmetric and Reductive Commutative Spaces 263 12.1. Commutativity Criteria 263 12.2. Geometry of Weakly Symmetric Spaces 264 12.3. Example: Circle Bundles over Hermitian Symmetric Spaces 268 12.4. Structure of Spherical Spaces 272 12.5. Complex Weakly Symmetric Spaces 275 12.6. Spherical Spaces are Weakly Symmetric 277 12.7. Kramer Classification and the Akhiezer-Vinberg Theorem 282 12.8. Semisimple Commutative Spaces 287
- 14. CONTENTS xi 12.9. Examples of Passage from the Semisimple Case 290 12.10. Reductive Commutative Spaces 293 Chapter 13. Structure of Commutative Nilmanifolds 299 13.1. The "2-step Nilpotent" Theorem 299 13.1A. Solvable and Nilpotent Radicals 299 13.1B. Group Theory Proof 300 13.1C. Digression: Riemannian Geometry Proof 301 13.2. The Case Where N is a Heisenberg Group 303 13.3. The Chevalley-Vinberg Decomposition 309 13.3A. Digression: Chevalley Decompositions 309 13.3B. Weakly Commutative Spaces 314 13.3C. Weakly Commutative Nilmanifolds 317 13.3D. Vinberg's Decomposition 318 13.4. Irreducible Commutative Nilmanifolds 319 13.4A. The Irreducible Case — Classification 320 13.4B. The Irreducible Case — Structure 321 13.4C. Decomposition into Irreducible Factors 326 13.4D. A Restricted Classification 327 Chapter 14. Analysis on Commutative Nilmanifolds 329 14.1. Kirillov Theory 329 14.2. Moore-Wolf Theory 330 14.3. The Case where N is a (very) Generalized Heisenberg Group 335 14.4. Specialization to Commutative Nilmanifolds 338 14.5. Spherical Functions 341 14.5A. General Setting for Semidirect Products N x K 342 14.5B. The Commutative Nilmanifold Case 342 Chapter 15. Classification of Commutative Spaces 345 15.1. The Classification Criterion 345 15.2. Trees and Forests 350 15.2A. Trees and Triples 350 15.2B. The Mixed Case 351 15.2C. The Nilmanifold Case 353 15.3. Centers 354 15.4. Weakly Symmetric Spaces 357 Bibliography 367 Subject Index 373 Symbol Index 383 Table Index 387
- 15. Introduction Commutative space theory is a common generalization of the theories of com- pact topological groups, locally compact abelian groups, riemannian symmetric spaces and multiply transitive transformation groups. This is an elegant meeting ground for group theory, harmonic analysis and differential geometry, and it even has some points of contact with number theory and mathematical physics. It is fascinating to see the interplay between these areas, as illustrated by an abundance of interesting examples. There are two distinct approaches to the theory of commutative spaces: ana- lytic and geometric. The geometric approach, which is the theory of weakly sym- metric spaces, is quite beautiful, but slightly less general and is still in a state of rapid development. The analytic approach, which is harmonic analysis of commu- tative spaces, has reached a certain plateau, so it is an appropriate moment for a monograph with that emphasis. That is what I tried to do here. Commutative pairs (G, K) (or commutative spaces G/K) can be characterized in several ways. One is that the action of G on L2 (G/K) is multiplicity-free. Another is that the (convolution) algebra L1 (KG/K) of if-bi-invariant functions on G is commutative. A third, applicable to the case where G is a Lie group, is that the algebra D(G, K) of G-invariant differential operators on G/K is commutative. The common ground and basic tool is the notion of spherical function. In the Lie group case the spherical functions are the (normalized) joint eigenfunctions of the commutative algebra D(G, K). The result is a spherical transform, which reduces to the ordinary Fourier transform when G = Rn and K is trivial, an inversion formula for that transform, and a resulting decomposition of the G-module L2 {G/K) into irreducible representation spaces for G. In many cases this can be made quite explicit. But in many others that has not yet been done. This monograph is divided into four parts. The first two are introductory and should be accessible to most first year graduate students. The third takes a bit of analytic sophistication but, again, should be reasonably accessible. The fourth describes recent results and in intended for mathematicians beginning their research careers as well as mathematicians interested in seeing just how far one can go with this unified view of algebra, geometry and analysis. Part 1, "General Theory of Topological Groups", is meant as an introduction to the subject. It contains a large number of examples, most of which are used in the sequel. These examples include all the standard semisimple linear Lie groups, the Heisenberg groups, and the adele groups. The high point of Part 1, beside xiii
- 16. xiv INTRODUCTION the examples, is construction of Haar measure and the invariant integral, and the discussion of convolution product and the Lebesgue spaces. Part 2, "Representation Theory and Compact Groups", also provides back- ground, but at a slightly higher level. It contains a discussion of the Mackey Little-Group method and its application to Heisenberg groups, and a proof of the Peter-Weyl Theorem. It also contains a discussion of the Cartan highest weight theory with applications to the Borel-Weil Theorem and to recent results on in- variant function algebras. Part 2 ends with a discussion of the action of a locally compact group G on L2 (G/T), where Y is a co-compact discrete subgroup. Part 3, "Introduction to Commutative Spaces", is a fairly complete introduc- tion, describing the theory up to its resurgence. That resurgence began slowly in the 1980's and became rapid in the 1990's. After the definitions and a num- ber of examples, we introduce spherical functions in general and positive definite ones in particular, including the unitary representation associated to a positive definite spherical function. The application to harmonic analysis on G/K consists of a discussion of the spherical transform, Bochner's theorem, the inverse spher- ical transform, the Plancherel theorem, and uncertainty principles. Part 3 ends with a treatment of harmonic analysis on locally compact abelian groups from the viewpoint of commutative spaces. Part 4, "Structure and Analysis for Commutative Spaces", starts with rie- mannian symmetric space theory as a sort of role model, and then goes into recent research on commutative spaces oriented toward similar structural and analytical results. The structure and classification theory for commutative pairs (G,K), G reductive, includes the information that (G, K) is commutative if and only if it is weakly symmetric, and this is equivalent to the condition that (GC,KC) is spher- ical. Except in special cases the problem of determining the spherical functions, for these reductive commutative spaces, remains open. The structure and classi- fication theory for commutative pairs (G, K), where G is the semidirect product of its nilradical N with the compact group K, is also complete, and in most cases here the theory of square integrable representations of nilpotent Lie groups leads to information on the spherical functions. The structure and classification in gen- eral depends on the results for the reductive and the nilmanifold cases; it consists of methods for starting with a short list of pairs (G, K) and constructing all the others. Finally there is a discussion of just which commutative pairs are weakly symmetric. At this point I should point out two areas that are not treated here. The first, already mentioned, is the general theory of weakly symmetric spaces, and the closely related areas of geodesic orbit spaces and naturally reductive riemannian homogeneous spaces. That beautiful topic, touched momentarily in Section 13.1C, has an extensive literature. The second area not treated here consists of certain extensions of (at least parts of) the theory of commutative spaces. This includes the extensive but somewhat technical theory of semisimple symmetric spaces, (the pseudo-riemannian analogs of riemannian symmetric spaces of noncompact type), the theory of generalized Gelfand pairs (G,H), and the study of irreducible unitary representations of G that have an iif-fixed distribution vector. It also includes several approaches to
- 17. NOTATIONAL CONVENTIONS xv infinite dimensional analogs of Gelfand pairs. That elegant area is extremely active but its level of technicality takes it out of the scope of this book. Acknowledgment s Much of the material in Parts 1, 2 and 3 was the subject of courses I taught at the University of California, Berkeley, over a period of years. Questions, comments and suggestions from participants in those courses certainly improved the exposi- tion. Some of the material in Part 3 relies on earlier treatments of J. Dieudonne [Di] and J. Faraut [Fa], and much of the material in Part 4 depends on O. Yakimova's doctoral dissertation [Y3]. In addition, a number of mathematicians looked at early versions of this book and made useful suggestions. These include D. Akhiezer (com- munications concerning his work with E. B. Vinberg on weakly symmetric spaces), D. Bao (discussions on Finsler manifolds), R. Goodman (advice on how to organize a book), I. A. Latypov and V. M. Gichev (communications concerning their work on invariant function algebras), J. Lauret, H. Nguyen and G. Olafsson (for going over the manuscript), G. Ratcliff and C. Benson (communications concerning their work with J. Jenkins on spherical functions for commutative Heisenberg nilmani- folds), and the three mathematicians who refereed this volume (for some very useful remarks). I especially want to thank O. Yakimova for a number of helpful conversations concerning her work and E. B. Vinberg's work on classification of smooth commu- tative spaces. Notational Conventions M, C, M and O denote the real, complex, quaternionic and octonionic number systems. If F is one of them, then x H-> X* denotes the conjugation of F over R, F m x n denotes the space o f m x n matrices over F, and if x G F m x n then x* e F n X m is its conjugate transpose. We write R e F n x n for the hermitian (x = x*) elements of F n x n and ReFp Xn for those of trace 0, and we write ImFn X n for the skew-hermitian (x + #* = 0) elements of Fn X n ; that corresponds to the case n = 1. In general we use upper case roman letters for groups, and when possible we use the corresponding lower case letters for their elements. If G is a Lie group then g denotes its Lie algebra. If I) is a Lie subalgebra of g then (unless it is defined differently) H is the corresponding analytic subgroup of G.
- 18. Part 1) GENERAL THEORY OF TOPOLOGICAL GROUPS))
- 19. Part 1 of this book is a crash coursein the theory of topological groups, together with a number of examples that will be needed in the sequel. Chapter 1 describes the basicstructureof topological groups, and Chapter 3 presents the basicfacts for analysis on those groups. The sophisticatedreaderwith some knowledge of topological group theory may wish to con357254201ne his attention to the examples in Chapter 2 and in Section 3.1, especially those of Section 2.10.)
- 20. ;,) CHAPTER 1) Basic Topological Group Theory) This 357254201rst chapter is an introduction to the basicstructural facts in the theory of topologicalgroups.Thereareno surprises.Thebasicanalytic facts are in Chapter 3. The conceptshereare essentialfor the rest of this monograph.) 1.1. De357254201nition and Separation Properties) A topological group is a group with a topology such that the group opera~ tions are continuous. In other words, the algebraicand topological structures are mutually consistent. We make the formal de357254201nition.) DEFINITION 1.1.1. Let G be both an abstract group and a topological space. Then G is a topologicalgroupif) the one point subsets of G are closedsubsets1and the map G X G 342200224> G by (g, h) H gh3422002301 is continuous. 0) 1.1.2. Now let G be a topologicalgroup. Notice that the maps G 342200224> G by g +342200224342200224> g3422002301 and G>< G 342200224> G by (g, h) H gh are continuous. Forthe 357254201rst, (h, g) +342200224> hg3422002301 is continuous and we set h = 1. Forthesecond,GX G 342200224> G by (g, h) 302273342200224+ gh now is the composition (g,h) r342200224> (g, h3422002301) 302273342200224> g(h'1)3422002341 of two continuous maps. Similarly these two conditions imply continuity of (g, h) 302273342200224> gh3422002301. So together they are equivalent to the secondcontinuity condition of the de357254201nition of topological group.) 1.1.3. If {g1,...,g,,} C G and {7*1,...,r,,} are integers, and if V is a neigh- borhood of h = g{1 gf," in G, then there are neighborhoods Uj of gj such that Uf1...U,342200231;" C V. ForthemapG><---><G342200224>G,givenby (g1,...,g,,) 302273342200224>g{1...gf,", is continuous. This fact is crucial to many arguments in topological group theory. If G = OK", +) it corresponds to 342200234e/n342200235 arguments from elementary calculus.) 1.1.4. The left translations Kg : h 302273342200224> gh on G, the right translations rg : h v342200224> hg, and the conjugations ag : h r342200224> ghg3422002301, all are homeomorphisms of G. In effect, they are continuous because joint continuity of (u, 11) o342200224> uv, of (u, v) 302273342200224342200224> ml, and of (u, 11)v342200224> uvufl each implies separate continuity, and because) -1 _ -1 _ 3422002241 _ lg 342200224 34220225494, rg 342200224 rg3422002241, and ag 342200224 O[g3422002241.) LEMMA 1.1.5. A topological group is a 7342200230egula'r2 topological space.) 1This is the Tychonoff separation condition T1. 2A topological space X is regular if, given a closed subset F CX and a point an 302242 F, there are disjoint open subsets U, V C X with as E U and F C V. This is the Tychonoff separation condition T3.))
- 21. 4 1. BASIC TOPOLOGICALGROUP THEORY) PROOF. Let F C X be a closedsubset and let g E G with g 302242 F. By applying 69.1 We can assume g = 1. Now W = GF is a neighborhood of 1. As G>< G 342200224> G is continuous We have an openneighborhood U of 1 with U >4 U C W. So U = UI"IU3422002301 is an open neighborhood of 1 with U = U3422002341 and U2 C W.) The closure U C W. For if h E U then every neighborhood of h meets U, so hU meetsU, say hb = a where a,b E U. Now h = ab3422002341 6 UU3422002301 = U2 C W.) Set V = GU. Then U and V are disjoint, g E U, and F C V. El) When G be a topological group and H is a subgroup, H carries the subspace topology unless weexplicitlyspecifyto the contrary. So the open sets in H are the U357254202 H where U is open in G. It is easy to checkthat H is a topological group.) COROLLARY 1.1.6. Let H be a closed subgroup of a topologicalgroup G. If G is locally compactthen H is locally compact. If G is compact then H is compact.) PROOF. A closed subspace of a compact Hausdorfftopologicalspaceis com- pact.GisHausdorff (Tychonofl separation condition T2) by Lemma 1.1.5. D) 1.2. Subgroups, Quotient Groups, and Quotient Spaces) The cosetspaceG/H carries the quotient topology: if 7r : G 342200224342200224> G / H denotes the projection,7r(g) = gH, then a set U C G/H isopen if and only if 7r3422002341(U) is open in G. Also, if U is openin G so is the union UH = Uheh Uh of open sets, and then 7r(U) is open in G/H because UH = 7r3422002301(7r(U)). So the map 7r is both continuous and open. The opensubsets of G/H are the sets 7r(U) = {gH ] g E U} where U runs over the opensubsets of G.) If H is a normal subgroup of G then G/ H inherits a group structure from G, and the map (gH,g342200231H) +342200224> (gH)(g342200231H)3422002311 = gg342200231_1H is continuous. So then G/H is a topological group if and only if its points are closed.In this connection, note that G / H is T1if and only if H is closed in G. SoWe have) LEMMA 1.2.1. Let H be a closednormal subgroup of G. Then G/H is a topo- logical group, the projection 17 : G 342200224> G /H is a continuous homomorphism,and H is the kernel of 7r. (Note that 7r : G 342200224> G/H is an open map.)) The converseis the topological version of the standard isomorphism theorem:) PROPOSITION 1.2.2. Let f : G 342200224> L be a continuous homomorphism between topological groups. Let H C G be the kernel of f and let M = f (G)C L, the image of f. Then f factors through a continuous injectiue homomorphism 7 ofG/H onto M.) In general one cannot expect 7 to beahomeomorphism. For example, let L be the multiplicative group of all 2 X 2 diagonal matrices diag{e342200230/:T0, e342200230/:i342200230l342200231} with 0,11) real. Let G be the additive group of real numbers. Choose an irrational number 357254202 and de357254201ne f : G 342200224342200224> L by f (t) = diag{e342200230/342200230_1342200230,e342200230/342200230_15342200231}. Since ,6342200231 is irrational, the image of f can342200231t be a closed curve, but its closuremust be a torus, so that closure is all of L. Thus the image M = f (G) is densein L. Evidently f is one to one.)
- 22. 1.3. CONNECTEDNESS 5) So Proposition 1.2.2 says that f de357254201nes a. continuous injective homomorphism of G onto M. But this cannot be a homeomorphism: G is locally compact, and M is not locally compact becauseit is not locally closed in L. For this last point, see the endof the proof of Lemma 1.6.1.) DEFINITION 1.2.3. The quotient space of G by the subgroupH is the quotient space G / H with the quotient topology. 0) DEFINITION1.2.4. The natural actionof G on the quotient space X = G/H consists of the translations rg 2 g342200231 H 302273342200224> (gg342200231)H where g, g342200231 E G. <)) PROPOSITION 1.2.5. The map <I> : G x G/H 342200224> G/H, given by <I>(g,g342200231H) = rg(g342200231H) = (gg342200231)H, is jointly continuous and satis357254201es (a) <I>(1,g342200231H) = g342200231H for all 9342200231 E G and (5) 342200230I342200231(g.<I342200231(g342200231.y342200235H)) = 342200230P(9g342200231.g342200235H) for g.g342200231,g342200235 6 G Each T9 is 0 homeomorphism of G/H and g +342200224a Ty is a homomorphism ofG into the group of all homeomorphisms of G / H .) PROOF. Joint continuity of (1) follows from continuity of the multiplication map (g,g342200231) v342200224> gg342200231 on G, and the algebraic conditions (a) and (b) follow from the de357254201nition of the identity in the group G and associativity of the multiplication in G. The assertions concerningthe 7-3 now follow. Cl) Essentially as in Lemma 1.1.5, we have) PROPOSITION 1.2.6. IfH is a closedsubgroup of the topological group G then the quotient space G/ H is a regular topological space.) PROOF. Let g E G and F C X with FH closed in G and gH357254202FH = (ll. Choose an open neighborhood W of 1with Wg357254202FH = (0. Let U be an open neighborhood of 1 with U: U3422002301 and U2 C IV.) UgH D UFH is empty. For otherwise We have ui E U, hi 6 H, f E F with ulghl = ugfhg. That would imply uglulg = fh2h1_1 6 Wg F342200230! FH = 9). That proves regularity of G/H. D) EXAMPLE 1.2.7. Let G = SO(n + 1),the special orthogonal group consisting of all (n + 1)X (n + 1) matrices a with Ta = a3422002341 and deta = 1. Let H % SO(n) denote the subgroup that 357254201xes the 357254201rst standard basis vector e1. In other words H 2 {aE G I a(e1) = e1}. We can identify G/H with G(e1) = {g(e1) I g E G}, which is just the n342200224sphere S" consisting of all unit vectors in Rn342200234. So ignoring topologies we have G/H % 5'". But in fact this identi357254201cation is topological. O) 1.3. Connectedness) We342200231ll look at some matters concerning the connectedcomponent of 1 in a topological group. Recall that a topological space is connected if it cannot be de- composed as the union of two nonempty disjoint open subsets, totally disconnected if its maximal connected subsets are onepoint subsets.) LEMMA 1.3.1. An open subgroup of a topological group is a closedsubgroup.))
- 23. 6 1. BASIC TOPOLOGICALGROUP THEORY) PROOF. Let H be an open subgroup of G. Then the complement of H is G H = Ug302242HgH, which is a union of open sets. [:1) PROPOSITION 1.3.2. Let G be a connectedtopological group. Let V be a neigh- borhood of 1 in G. Then G = U:302260=1 V".) PROOF. V contains an open neighborhood U = U3422002301 of 1, so U:302260=1 U342200235 is an open subgroup of G. But then it is a closed subgroup, so it must be all of G. D) LEMMA 1.3.3. Let G be a connectedtopological group. If H is a subgroup then G/H is connected.) For G/H isthe continuous image of a connected set.) THEOREM 1.3.4. Let G be a topologicalgroup. Let G0 denote the topological component of 1 in G. Then G0 is a closednormal subgroup of G, and G/G0 is totally disconnected.) PROOF. G0 is connected by de357254201nition, is closed because topological compo- nents are closed.Themap G0 XG0 342200224> G, given by (g, h) r342200224342200224> gh"1, maps G0 ><G302260 342200224> G to a connected set that contains 1,soit maps G0 X G0 into G0. Thus G0 is a sub- group of G . It is normal in G because eachgG302260 g"1 is a. connected subset of G that contains 1, henceis contained in G0. We have proved that G0 is a closednormal subgroup of G.) Let -rr : G 342200224> G/G0 be the projection and let X be a non342200224empty connected subset of G / G0. Then 7r3422002301(X) C G is connected. For otherwise 7r3422002301(X) = A U B with both A and B open in 7r3422002301(X) and with AFWB = 0. lfa: E X then7r3422002301(:c) % G0, which is connected, while 7r3422002301is the disjoint union of its opensubsets 7r3422002301357254202A and 7r3422002301(:c) H B; so either 7T_1(CL342200230) C A or 7r3422002301(:t) C B. Thus X is the disjoint union 7r(A) U 7r(B) of non342200224empty open subsets. That is a contradiction. We conclude that 7r3422002301(X) C G is connected.) The topologicalcomponents of G are the cosets gG302260, because G0 is a topological component and the left translations are homeomorphisms. So every connected subsetof G is contained in some coset gG0. If X is a non342200224empty connected subset of G/G0, we just saw that 7r3422002351(X) is connected, so 7r3422002301(X) C gG0 for some g E G. It follows that X = 7r(7r"1(X)) = 7r(gG0), a single point. Thus G/G0 is totally disconnected. D) The group G0 is called the identity componentof G. For example, S0(n) is the identity component of O(n), and O(n)/SO(n) E Z2, for n 2 1.) A complement to this result, which we will need in order to study covering groups,is) LEMMA 1.3.5. Let G be a connectedtopological group and let H be a totally disconnected normal subgroup. Then H is central in G.) PROOF. Consider the map qh : G 342200224342200224> G, de357254201ned by qh(g) = ghg3422002301 where h E H is 357254201xed and g varies over G. It has image in H because H isnormal in G, and the))
- 24. 253) ,s.a,302273vP) 1.4. COVERING GROUPS 7) image is connected because G is connected,so the image is the single point {h} because H is totally disconnected. D) 1.4. Covering Groups) Let G be a topological spacethat is connected, locally arcwise connected, and semi342200224locally simply connected in the large3. Fix a basepoint g E G. Every sub- group I342200230 C 711 (G, g) of the fundamental group correspondsto a covering space fr : (G,_?j) 342200224> (G, g) such that fp maps 7r1(G,g)isomorphically onto I342200230. In this section we will prove) THEOREM 1.4.1. Let G be a connected, locally arcwise connected topological group that is semi342200224loca.lly simply connected in the large. Then the fundamental group 7r1(G, 1) is commutative. Suppose that I342200230 C 7r1(G,1) is a subgroup and let fr : (G, T) 342200224+ (G, 1) denote the corresponding coveringspace.Then G has a unique structure of topological group such that T is the identity element and fr is a group homomorphism.) PROOF. We will write 1g rather than 1 for the identity element ofG,in order to avoid confusion. First recallone of the standard constructions of the coveringspace fp : (G, 342200224342200224> (G,1g). The elements of G are the homotopy classes (endpoints 357254201xed) [0] of continuous arcs 0,) 0 : [0,1]342200224> G such that 0(0) = lg) modulo the equivalence relation) [0] 2 [T] <:> {0(1) = 7(1)and) I342200230 contains the homotopy class (endpoints 357254201xed) of 'r3422002301 - 0.) Here, as usual, r3422002351(t) = r(1342200224t) and p-0(t) = 0(2t) for 0 302247 t g %, p-0(t) = p(2t3422002241) for % 302247 t 302247 1. We write [[0]] for the equivalenceclassof [0]. The base point 1}} is the class[[01]] of the trivial arc, 01 (t) = 1g for 0 302247 t 302247 1, and the projection is fr([l0ll) = 0(1)- We first suppose that F = {I}, so we are dealing with the universal covering space f : (G, 342200224> (G, lg).) De357254201ne inverse in G by [[0]]3422002301 = where n(t) = 0(t)3422002301. If [[0]] = [[0342200231]] then we have a homotopy H :0 2 0342200231 with endpoints 357254201xed. In other words, if yo = 0(1) then H: [0,1]><[0,1] 342200224> G is continuous, H(s,0) = lg, H(s, 1)= go, H(0,t) = 0(t), and H(1,t) = 0342200231(t) . Now K 2 7; 2 n342200231 where n(t) = 0(t)3422002311, n342200231(t) = 0342200231(t)3422002301, and K (s,t) = H (5,t)3422002301. Thus inverse is well de357254201ned, and the de357254201nition includes the fact that f([[Ull3422002301)= f(l[Ull)3422002351~) De357254201ne composition in G by = where n(t) = 0(t)r(t). If [[0]] = [[0342200231]] and = [[r342200231]], so we have homotopies H : 0 2 0342200231 and K : r 2 7" with endpoints 357254201xed, then J : r] 2 77342200231 where r](t) = 0(t)r(t), n342200231 (t) = 0342200231 (t)r342200231 (t) and) 3This last condition means: if g 6 G there is an arcwise connected open neighborhood U of g such that, if 'y is a closed curve in U based at g then 1 is nullhomotopic (endpoints 357254201xed) in G.)
- 25. 8 1. BASIC TOPOLOGICAL GROUP THEORY) J (s, t) = H (s,t)K (s,t). Thus composition is well de357254201ned, and the de357254201nition in- cludes the fact that f([lUllllTll = f([[Ullf([lTll)-) We have proved that G has a well de357254201ned group structure such that T is the identity element and fp is a group homomorphism, in the case where I342200230 = There the group operations are continuous by construction and because fp is a local homeomorphism.Thus, in fact, G is a topological group and fp isa homomorphism of topological groups. The group structure is unique, by the co~vering homotopy property, sowemay speak of the universal covering group f :(G,1g)342200224> (G, lg).) Let A denote the kernel of~the universal covering f 2 G 342200224342200224> G. T hen342200231 A is a discrete normal subgroup of G. Lemma 1.3.5 says that A is central in G. As 7r1(G, lg) % A now 7r1 (G, 1g) is commutative.) We 357254201x the notation f : G 342200224> G for the universal covering group. Let I342200230 be any subgroup of 7r1(G,1g). We identify 7r1(G,1g) with A = Ker(f). Then f factors through a topological group homomorphism fp : G/I342200230 -9 G, and fr maps 7r1(G/I342200230, E11342200230/1342200230) isomorphically onto F. This completes the proof. Cl) COROLLARY 1.4.2. Let G be a connected, simply connected, locallg arcwise connected topological group. Then the universal covering groups f : G 342200224> G are just the G 342200224+ G /I342200230 where F is a discrete central subgroup of G.) 1.5. Transformation Groups and Homogeneous Spaces) We now take a close look at an extensionof the idea of the action of G onG/H by translations.) DEFINITION 1.5.1. A topological transformation group consistsof a topo- logical group G, a topological spaceX, and a continuous map <I> : G X X 342200224> X such that) (a) <I>(1,:1:) = as for all m 6 X, and (b) <I>(g, <I>(h, 2 <P(gh,a:) for all g,h E G and 9c 6 X.) This is also called a topological action of G on X. Conditions (a) and (b), togetherwith continuity of <I> in its second variable, say that the map) 9 r-342200224> 7'9 where rg(:r) = <I>(g, so)) is a homomorphism of G into the group of all homeomorphisms of X. In this context T9is often called translation by g. 0) EXAMPLE 1.5.2. Let X = G/ H with the quotient topology, as above, where H a subgroup of G. De357254201ne J) : G X X 342200224342200224> X by <I>(g,g342200231H) = (gg342200231)H, so that 79 is the map of X induced by the left translation 69. Then (G,X, <I>) is a topological trans- formation group. For continuity of CI) follows from continuity of the multiplication map (g,g342200231) 302273342200224> gg342200231 on G, and the algebraic conditions (a) and (b) follow from the de357254201nition of the identity in the group G and associativity of the multiplication in G. O) y I at 302253m
- 26. 1.6. THE LOCALLY COMPACTCASE 9) DEFINITION 1.5.3. Fix a topological transformation group <1) : G X X 342200224-> X _ If as E X then the set Om = <I>(G,9c) C X is the orbit of m under G, and G5, = {g E G | <I>(g,w) 2 ac} is the isotropy subgroup of G at :13. The space X is a homogeneous space of G if there is just one orbit,i.e.if X = 093 for some (hence all) :17 E X. 0) The orbits partition X, by the group propertiesof G. Unless we explicitly say otherwise, we usethe subspace topology for an orbit of G in X.) If the space X is T1 (points are closed subsets), then the isotropy subgroups are closedin G. For <I> is continuous and Gm is the projection on G of<I>3422002301(a:)357254202(G ><) The isotropy subgroups at the variouspointsof an orbit are conjugate: G,.g(m) = as(Gav) -) LEMMA 1.5.4. Fist a topological transformation group <I> : G X X -> X and a point :1: 6 X. Let 0 denote the orbit (93, and let H denote the isotropy subgroup Gm. Then <13 : G X (9 342200224> O is a topological transformation group, O is a homogeneous) space of G, H is a closedsubgroup, and g r342200224> 7'g(.r) induces a continuous one to one map of G/H onto (9.) PROOF. <I> : G X C342200231) 342200224> O is a topological transformation group because <I>(G, O) = (9, (ii) <I> is continuous, and (iii) (9 carries the subspacetopology.) Let 31: E X and H 2 G95 as in the statement of the Lemma.Then7'g(:t) = rgz just when 79-19, = at, in other words when gH = g342200231 H . So g +342200224> 7-g induces a one to one map 711$) : G / H 342200224> O of G / H onto (9. Continuity of 71193) follows from continuity of g u342200224> 7-g E!) The discussion following Proposition 1.2.2 givesan examplewhere the contin- uous one to one map of G / H onto 0 is not a homeomorphism. However, it also depends on (9 not being locally compact. We now address this issue.) 1.6. The LocallyCompactCase) We 357254201rst note that locally compact subgroups are always closed.) LEMMA 1.6.1. Let H be a subgroup of a topological group G. Then the closure H of H in G is a closedsubgroup of G. Further, if H is locallyclosedin G, in particular if H is locally compact,then H is closed in G. PROOF.Let g,g342200231 E H. We must prove that g342200231gffl E H. Let W be an open neighborhood of g342200231 g3422002301 in G. Then we have open neighborhoods U of g and V of g342200231 with VU3422002301 C W. As g,g342200231 E H, both U and V meet H, so VU3422002301 meets H and thus W meets H. We have proved that every neighborhood of g342200231 g3422002351 meets H , so g342200231g"1 E H. Thus H is a subgroupof G.) Now suppose that H is locally closed in G. In other words, if h E H then there is an openset Vh C G with h E Vh and H H Vh closed in Vh. We can assume)
- 27. 10 1.BASIC TOPOLOGICAL GROUP THEORY) V}, = Uh for some open neighborhood U of 1 independent of choice of h E H, and that H {W7}, is closed in G for all h E H. To seethat, choose an open neighborhood U of 1 with U = U *1 and U2 C V1. Then U C V1 as in the argument of Lemma 1.1.5,so H 357254202 U is closed in G. Now replace V}, by Uh.) If g E U_357254202 H, then every niighborhood of g m_e_ets H (WU, which is closed in G, sogEHF342200230IUC H. Thus U357254202H C Hand so U357254202H=U357254202H. NowW=UhEHUh is open in G, contains H, and satis357254201es W (342200230I H = W H H.) Let g E H and h E H. Choose anopenneighborhood V of 1 such that hV C W. As g E H there existsk E H such that kg E V. Now hkg E W. And since H is a subgroupof G we have hkg E H. Thus hkg E WFWH = WFWH C H. Consequently g E k:3422002301h3422002301H = H. That proves H = H.) The remaining statement of Lemma 1.6.1 follows from the general fact) LEMMA 1.6.2. Let H be a locallycompact subspace of a Hausdor357254202 topological space G. Then H is locally closed in G.) PROOF. Every h E H has a compact H342200224neighborh0od K. There _is a G~open neighborhood_Uof it such that h E U H H C K. Now the closure_Uof U in G satis357254201es H O U C K and we cut K down to its compact subset H (W U.) Fix g E clU(H357254202U). Since HHU C K and K is compact,we have g E K C H. But we chose g E U. So g E H357254202 U. This proves HDU = clU(H357254202U), so H is locally closedin G. El) We now need two technical facts:) LEMMA 1.6.3. Let G be a topological group, K a compact subset, and F a closed subset. Then FK is closed in G.) PROOF.Let g E G FK, so F3422002301g 357254202 K = Q). F"1g is closed in G because F is closed.If k E K regularity of G provides an open neighborhood Vk of 1 such that l<;Vk2 357254202 F3422002301g = 0. As K is compact We have {k1,...,k.,,} C K with K C U139, kt,-Vki. The intersection V9 = 357254202lgign Vk, is a neighborhood of 1, and) my c (ulgm k,-1/,.,)'Vg c U,S,.g,, k,V,.,V,.,, which is disjoint from F-1 g. Thus FK is disjoint from the neighborhood gVg3422002301 of g, for every g E FK. In other words, G FK is open in G, so FK is closed. El) LEMMA 1.6.4. Let X be a locally compact Hausdor357254201342200231 space. Suppose X = U211 F,, with each F342200235 closed in X. Then at least one Fn contains a non342200224empty open set.) PROOF. Choose 342200230U0 E X. It has an open neighborhood V0 with V0 compact. If V0 302247Z F1 there is at least one point 111 E V0 F1. Since V0) U (70 F1 F1) is compact, 121 has an open neighborhood V1 such that 71 C 70 and V1 0 F1 = (0. If V1 302247Z F2 we proceed as before. Unless someFn contains a non342200224empty open set V,,_1 we construct in this way a decreasing sequence {V,,}of non342200224empty open sets with each V,; compact and V; F1 F1, = (D for is 302247 n. But then 357254202 V7, 75 0 while 357254202V,, is disjoint from every Fk. That is a contradiction. D)
- 28. 1.6. THE LOCALLY COMPACT CASE 11) Now we can start to settle someproblems raised in Proposition 1.2.2 and Lemma 1.5.4:) PROPOSITION 1.6.5. Let G be a topologicalgroup, L a closed subgroup, and K a compact subgroup. Let 71' 2 G 342200224> G/K denote the projection onto the quotient space, let 7rL : L 342200224342200224> 7r(L) denote 7rlL, and let 7;: L/LFWK 342200224> 7r(L) be the pushdown of 7r]L to L/L H K. Then 7rL is both open and closed, and 357254201 is a homeomorphism.) PROOF. We 357254201rst prove that 7rL is a closed map. LetT bea closed subset of L. Then T is closed in G. By Lemma 1.6.3TK isclosed in G. Now G TK is open in G. The map 7r is open so 7r(G TK) is open in G/K. Thus 7r(T) is closed in G / K, soit also is closed in 7r(L). This provesthat 7rL is closed.) Let u : L 342200224> L/L O K denote the projection. If Y is closedin L/L H K then ,u.3422002301Y is closed in L, so 357254201(Y) = 7rL(u3422002301Y) is closed in 7r(L). Thus if is a closed map. But 357254201 is continuous and bijective, so now it is a homeomorphism. It follows that in; is open. El) COROLLARY 1.6.6. If K is a compact normal subgroup of G and L is a closed subgroup of G, then 7r : G 342200224> G/K induces a topological group isomorphism4 of L/L 0 K onto LK/K closed in 7r(L).) We can also handle a question of separateversus joint continuity:) THEOREM 1.6.7. Let G be a locally compact and o~compact5 topologicalgroup. LetX be a locally compact Hausdor357254201342200230 space. Suppose that CI) : G><X 342200224> X satis357254201es all the conditions for a topologicaltransformation group such that X is a homogeneous space of G, except that <I> is only assumed to be separately continuous. Then {D is jointly continuous. Let me E X and let H denote the isotropy subgroup of G at 930. Then g s342200224> T9 (:30) de357254201nes an open map 7 : G 342200224> X , and 7 induces a homeomorphism of G / H onto X.) PROOF.We need only check that if U is a neighborhood of 1 in G then <P(U,9:0) is a neighborhood of mg in X. For then 7 is open.Since we already know that 7 is continuous and surjective, it will follow that 7 induces a homeomorphism 7 of G / H onto X. Thenjoint continuity of <I> will follow from joint continuity of the map G X G / H 342200224342200224> G / H induced by multiplication in G.) Let U be a neighborhood of 1 and choose an open neighborhood V = V3422002341 of 1 with V2 C U. As G is locally compact, V contains a compact neighborhood F of 1. As G is o~compact it contains a sequence {gn},n = 1,2,3, ..., such that G = U gnF. Since7 iscontinuous and surjective, the 7(gnF) are compact and they cover X. Now some 7(gnF) contains an opensubset Z C X. If f E F such that 7(gn f) E Z then also belongs to the open set <I>((gn f )3422002311, Z), which is contained in Q(f_1F,l'0) C 302242(l/2,1130) C .719) D) COROLLARY 1.6.8. Let f : G 342200224> G342200231 be a continuous surjectioe homomorphism of locally compact topological groups. If G is a342200224compact then f is open.) 4By topological group isomorphism we mean an isomorphism in the category of topological groups, that is, a. group isomorphism which is a homeomorphism. 5A topological space is called o342200224compact if it is a countable union of compact subsets.))
- 29. 12 1. BASIC TOPOLOGICAL GROUP THEORY) Finally we note something that we can342200231t do without:) PROPOSITION 1.6.9. Let H be a subgroup of a topological group G. If H and G / H are locally compact, then G is locally compact. If H and G/H arecompact, then G is compact.) PROOF. Let 7r : G 342200224> G / H be the projection,U a closedneighborhood of 1 in G such that H (1U3422002301U is compact, and U1 a closed neighborhood of 1 such that Uf1U1 C U. As G /H is locally compact there is a compact neighborhood 0 of 7r(1) such that C342200231 C 7r(U1). Set V = U1 0 7r3422002301(C), closed neighborhood of 1 in G. We are going to prove that V is compact.) Let B be any family of subsets of V that is maximal for the property that 357254201nite intersections are non-empty. We will prove compactness of V by showing that HBEBB # (ll. As 7r(V) = C, which is compact, 357254202BEB7r(B) 79 (7), say with) v E V such that 1r(v) E 7r(B) for all B E B. H 0 v3422002301U is compact because it is closedin the compact set H 0 U3422002301U. Now UH O U is compact.) If HBB were empty, to every g E UH (1U we could associatea neighborhood V9 of 1 and some By E B such that Vg2g (1 B9 = (0. As UH Fl U is compact it is covered by a 357254201nite union Vglgl U U Vgngn. Set B = By, F1 (1 B9342200235. Then B E B by maximality of the latter. W = U1 0 V91 0 302253 - - (1 V9" is a neighborhood of 1 in G such that W(vH F] U) F1 B = 0. But 7r(Wv) meets 7r(B)becauseit is a neighborhood of 7r(v). Sothere existin 6 VV, b E B, h E H such that wvh = b. Now ch = w3422002311b E Uf1U1 C U, so b E W(vH (1 U) H B = (ll. This contradiction proves 357254202BB 75 9}. Thus V is compact, and it follows that G is locally compact.) If H and G/H are compact, wejust take C = G/H so U = G and V = G is compact. El) 1.7. Product Groups) In this Section wediscussdirectproducts and semidirect products of topological groups.) DEFINITION 1.7.1. Let {G1-}i342202254A be a family of topological groups indexedby some set A. The direct product G = H,-EAG, of topological groups is direct product G = H,-GAG, as abstract group with the direct producttopology. Thus the group composition in G is) (1-7-2) (gt-)(y302253E) = (9.-gl) and <90342200235 = (95 ) and a basis for open sets in G is given by the products 1'I,EAU,- where U, = G,-) except for a 357254201nite number of indices 1342200231 E A, and for those indices U, is an open subset of G,-. O) LEMMA 1.7.3. G = H,-EAG, is a topological group, and each G, is a closed normal subgroup.) PROOF. Let (g,-), E G and let W = ILEAW, be an open subset of G that contains (g,-)(h,~)3422002301. For each 2' with VV, 75 G, choose open neighborhoods U, of g, and V, of h, in G, such that U,-V171 C I/V,-. If W, = G, let U, = V; = G,-. Then))
- 30. 1.7. PRODUCT GROUPS 13) U = l'I,~.gAU,-and V = H,-GAVE are G342200224neighborhoods of g; and hi, respectively, and UV3422002301 C W. Thus the group operations are continuous. The one342200224point subsets are closed because G = UNA Pj where Pj is the openset H1-EAUz~,j with UM = Gi for 2' 75 j and U342200234 = Gj {gj}. Thus G is a topological group.) For the last statement we identify G,~ with the subgroup consisting of all (gj) such that gj = 1 for j 7E 1'. It is normal because each of its elements commutes with every element of every other factor. It is closedbecauseits complement is the union of the open subsetsSj = {(gk) 6 G I gj 75 1} C G forj #1". CI) Note that a direct product of compactgroups is compact. Direct products of locallycompact groupsarelocally compact only when all but 357254201nitely many factors are compact.) The proofof Lemma 1.7.3 is easily adapted to the boxtopology on the abstract product group. That, and the problemof dealing with in357254201nite products, force the 357254201nitenessrestriction in the converse to Lemma 1.7.3:) LEMMA 1.7.4. Let G be a topologicalgroup, {G1,...,G,} be closed normal subgroups. Suppose that every g E G has unique expression g = g1...gT with gi E G,-. Then (g1, . ..,g,) 302273342200224-> g1...gr de357254201nes a topological group isomorphism of G1 >< ---XGT ontoG.) Finite direct products of matrix groups can be realized as block matricesdown the diagonal. Let G) be a multiplicative group of m X ni real matrices in the topology ofIR"342200231'342200231342200230"* from the matrix entries, say for 1 302247 1' 302247 7342200234. Then G = H131-9G,342200231 is realized as the group of TL X n real matrices, n = Xlgigr m, of the form 342200224 _) 0 gg . . . 0 (1.7.5) 9 = . . , where g,- E G.342200231 for 1 302247 1' g r, 0 0 gr) again in the topology of lR""" from the matrix entries.) However product groupscan be a bit more subtle. Let SO(4) denote the group of proper rotations of R4 and let ll-ll denote the (noncommutative) 357254201eld of real quaternions. We identify llil with R4; the real inner product becomes6z-w = Re(302247w). Let S3 = {z E H I |z| = 1}. Topologically it is a 3342200224sphere, and evidently it is a multiplicative group. Now we have two topological groups isomorphic to 33: the group G1 of R342200224lineartransformation E(z) : CL342200230 r342200224342200224> zzr of H and the group G2 of R342200224linear transformation r(w) : at 302273342200224> :rw3422002301 of H. These actions commute, and the resulting transformations preserve the inner product, so we have homomorphism) (1.7.6) 7r 2 S3 X S3 -342200224> SO(4) given by 7r(z,w) : ac +342200224> zwufl of topological groups. One easily checksthat it is surjective and its kernel consists) of {(1, 1), (-1, -1)}, so 30(4) 2 (S3 >< S3)/Z2.) 6There is a reason for placing the conjugation of H over R on the first variable of the inner product. This will surface in Chapter 2.))
- 31. 1. BASIC TOPOLOGICAL GROUP THEORY) There is an interesting variation on this isomorphism, important in differential geometry. Let K be a topological group.Then) (1.7.7) G = K x K actson K by <I>((k;,k2), k) = klkkgl.) This is a topological transformation group <I> : G342200231 X K 342200224> K . The situation described above specializesto <I> : (S3 X 5'3) X S3 342200224+ S3.) DEFINITION 1.7.8. Let K and N be topological groups. Let Aut(N) denote the group of all continuous automorphisms of N. Let <25 : K 342200224> Aut(N) be a group homomorphism such that) (179) CD : K X N 342200224342200224> N, de357254201ned by <I>(k,n) = <b(k)n, is continuous.) Let N ><302242 K denote the topological space N X K with composition de357254201ned by (n, k:)(n342200231, k342200231) = (n302242(k)n342200231, kk342200231). Then N ><1302242,K is the semidirect product of N and K de357254201ned by go. 0) 1.7.10. If K is locally compact and o342200224compact, and if N is locally compact, then (179) is equivalent to the condition that (k, n) +342200224> q5(k)n is continuous in k. In effect,the de357254201nition of Aut(N) gives continuity in n, so continuity in lc provides separate continuity, and then Theorem 1.6.7provides joint continuity.) LEMMA 1.7.11. Let K and N be topological groups and let gt : K 342200224> Aut(N) be a group homomorphism that satis357254201es (179). Then N >44, K is a topological group, N is a closednormal subgroup, and N ><302242 K 342200224342200224> (N ><1302242 K ) /N induces a topological group isomorphism ofK onto (N ><1302242) PROOF. First we check that (n, k)(n342200231, k342200231) = (nq5(k)n342200231, kk342200231) de357254201nes a group structure on N x K . The multiplicative identity is just (1,1). Two342200224sided inverses are given by (n, k)3422002351 = (q5(k3422002351)(n3422002301),k3422002341). For associativity, compute ((7% k)(n'> 76'302273 ("W") = (W5(k)n'a kk')(n"k = (n<15(k)(n')302242(kk')(n,") kl342202254'k = (n, k><n'<z><k'><n">, W) = (n,is) (me k='><n"k">>. Now N ><1302242 K is a group. We have to checkcontinuity of the map N><KxN><K 342200224> NXK de357254201ned by (n342200231,k342200231,n,k) H (n'302242(k342200231)((q5(k3422002301)(n"1)),k342200231k3422002301).) First, the map (n342200231, k342200231, n, k) +342200224> k342200231k3422002301 is continuous because K is a topological group. Second, the map (n342200231, k342200231, n, k) r342200224342200224> (n342200231302242>(h342200231)((q5(k3422002301)(n3422002341)) is continuous by (1.7.9) and the fact that both N and K are topological groups.Thus N >445 K is a topological group. The remaining statements are obvious. D) The 342200234converse342200235 construction is) LEMMA 1.7.12. Let G be a topological group, N a closed normal subgroup, and K a closed subgroup such that G 342200224> G /N maps K isomorphically onto G/N. Then G %342200231 N >4302242 K where q5(k)(n) = l3422022547342200231Ll342202254_1.) ,.., ;:;;30
- 32. 1.8. INVARIANT METRICS ON TOPOLOGICAL GROUPS 15) There are many important examples ofsemidirect products.Thebestknown is the euclidean group, the group consistingof all rigid motions of euclidean n342200224space.It is the semidirect product R" >4 O(n) where O(n) is the usual orthogonal group:) (o,k)(v',k') = (v+ki1',kk342200231) and (v,k) :11342200231 302273342200224342200224> v+kv342200231 for 12,12342200231 6 R" and k, k342200231 E O(n)) Thus euclidean space is identi357254201ed with the translation subgroup R" of and the actions is, in effect, by left multiplication. The proper euclidean group is the subgroup R" >4 SO(n) of orientation342200224preserving rigid motions of R". Along the same lines one has the affine group A(n) = R" >4 GL(n; R) of euclidean n342200224space; here GL(n; R) consists of the invertible linear transformations of R342200234. We will see a number of lessclassicalexamplesin Chapter 2.) 1.8. Invariant Metrics on Topological Groups) In this Section we construct invariant metrics on topological groups and look at the consequences for completionand covering.) THEOREM 1.8.1. Let G be a topologicalgroup with a countable basis for open sets at 1. Then the topology on G is the underlying topology of a metric that is invariant under all right translations.) REMARKS. The topological condition on G, usually called 342200234357254201rst countable342200235, is necessary because the open balls of rational radius about a point in a metric space form a countable basisfor open sets at that point. The conclusionof the Theorem is usually phrased: G342200231 has a right342200224inva.riant metric. Note that the map g +-> g3422002301 carries a right342200224invariant metric to a left342200224invariant metric.) PROOF. Let {U1-},342200230=g,1,_, be a countable basis for open sets at 1. Then the On = 357254202fzo U, form a countable, monotone decreasingbasisfor open sets at 1. Recursively chooseopenneighborhoods Wn of 1 such that (0) W0 = 00 and W342200235 = W; 1, and (1) W,2,+1 C W" C On. Then de357254201ne open neighborhoods V, of 1, where r = 19/2" is dyadic rational With 1 302247 k g 2342200235, by (2) V1/2342200235 = Wm (3) l/Qk/2n+1 = Vk/27. and (4) l/(2k+1) /271+; = V1/2.1+; Vk/2...Then V} depends only on r, not on its representation as a dyadic rational. One also has (5) V1/2nV,,,/2n C V(,,,+1)/27.; if m = 2k this is immediate from (3) and (4), and if m = 2k: + 1 it follows from) V1/gnvm/2n Z l/1/271. (l/1/2nVk/2n~1) C V1/2n342200224lVk/2n3422002241 and l/V(m+1)/2n = l/(k+1)/271-1 by induction on n. Note that (5) implies (6) VT C V3 for 0 < r < 8 302247 1.) De357254201ne f : G X G342200231 342200224> R by f(g, h) = 0 if hg3422002301 E VA/fl for every dyadic rational r, 0 < r 302247 1, and by f(g, h) = lub{r l hg3422002301 E VrV,.3422002301} otherwise. Then f(g:1:, hr) = f(g, it) because (h:r)(g:r)3422002351 = hg"1, f(g, h) = f(h, g) becauseV,Vf1is symmetric, and f(g, h) = 0 302242> g = h because l/1/27302242342200230/1-/$7, = 12/2,. C l/1/2~n3422002243422002241 = W,,_1 C On_1 and 357254202On_1 = But the triangle inequality is a seriousproblem here, so we now de357254201ne) 9 = G X G 342200224> R by P(9;h)=111bzeGlf(9302273-T)342200230 f(h,rv)l-))
- 33. 16 1. BASIC TOPOLOGICALGROUP THEORY) Then p(g,h) = p(h, g) 3 O and p(g,h) 2 f(g,h) by construction, and it follows that p(g, h) = 0 <=> g = h. Right invariance comesfrom) ply342200234: = 1U-b0L342200230EG|f(.gu7x) _ = 1ubzeG|f(9,wu'1) 342200224 f(h,93u'1)| 1342200230-1b97342202254Glf(gv37) 342200224 : p(ga h)302273 and the triangle inequality comes from p(g:k) = h1bacEGlf(gam) 342200224 + 342200224 302247 p(g9h)) Thus p is a metric on G.) To see that the topology induced by p is the original topology on G, we need only show that they have the same neighborhoods of 1 E G, because both are right342200224invariant. De357254201ne BT to be the metric ball {gE G 1 p(g, 1) < 7"}. Unwinding the de357254201nition of p one sees that V1/2n C B1/2n . Thus the open metric ballsB1/2342200235 are neighborhoods of 1 in the original topology. And if 0 is a neighborhood of 1 in the original topology, then some Ok C 0, soB1/2:c+1 C 0 because g E B1/2k+1 implies p(g,1) < 1/2'342200234+1, which implies f(g, 1) < 1/2k342200234,which in turn implies g E l/12/2k+1 C V1/2:: C Ok C 0. That completes the proof. Cl) DEFINITION 1.8.2. Let (G, p) be a topologicalgroup with a right342200224invariant metric. A sequence {gn} C G is Cauchyif, given 6 > 0, there is an no > 0 such that p(gm,gn) < 6 for m,n > no. A sequence {gm} C G converges to g E G if, given 6 > 0, there is an no > 0 such that p(gn,g) < e for n > no. One then says that g is the limit of {gm},written g = lim gn and also {gn} 342200224> g. This forces {gm} to be Cauchy. (G,p) is complete if every Cauchy sequencein G converges in G. (G,p) is the completion of (H, pH) if H is a subgroup of G, (ii) pH = p|HXH, and (iii) every g E G is the limit of a Cauchy sequence from (H, pH). 0) We will now construct the completionof (G, p) and show that it is a topological group. In effect we follow the Cauchy sequence construction that yields the real number 357254201eld as the completion of the rationals.) Recall that Cauchy sequences {gn}, in a metric space (G,p) are called equivalent if {p(gn, 342200224> 0 in the real numbers.) EEMMA 1.8.3. Let (G, p) be a topologicalgroup with a right342200224in1Ja7iant metric. get G consist of the equivalenceclasses of Cauchy sequences {gn} in G. Then G is a group with well de357254201ned composition = and inverse) l{.9n}l_1= [{9;1}]-) PROOF. De357254201ne F : G X G 342200224+ G by F(g, h) = gh3422002301. If {gn} C G is Cauchy, so is {g;1}.To see this, note that the Cauchy condition for {gn} just says: if U is a neighborhood of 1 E G then F3422002301(U) contains {(gm, gn) | m,n > no} for some no > 0. Since g 1-> g3422002301 is a homeomorphism of G, the sameholds for the map H (g, h) = g3422002301h, not necessarily with the same no. Thus {gg1}isCauchy.) If {gn}, C G are equivalent Cauchy sequences,then {gg 1}and {g357254201fl} (which we just saw to be Cauchy) are equivalent. For the equivalence condition on {gm} and just says: if U is a neighborhood of 1 E G then F3422002301(U)contains) r1;3
- 34. 00224302242~ DQ_,) 1.8. INVARIANT METRICS ON TOPOLOGICAL GROUPS 17) 342200230l(9m 9:1) I n > no} for some no > 0. Now the same holds for H3422002311(U),for some no. Thus {gg 1} and {g357254201fl} are equivalent.) If {gm}, C G are equivalent Cauchy sequences, and {h,,}, C G are equivalent Cauchy sequences, then {gnhn} and {g;,h;,}are Cauchy sequences and are equivalent. To see this, let U be an open neighborhood of 1 in G. We must show that there is a number n1 such that m,n 2 n1 implies(gmhm)(g,,h,,)3422002301,(gmhm)(g;,h;,)'1 E U. First, let E = E3422002311 be an open neighborhood of 1 such that E3 C U and let no > 0 such that, if E 2 no then gg E Egno. Now W = g;01Eg,,0is an openneighborhood of 1 such that, if E 2 no then ggWg[1 C Egno Wg;O1E = E3 C U. At this point we choose an open neighborhood V = V3422002341 of 1 in G such that V3 C W, and we let n1 2 no such that) m,n 2 n1 implies hmh;1,hmh302247,'1 E V and a,b 2 n1 impliesg302247,_1ga,g;1ga E V. If a,b, m, n 2 n1 now) (9ahm)(9bhn)_1 : gm ' (91l119a)(hmhr_z1)(9b34220022419n1) 342200230.9111 C gm Wrlil C E3 C U and (9ahm)(gbh342200231,n)_1 2 gni ' (g'r:11ga)(hmhl1,_1)(gb3422002301gn1) ' 97:11 C gni W977: C E3 C U342200230) That proves our assertions.) We have now proved that the rules of composition, describedabove, specify a well342200224de357254201ned group structure on G. [3) LEMMA 1.8.4. Let (G, p) be a topological group with a right342200224invariant metric. Construct the group G as in Lemma 1.8.3. De357254201ne o5 : G 342200224> G by 302242(g) = where gn = g for all n. Then (1) is an lnjectiue homomorphism of G into G. De357254201ne o : G X G 342200224> R by 357254201([{g,,}], = limg(gn,hn). Then o is a well342200224de357254201ned) right342200224invariant metric on the abstract group G, and 342200230p342200230(q$(g), = p(g, h) for all g, h E G.) PROOF. It is clear from the de357254201nition that (Z) : G 342200224> G is an injective homo- morphism.) We check that 357254201([{g,,}], = limp(gn,hn) is well342200224de357254201ned on G. Let {gn}, C G be equivalent Cauchy sequences, and let {hn}, C G be equiv- alent Cauchy sequences; we must prove that limn_,oo p(g,,, hn) exists and is equal to limnnoo p(g;,,h302247,). For the existence, let 6 > 0 and chooseno > 0 such that E, m 2 no implies p(-9579771) < 6/2302273 and < Then if 342202254,m 2 no we have p(gg,hg)342200224 p(gm,hm) < 6, so limp(g,,,h,,) exists. Now choose n1 > 0 such that m,n 2 n1implies p(gm,g302247,) < 6/2 and ,o(hm,h;,) < 6/2. Then m, n 2 n1implies p(gm, hm) 342200224 p(g;,, < 6. So the two limits are the same, and 5 is well de357254201ned.) By construction, if g, h E G then o(q5(g), gb(h)) = p(g, h).) Now we check that ]6 is a right342200224invariant metric on G. The metric properties p(gn7hn) : /7(hn3022739n) and p(gn:k342200231rL) 302247 p(9nvhn) + p(h342200231n=kn) Survive to the limit342200230)
- 35. 18 1. BASIC TOPOLOGICAL GROUP THEORY) Note : 1imp(gnagn)= 0- If : 03 then {.971} is equivalent to {hn}, i.e. = Th_us p is a metric on the underlying set) of G. It is right342200224invariant because [{an}] E G gives us 357254201(l{9n}ll{an}l, l{hn}ll{an}l) = 1imp(9nan,gnan) = 1imp(9m on) = /_7(l{9n}l302273 l{hn}l)- Thus p is a right342200224invariant metric as asserted. Cl) THEOREM 1.8.5. Let (G, p) be a topological group with a right342200224inuariant metric. Construct the group G as in Lemma 1.8.3,the injection 302242 : G 342200224+ G as in Lemma 1.8.4, and the metric p on G as in Lemma 1.8.4.Then the underlying topology from p givesG the structure of topological group, (p becomes an injectiue homomorphism, and (p is a homeomorphism of G onto 302242>(G). Thus we can view G as a subgroup of G, p as p|gXG, and (G,p) as the completion of (G, p).) PROOF. We use Lemmas 1.8.3 and 1.8.4 to identify g e G with 302242(g) e 6, thus viewing G as a metric subgroupof G. Here p = plgxg and the p342200224-metric balls in G are just the intersections of G with the p342200224metricballs in G. So the original topology of G is its subspace topology in G. This completes the proof that (Gj) is the completion of (G, p). D) 1.8.6. Different invariant metrics can give different completions. For example, considerthe metricsonthe additive group of the rational number 357254201eld Q given by po302260(.1:, y) = [ac 342200224 y] (usual absolute value) and pp(:17, y) = 342200224342200224 jg/Hp (p342200224adic norm, p prime). The corresponding completions are the additive group of the real number 357254201eld R and the additive groups of the p342200224adic number 357254201elds Qp.) 1.8.7. Everything is essentially the samefor_left342200224invariant metrics, and the homeomorphism g n342200224342200224> g'1 carries the completion (G,p) for a right342200224invariant metric) p over to the completionfor the corresponding left342200224invariant metric.)
- 36. CHAPTER 2) Some Examples) Lie groupsand linear algebraic groups are the most interesting and useful topo- logical groups. Without going into any structure theory here, we summarize some basicfacts on linear Lie groups and then turn to a number of important exam- ples.Thoseexamplesinclude general linear groups G'L(n; IF), special linear groups SL(n;IF), unitary groups U (p, q; IF) of various signatures over the real, complex and quaternion 357254201elds, real and complex symplectic groups S'p(n;IF), p342200224adic completions of linear algebraic groups de357254201ned over the rationals, and various types of Heisen- berg groups. Other examples are concernedwith application to harmonic analysis on spheres, projective spacesand Grassmann manifolds. All of these exampleswill be used later in the book.) 2.1. General and SpecialLinear Groups) Let IF be the real number 357254201eld IR, the complex number 357254201eld (C, or the quaternion division algebra ll-II. Let V be a 357254201nite dimensional vector space over IF. Since we342200231ll be concentrating on groups of linear transformations of V it will be convenient to have linear transformations act from the left, so that the correspondence between matrices and linear transformations does not reverse order of products. Thus we want scalars to act from the right, that is, we take V to be a right vector space over IF. Now a choice ,6 = {e1,...,en} of basis of V gives an isomorphismof V with the right vector space IF" of n X 1 (column) vectors with entries from IF, and here every lF342200224linear transformation of V corresponds to an n X n matrix with entries from IF acting on IF342200235 by g : v 302273342200224+ gv. Now we have the general linear group) (21.1) GL(V) E GL(n; IF) : invertible IF342200224linear transformations of V E IF".) Here we think of G'L(V) as a group of linear transformations and GL(n; IF) as a multiplicative group of matrices.) Recall the notation IF""" for the space of n X n matrices over IF. If IF is IR or (C then IF342200235X" is a vector space of dimensionn2 over IF. If IF = H then it is a vector spaceof dimension 477,2 over IF.) We can View V as a vector space of dimension m342200230 over R where 1" = dim], IF. Then GL(V) consists of all lR342200224l'1near transformations g : V ~> V with det,R (g) a303251 0 that also are lF~linear, i.e. that commute with the (right) scalar action of IF. This exhibits GL(n; IF) as an opensubset of the real vector space IF "X" ER R342200231"342200231342200230342200231 In the subspace topology the matrix group GL(n;IF) now is a locally compact topological group, and GL(V)acquires the same structure from the isomorphismof (2.1.1).) 19))
- 37. 20 2. SOME EXAMPLES) We can also view GL(V) as a closedsubmanifold of lF2"X2", where (2.1.2) on E GL(V) corresponds to (3 (L91)6 lF2""2".) When IE342200230 is R or C, so that one has the determinant, it is more usual to use) GL0/) 3' 3 (det(3z)3422002311 ) i 3422002301 E Fnxn}) with the group isomorphismgiven by a I342200224> (3 det(;_1 The general linear group has someobvious and useful subgroups, such as (2.1.3) GL'(V)% GL'(n;lF) : all elements that preserve Lebesgue measure on V. So GL'(n;lR) = {g E GL(n;lR) | det(g) = :l:1}, GL342200231(n;<C) = {g E GL(n;C) | |det(g)| = 1}= {gE GL(n; (C) |detRg = 1},and GL'(n;H) = {.9 E GLWH) I detm(9) = 1}- A slightly smaller but more famous subgroup is the special linear group (2.14) SL(V) 2 SL(n;lF) : the derived group of GL(V) 302247 GL(n;) All these are closedsubgroupsof general linear groups, hence also are locally com- pact topological groups.) These groups have an interesting involutive automorphism, the Cartan invo- lution given on the matrix level by) (2.1.5) 0 : g v342200224-> tg3422002301 where the bar designates conjugation of IF over R. It has 357254201xed point set (2-1-5) {9 E GLWJF) I 9(9) = 9} = U(n;F),) which will be de357254201ned below and which turns out to be a maximal compact subgroup.) 2.2. Linear Lie Groups) For our purposes, a linear Lie group is a group G of linear transformations of a (real, complex or quaternionic) vector space V de357254201ned by some polynomial equations (in the matrix entries for a choice of basis of V). Thus G is a linear algebraicgroup by de357254201nition, and we will see just how G is a Lie group. By de357254201nition, a Lie group is a C342200234 (real analytic) differentiable manifold with a group structure such that the group operationsare C342200234.) First, we may always regard V as a real vector space, perhaps at the cost of doubling or quadrupling its dimension,sowe may View G as a real linear Lie group perhaps at the additional costof increasing the number of de357254201ning equations. That done, 71 = dim V, and a basis chosenfor V, now G may be viewed as the group of all n X n real matrices (gm) that satisfy a collection of polynomial equations f;,(g171, . . . ,g,,,,,) 2 0. The implicit function theorem now ensures that generically and locally G is a C342200234 submanifold of the vector space ]R"X" of n X n real matrices. Applying group translations G is globally a C342200235 submanifold of Rm342200235. The group operations are polynomial in lR""", thus C342200235 on the C"" submanifold G. Thus, by de357254201nition, G is a Lie group.) iaw
- 38. 2.3. GROUPS DEFINED BY BILINEAR FORMS 21) Equation (2.1.2) shows that the general linear groups GL(n;lF) are linear Lie groups. It follows that the GL342200231 (n;lF) and the special linear groups SL(n;lF)are linearLiegroups because we obtain them by just enlargingthe setof lR342200224polynomial equations used to define GL(n;) We will want the Lie algebra g of a linear Lie group G. By de357254201nition (only for linear Lie groups!) g consists of all linear transformations E of V such that exp(t302247) E G for all real t. Hereexp(t302247) means the exponential series 2,20 %302247", and it is easy to seethat this series converges for all real t. Onecanprove thz357254201; g is closed under linear combination and alsounder the composition [5, 77] = 517 342200224 775. Thus g is a vector spaceof linear transformations of V with the Lie algebracomposition) [30224377]-) DEFINITION 2.2.1. The automorphism 0 : GL(n;F) 342200224+ GL(n; lF), given by 0(g) = E? as in (2.1.5), is a Cartan involution of GL(n;lF). If a linear Lie group G C GL(n;lF)is stableunder 0 (or one of its conjugates in the group of automor- phisms of GL(n; then 6|G(orthe restrictionof the conjugate that stabilizes G) is a Cartan involutionof G. If I357254202 is a Cartan involution of G then dzp : g 342200224> g is called a Cartan involution of the Lie algebra g, and the 357254201xed point set K = G342200230(342200231 is a maximal compact subgroup of G. O) 2.3. Groups De357254201ned by Bilinear Forms) Let b be a bilinear form on V. That means b: V X V 342200224+ ]F is an lF342200224bilinearmap. In general an lF~bilinea.r map T : U x V 342200224> W factors through an lF342200224linear map U (8) V 342200224+ W, so U 302256 V must be de357254201ned and must be a (right, in our case) vector space over IF. This requires F to be commutative. Sowe only consider bilinear forms in the real and complex cases.) Fix a bilinear form b on V. Then a choice,5342200231 = {e1, . . . , en} of basis of V gives a matrix B = (b(e,-,e,-))such that, identifying a vector v = Ev,-6," E V with the column vector 342200230(vb . . . , 11,") we have (2.3.1) b(u,1J) = Z u,-b,-7]-vj = t'u,B'u. 1302247i,j302247n 7 The bilinear form now de357254201nes a group O(V, b) = {g E GL(V)|b(gu,gv)= b(u,v) Vu,v E V} -% {9 E GL('n;TF) l *9Bg = B}- Each of the groups(23.2)is a subgroup of the general linear group de357254201ned by a system of quadratic equations in the matrix entries, thus a closed subgroupof the general linear group, and thus a locally compact topological group.Thegroup) (2.3.2) is called the orthogonal group of b when b is symmetric and nondegener- ate,calledthesymplecticgroupof b when b is antisymmetric and nondegenerate.) (2.32)) Suppose that b is symmetric and nondegenerate.In the complexcasethere is a basis in which the matrix B = I, the identity matrix, and the corresponding group is the complexorthogonal group) (23.3) O(n;C) = {g E GL(n;<C) I tgg = I}.)
- 39. 22 2. SOME EXAMPLES) In the real case there is a basis in which B is the matrix Ipjq = (15 jg ), 0 302247 13 302247 n and p + q = n, and the corresponding groups are the real orthogonal groups) (23.4) 0(p,q) = {gE GL(n;lR) | tgIp,qg = Ipyq} Where Ip,q = (15 _0Iq)) of 342200234signature342200235 (p, q). They are called inde357254201nite orthogonal groups when pq a303251 0. If pq = 0, then O(p,q) is the ordinary orthogonal group O(n) = O(0,n) = O(n, 0), and the condition of (2.3.4) is the standard condition tgg = I.) Taking determinant on the de357254201ningconditions of (2.33) and (2.34) we see that it forces det(g)2 = 1. The additional condition det(g)= 1de357254201nes the special orthogonal groups SO(n,C) C O((n,(C)and SO(p,q) C O(p,q). In all cases the specialorthogonal group is a subgroup of index 2 in the orthogonal group. S O(n, (C) and SO(n) are connected, but S'O(p, q) has 2 components when pq 75 0.) Now suppose that b is antisymmetric and nondegenerate. Then 71 = dimp V is even, n = 2m, and there is a basis in which B is the matrix J = (_?m 15" The corresponding groups are the complex symplectic group) (2.3.5) Sp(m;(C) = {g E GL(n;<C) | tgJg = J} where J = (_(}m 15342200235). and the real symplectic group (23.6) Sp(m;R)= {ge GL(n;R) 5 guy = J} where J = (_9m 15302273).) The groups Sp(m, ]F) are connected. Note that S'p(m; R) simply consists of the real matricesin Sp(m; C).) In all cases, we have the Cartan involution 6 of G given by the restriction of the Cartan involution (2.1.5) of the general linear group. It has 357254201xed point sets) (2-3-7) {9 E 0(n;<C) I 9(9) = 9} = 0(n; C) (7 GL(n;R) = 0(9), (2-3-8) {9 6 0(p.q) I 0(9) = 9} = 0(1), 9) 0 0(9) g 0(9)X 0(9), (2-3-9) {9 6 Sp(m;<C) I 9(9) = 9} = Sp(m; C) 0 U(2m) 342200231302243 Sp(m),) (2-3-10) {9342202254Sp(m;R)|0(9)=9} = S;v(m;1R)3572542020(2m) % U(m),) respective maximal compact subgroupsof O(n; (C), O(p, q), Sp(m; (C) and S'p(m; R). Here Sp(m) will be de357254201ned later.) 2.4. Groups De357254201ned by Hermitian Forms) Let h be an hermitian form on V. This will mean that h : V X V 342200224342200224> F is linear in the 357254201rst variable, conjugate3422002241inear in the second variable, and satis357254201es h(v, u) = h(u, 11).This isthe usual (for mathematicians 342200224 not for physicists) role of the 357254201rst and second variables. It is slightly inconvenient in the quaternionic case, where conjugation reverses the order of products, but we meet that problem with the formulation (2.4.1) for the matrix expressions of hermitian forms.) Now, as for bilinear forms,a choice,6 = {e1, . . . , en} of basis of V gives a matrix H = (hm) = (h(ej,e302242)) such that, identifying a Vector v = 261% E V with the column vector 342200230(v1,. , . ,vn), we have) (2.4.1) h(u,v) = Z v_ih,-,3-uj = v*Hu, 11* = 342200230E. 1302247i,j302247n) 6
- 40. 2.4. GROUPS DEFINED BY HERMITIAN FORMS 23) The hermitian form now de357254201nes a group (2.42) U(V, h) = {gE GL(V) 1 h(gu,gv) = h(u,v) 1u,v E V} % {9 E GL(n;1F) lg*Hy = H}, 9* = 342200230@-) As mentioned above, there is a delicatepoint here in the quaternionic case, where it usesthe calculation) h(9ua!]U) = Z 91312112) hm" !Jj,mUm> I m) id342200231) : Z(g*)342202254,'ihi,jgj,m um = If m) i,j l,m) Each of the groups (2.4.2)isa subgroup of the general linear group de357254201ned by a system of quadratic equations in the real components of the matrix entries,thus is a closed subgroup of the generallinear group, soit is a locally compact topological group and a linear Lie group.) V has a basis that is orthonormal in the sense that H = Ipvq = (if _[}q ),) 0 302247 p 302247 n and p + q = m, the matrix we met while describing the real orthogonal groups. The corresponding groups are the (inde357254201nite) unitary groups over F,) (2-4-3) U(p, q;F) = {9E GL(n;F) I 9* my = Im}, 0 E p,q and :0 + q = 7302273- When h is de357254201nite, i.e. when pq = 0, these are the usual unitary groupsover F, U(n;lF) = U(n,0;lF) = U(0,n; This notation is standard, but unfortunately there are several standards.Another standard notation is to write (2-44) 0(1),q) for U (p,q;1R), U (P, 4) for U(:0,q;C), and 317(1), q) for U(:D,q;1HD,) reflecting the respective names orthogonal, unitary, and symplectic unitary, of 342200234signature342200235 (p, q), for these groups. In the de357254201nite case one has the standard notation) O(n) for U(n;lR), U(n) for U(n;(C), and S'p(n)for U('n;lHl) and one usually refers to thesegroups as the orthogonal, unitary and symplectic unitary groups. Thecorresponding (inde357254201nite) special unitary groups are the (2.4.5) SU(p,q; IF) = U(p, q;lF) 357254202 SL(n;lF) where 0 302247 1), q and p -1-q = n. In the real and complexcasesS'U(p,q;lF) = {g E U(p,q;lF) I det(g) = 1}. There is no distinction in the quaternion case: SU(p, q; H) = U(p,q; llil). Except in the case where F = R and n = 2, SU (19, q; IF) is the derived group of U(p, q;lF As before, when h is de357254201nite these are the usual special unitary groups over F, SU(n; F) = SU(n, 0;lF) = S'U(0, n;F), and one has the other standard notation (2-46) 30(1), c1) = 5U(;v,q;R), 571(1), <1) = 3U(p,q;<C), Sp(p,q) = 3U(P,q;IH1))
- 41. 24 2. SOME EXAMPLES) re357254202ectingthe respective names special orthogonal, special unitary, and sym- plectic unitary, of 342200234signature342200235 (p, q), for these groups. And of course in the de357254201nite case we have the notation) SO(n) for SU(n;]R), SU(n) for .S'U(n; (C), and Sp(n) for SU(n;lHI), calledthe special orthogonal,specialunitary and symplectic unitary groups.) Again the Cartan involutions of our groups are of the form (21.5), that is, 9(g) = (g*)'1. It has 357254201xed point set) (24-7) {9 E U(p,q;1F) I 9(9) = 9} = U(p,q;1F) F7 U(n;1F) % U(p;F) >< U(9;IF)) in the full unitary case, but is slightly more complicated in the special unitary case. Of course nothing new happens in the quaternionic setting, but) (2-4-8) {9 E SU(p,9;1F) 19(9) = 9} = 5U(p,9;1F) F) U(n;1F) ES(U(p;lF) >< U(q;lF)), l6342200230=R,(C.) The 357254201xed point set of 9 is a maximal compact subgroup in all cases.) One can alsoconsidergroups de357254201ned by skew342200224hermitian forms s on V. So s:V x V 342200224+ F is linear in the second Variable, conjugateslinear in the 357254201rst variable, and satis357254201es s(u, v) + s(v, u) = 0. Inthe real case, skew342200224hermitian is the same as antisymmetric, so We get the groups S'p(n;lR). In the complex case,a matrix S is skew342200224hermitian if and only if it is of the form 5' == /3422002243422002243422002241H where H is hermitian. In that case, g*Sg = S if and only if g*Hg = H, sowe get the groups U(p, q) and SU(p,q). But in the quaternionic case we get something new,) (24.9) SO*(V, s) 2 {g E GL(V) | s(gu,gv)= s(u,v) for all 14,11 6 V}. We can choosea basis of V in which s has matrix U342200235, and then this group is written (24.10) S'O*(2n) = {gE GL(n;lHI) | g.*Sg = S} where S = H342200235. We embed GL(n; H) 342200230342200224> GL(4n; R) by the usual map) 91 -92 -93 -94 92 91 -94 93 93 94 91 -92 94 -93 92 91) (24-11) 342200230P I 91 + 9'92 +993 + 1994 302273342200224>) That is induced from) U1 V U2 U3 U4) ll-ll342200224>lR4 by ul +ugi+u3j+u4k r342200224>342200231 Similarly GL(2n; (C) => GL(4n;lR) by 342200230I> : a + ib 302273342200224> (_342200234b One can check that) (2.4.12) II(S'0*(2n))= II(GL(n; 11-11))(W <I>(U(n, n)),) which is another standard way of describing SO*(2n).)
- 42. 2.5. DEGENERATE FORMS 25 2.5. Degenerate Forms Let b be a symmetric bilinear form on V, not necessarily nondegenerate. Thus the subspace (2.5.1) U = Ker(b) == {1} E V I b(v, V) = 0} C V) can be nonzero. In any case b inducesa nondegenerate symmetric bilinear form b on the quotient spaceV/U. LetW be any vector space complement to U in V, so V : U {B W. Then b is nondegenerate on W. Let O(V, b) denote the group of (2.32) and choose a basis H = {u1, . . . ,u,.;'w1, . . .,ws}of V that starts with a basis of U and 357254201nishes with a basis of W. Relative to 357254202 the matrix group O(V, b) is) (2.52) O(V, b) = (9 =) Let X = ]F342200231342200234"3 E Hom]p(l/V, U A quick computation with (2.5.2) shows that O(V,b) = X >4 (GL(U) >< O(l/V,b|WXW)), semidirect product for the action (3 2) ::cI342200224> aaslfl of GL(U) X 0(W,b|WXW) on X) (1 E and b E .) (2.53)) Next, let b be an antisymmetric bilinear form on V, not necessarily nondegen- erate. Again, the subspace) (2.5.4) U = Ker(b) = {U E V I b('v, V) = 0} C V) can be nonzero, and b induces a nondegenerate antisymmetric bilinear form b on the quotient spaceV/U . As before, let W be any vector space complement to U in V, so V = U EB W. Then b is nondegenerate on W. Let Sp(V, b) denote the group of (2.32) and choose a basis 3 = {u1, ...,u,;w1, ...,w3} of V that starts with a basis of U and 357254201nishes with a basis of W. Relative to ,8 the matrix group O(V, b) is) (26.5) Sp<V,b>=(g= (3 if)) Let X = IFTX342200231 E Hom[(.~(I/V, U As before, computation with (255) shows that Sp(V, b) = X >4 (GL(U) >< Sp(VV, b[WXW)), semidirect product for the action (3 E) : a: 302273342200224> axlfl of GL(U) >< Sp(W,b|WXW) on X) a E GL(U) and b E Sp(VV,b|WXW)}.) (25.6)) Now let h be an lF342200224hermitian form on V, not necessarily nondegenerate. Yet again, the subspace) (25.7) U = Ker(h) = {U E V | h(v, V) = 0} C V can be nonzero, and h inducesa nondegenerate lF342200224hermitian form H on the quotient space V/U. As before, let W be any vector space complement to U in V, so V = U 69 W. Then h is nondegenerate on W. Let U (V, h) denote the group of (2.42) and choose a basis, 357254202 = {u1, . . . ,u,.;'w1, . . . , ws} of V that starts with a basis of U and 357254201nishes with a basis of W. Relative to B the matrix group U(V, h) is) (2.58) U(V, h) = {g = (3 aeGL(U) and b e U(W, h|WxW)}.)
- 43. 26 2.SOME EXAMPLES) Let X = IFTXS 302247 Hom11.~(I/V, U Once again, computation with (2.5.8) shows that) U(V, h) = X )4 (GL(U) >< U(l/V, h|WXw)), semidirect product for) 2.5.9 ( ) the action (3 2) ::cH axlfl of GL(U) >< U(W,h|WxW) on X Finally, let s be a skew342200224hermitian form on a vector space V over lHl, not neces- sarily nondegenerate. Yet again, the subspace) (2.5.10) U= Ker(s) = {v E V I s(v, V) = 0} C V) can be nonzero, and s induces a nondegenerateskew342200224hermitian form 302247 on the quotient space V/ U. As before, let W be any vector space complement to U in V, so V = U 69 W. Then s is nondegenerate on W. Let SO*(V, s) denote the group of (2.4.9) and choose a basis 357254202 = {u1, . . . ,uT;w1, . . .,ws}of V that starts with a basis of U and 357254201nishes with a basis of W. Relative to (3 the matrix group SO*(V,s) is) (2.511) SO*(V,s)={g=<3) a E and b E S|WXW)} .) Let X = ll-ll342200231""3 % Hom]F(W, U). Once again, computation with (25.11) shows that) S0*(V, s) = X ><1 (GL(U) >< SO*(W', s|wXW)), semidirect product for) (2.5.12) th _ Q 0 _ _, , e action (0 b) . x 1342200224> axb of GL(U) X 5'0 (W,s|W,(W) on X) 2.6. Automorphism Groups of Algebras) Let A be a 357254201nite dimensional algebra over R or C, not necessarily associative. In other Words, the multiplication on A c011ld be any bilinear map A X A 342200224-> A. Then one has the automorphism group of A,) (2-6-1) Aut(A) = {9 6 G342200231L(3022534) l9(:vy) = g(rv)g(y) V :0. y 6 A}-) It is a closedsubgroup of GL(A) defined by some quadratic equations,so it is a topological group.) For example, if A is the associativel algebra Mn of n X n matrices over 11''342200230, then every automorphism is inner, so Aut(A) is the projective general linear group PGL(A) = GL(A)/lF*. If A is a semisimple Lie algebrathen Aut(A) has identity component that is the adjoint group of A.) Let (0) denote the algebra of octonions (Cayley division algebra) over R. It can be de357254201ned as the algebra with basis {(30,. ..,e7} over R with multiplication de357254201ned by (a) eoej = ej = ejeo for 1 302247 j 302247 7, (b) e? = 342200224e0 for 1 302247 j 302247 7, (c) ejek +ekej = 0 for 1 302247 j,k 302247 7 with j a303251 k, (d) 6162 = e3, 6365 = e5, e5e7= 61,(3164 = 65, e3e4 = 342200224342200224e7, 6564 = eg and e2e5 = e7, and (e)eachequation in (cl) remains true when the subscripts involved in it are cyclically permuted. This) 1Thealgebra A is called associative if its multiplication is associative, i.e. if (ab)c = a.(bc) in A.))
- 44. 2.6. AUTOMORPHISM GROUPS OF ALGEBRAS 27) multiplication table is summarized in the diagram) (2.6.2) (M3)) (3,5,6) (6,7, 1) (1, 4, 5) (3, 4, 7) (6,4,2) (2,5,7)) Then Aut(<U)) is the compact 14342200224dimensional simple Lie group G2. Elements of Im (U) = Spank {e1, . ..,e7} are calledpure imaginary, and Im (D) has a positive de357254201nite inner product given by: u - 11 is the eg-component of 342200224-uv. An element g E G'L(302256;lR) is an automorphism just when (say with g(ej) = e3) e0 = ea, e3422002311 and eg are orthonormal in 1m (1) and eg = e302247e'2, ea is a unit vector in Im C) that is orthogonal to e3422002311, eg and eg. Then cg, eg and e3422002317 are given by e3422002311e3572542011 = cg, ege357254201l = 342200224eg and ege357254201l = e302247. Thus elements of Aut(<O>) are speci357254201edby the choice of pure imaginary unit element e'1, pure imaginary unit element e3422002312 orthogonal to 6'1, and pure imaginary unit element 6; orthogonal to 6'1, eg and e'1eg.This is how one understands the group G2 = Aut(<U>).) A Jordan algebra over a commutative 357254201eld ll342200231 of characteristic 75 2 is a com- mutative algebra J over IF such that (a2b)a = a2(ba) for a,b E J (the Jordan identity). A standard construction: let A be an associative algebraover IF, let J be a subspaceof A that is stable under the composition a ob = %(ab -1- Em), and let J be viewed as an algebra with that composition. A Jordan algebra J is called special if it can be obtained in this way, exceptional if it cannot. In addition to the automorphism group, a Jordan algebra J hastwo larger groups of transforma342200224 tions, the structure group Str(J) and the reduced structure groupStr0(J They are de357254201ned as follows. If at E J then L(x) : J 342200224+ J denotes the left transla- tion, L(1')y= my. It de357254201nes the trace form T(a:,y) = trace L(:1r:y) on J. One says that J is semisimple if 7' is nondegenerate as a bilinear form. This is equivalent to J beingthe algebradirectsum of simple ideals. Suppose that J is semisimple. If g is a linear transformation of J let g342200231 denote its transpose relative to 7342200230. The 342200234quadratic representation342200235 of J is :1:v342200224342200224> P(.1;) := 2L(:r)2 342200224 L(x2). By de357254201nition the structure group Str(J) = {g E GL(J) | P(gx)= gP(:r)g342200231 for .342200231L342200230 E J}. Note that Str(J) contains the scalardilations st : so 302273342200224> tr, t E F)342200230, as a normal subgroup. The reduced structuregroup Stro (J) = Str(J) F1 SL(J) is a 357254201nite cover of the quotient of Str(J) by the scalardilations.) Fact: Aut(J) = {g E Str(J) |g(I) = I}.) A Jordan algebra J over R is calledformally real oreuclidean if it has the property :52+ y2 = 0 302242 :1: = 0 = y. These algebrasare important in analysis on homogeneous cones and in the geometry of complex bounded symmetric domains. They have the property that Aut(J) is a maximal compact subgroup of Strg(J They are given as follows.) First, J could be the specialreal simple Jordan algebra of n X n hermitian matrices over IF (= R, C or H) with composition as oy = %(a:y+'y:302243). Then Str(J) % GL(J)/{d:I} and Aut(J) E U(n;lF)/{:tI}, each acting on J by :l:g : a 342200224> gag*.)
- 45. 28 2. SOME EXAMPLES) Second, J could be the real simpleJordan algebraV (B R, where V is a real vector space of dimension n < oo with a positive de357254201nite symmetric bilinear form f. .7 has composition(u,a) 0 (1), b) = (av + bu, ab + f(u,v). liere Str0(._7) is the orthogonal group of signature (n, 1) associated to the form f ((u, a), (11,b)) = f (u,v) 342200224 ab and Aut(J) is the orthogonal group of (V, f). It is not obvious, but in this case J is special.) Finally, third, we have the exceptional simplereal JordanalgebraJ consisting of 3 x 3 hermitian matrices over the octonion division algebra (D). It has composition :v o y = %(acy + Then Str0(J) is the 78~dimensional simple Lie group E673, Whosemaximal compact subgroupisthe 52342200224dimensional compact simple Lie group F4, and that F4 is Aut(._7 We will see these groups later in connection with projective planes. (One can try this construction for the n X n hermitian matrices over (0),but that does not result in a Jordan algebrawhen n > 3.)) Summarizing, the simple formally real Jordan algebras and their automorphism and structuregroups are) .7 Str(..7) Aut(.7) Sym lR""" G'L(n; R) SO(n) Re (C"342200231342200230342200235 GL(n; CC) SU(n) Re lHI"X" G-'L(n; ll-ll) .S'p(n) IR342200235 GB R O(n, 1) O(n) Re 3022513X3 E67F4 X RX F4) where the hermitian and skew342200224hermitian parts of a square matrix m are given by) (2.64) Rem := %(m+m*) and Imm := %(m 342200224 m*).) (2.6.3)) (These are real and imaginary parts only in a rather generalizedsense.)See[F-K] for complete proofs.) 2.7. Spheres, Projective Spaces and Grassmannians) The underlying set of the projective space P(V) of a vector space V over IE342200230 is the set of all 1342200224dimensional linear subspaces of V. Thus it can be described in terms of compact groupsas a homogeneous space of a unitary group, or in terms of noncompact groups as a homogeneousspaceof a general linear group. If n = dirny V then (2.7.1) P(V) % U(n;lF)/U(1;lF) X U(n 342200224 1;lF), which can also be described as G'L(n;E4342200230)/{(3%) Ia e GL(1,]F), b e GL(n 342200224 1;]F) and .1342200230 e rm"-1)}.) For U(n;lF) is transitive on P(V) and the stabilizer of e1lF is U(1;lF) X U(n 342200224 1;lF), While GL(n; F) is of coursealsotransitive and the isotropy subgroup of GL(n;lF) at e1lF is the subgroup (call it described just above. Thus we have real, complexand quaternion n342200224space given by) (2.72) P"(]R) = O(n + 1)/0(1)>< O(n) = GL(n + 1;R)/Q,,+1(]R) (2.73) P"(<C) = U(n + 1)/11(1) x U(n) GL(n + 1;C)/Q,,+1(<C) (27-4) P"(H) = Sp(n + 1)/317(1)X 52001) = GL(n + 1;1H1)/Qn+1(1H1)))
- 46. 2.7.SPHERES, PROJECTIVE SPACES AND GRASSMANNIANS 29) For n > 1 the universal cover of P" (R) is the sphere S"with the covering given by .S'O(n + 1)/S'O(n) E342200231 0(n + 1)/O(n) 342200224> O(n + 1)/0(1) ><) The complex and quaternion projective spaces are simply connected.Thereis an- other locally compact projective space in the background, the octonion(orCayley) projective plane, (27.5) P2(o) = F4/Sp2'n(9) = Em,/Q3(o). HereE534 and F4 are the groups mentioned above in connection with the excep- tional simpleJordanalgebraJ, which is used to construct2 P2 ((0)), and Q3((D)) and Sp2'n(9) are certain subgroups.P2(302251) is simply connected. The projective lines in P" (R) are circles S1, in P"(C) are Riemann spheres S2, in P" (H) are 4342200224spheres S4, and in P2((U>)are 8342200224spheres S8.) More generally, the Grassmann manifold of k342200224dimensional linear subspaces of V is given by) Gk(V) E U(n;lF)/U(k;lF) >< U(n 342200224342200224 k;lF)) 2.7.6 ( ) E GL(n;lF)/ {(3 3;)| a e GL(k,lF342200230), b e GL(n 342200224 k;lF) and :10 e r342200231342200235<"342200224342200 For again U (n; IF ) is transitive on P(V) and the stabilizerof e1lF EB e2lF EB - - - EB ek]F' is U (k; IF342200230 ) X U (n 342200224 k; F), while GL(n; F) is of course also transitive and the isotropy subgroupof GL(n;lF) at e1lF EB e2lF G3 EB ekl357254201342200230 is the subgroup (call it Qk,,,(lF)) described just above. Thus we have real, complex and quaternion Grassmann manifoldsgiven by) (27-7) Gk,n(R) = 001)/0(k) X 001 ~ 1?) = GM342200235 + 1; R)/Qn+1(1R) (3-7-8) Gk,n(C) = U(n)/U (k) X U (71 - k) = GL(n + 1; C)/Qn+1(342200230C) (2-7-9) Gk,n(H) = Sp(n)/510(k) X 31001 - k) = GL0?302273 + 1;1H1)/Qn+1(1H1)) For k(n 342200224 k) > 1 the universal cover of Gk7n(lR) is the 2342200224sheeted cover by the) grassmannian C~302245k7n(lR) of oriented kaiimensional linear subspaces of an oriented R", and that covering is given by) SO(n)/S'0(k) >< S'O(n 342200224 k) 342200224> S'O(n)/{g E O(k) >< O(n 342200224 k) I det(g) = 1}. The complex and quaternion Grassmann manifolds are simply connected.) There is an important natural generalization of the real, complex and quater- nion Grassman manifolds, the compact riemannian symmetric spaces.SeeSection 11.2 for a quick description and Cartan342200231sclassi357254201cation. In differential geometry, in a certain sense 342200234most342200235 of the compact riemannian symmetric spaces are grass- mannians. By this I mean that if a theorem can be veri357254201ed, or a phenomenon can be understood, for the Grassmann manifolds mentioned above, then this often is enough of an indication to carry it through for compact riemannian symmetric spaces.) 2Here is a uniform construction of all four types of projective spaces P"3422002311(lF Let .7 denote the Jordan algebra of n X 17, hermitian matrices over F, with n = 3 in the case 17 = (0). An element e E J is idempotent if e2 = :3. An idempotent e E .7 is primitive if e 75 0 and 6 cannot be expressed in the form e1 + (:2 where the e,- are nonzero idempotents that are orthogonal in the sense e1e2 = 0. Then the set I(J) of primitive idempotents is a compact submanifold of .7 that is Stro(.7)342200224equivariantly diffeomorphjc to P"3422002301 See [F-K, Exercises 4 & 5, Chapter IV] and [F-K, Exercise 5, Chapter V].))
- 47. 2. SOME EXAMPLES) For noncompact symmetric spacesthere is something similar. Let h be the) hermjtian form on F" with matrix (I3 In0_k) Then U(k,n 342200224 k;lF) is its unitary) group. The orbitsof U(k,n 342200224 k;lF) on Gk,n(lF) are the various subsetson which h has equivalent restriction. The possibilities for that restriction are given by triples (p,q,u) where p is the number of +342200231s, q is the number of -342200231s, and u is the dimension ofthe kernel,of the restriction of h to our k-dimensionalsubspace.Thus p 302247 k,q 302247 71 342200224 k,u 3 min{k,n 342200224 kt}, andp+ q + u = k. The most important of these orbits, for riemannian geometry,is) (27.10) U(k,n 342200224 k; lF)([e1 / ~ - - / ek]) % U(k,n 342200224 k';lF)/U(k;lF) >< U(n 342200224- k;lF),) where [e1 / - - - / ek] = Span1g{e1, e2, . . .,ck}.The manifold (2.7.10) can be realized as the boundeddomain of k X (n 342200224342200224 k) matrices Z over IF such that I 342200224 ZZ* >> 0. The other orbits, at leastin the caseIF = C, are useful in complex function theory and in the unitary representation theory of the group U(k,n 342200224 lc;) 2.8. Complexi357254201cation of Real Groups) Certain of our groups are obtained from others by a processthat could be called complexi357254201cation. For example, GL(n;C) is the complexi357254201cation of GL(n;lR) and O(n; C) is the complexi357254201cation of O(n; R). This notion, however,goeswell beyond those obvious cases, and it has a number of geometric and analytic consequences. We will formalize it from the viewpoint of linear groups.) DEFINITION 2.8.1. Let G be a topological group.We say that another topolog- ical group GC is a complexi357254201cation of G if there exist n > 0 and homomorphisms 45: G 342200224+ GL(n;lR) and Lb : GC 342200224-> GI/(n; (C) such that) o ()5 is a homeomorphism of G onto 302242>(G) and 1/} is a homeomorphism of GC onto 1,b(GC), o qb(G) = {g E RM" 1 F(g) = 0 for all F E I} for some set I of real polynomial functions on Rnx", and 0 1/J(GC)= {gE C"342200231342200230342200235 | F(g) = 0 for all F E I} when we view the elements of I as polynomial functions on <C"X".) If GC is a complexi357254201cation of G then we also say that G is a real form of GC. 0) From now on, when we write GCand G, it is understood that Gc is a complex- i357254201cation of G.) We now make sure that the de357254201nition of complexi357254201cation is what we want, by verifying (2.82) GL(n;R)c = GL(n; (C)and S342200231L(n;lR)C = SL(n; (C).) For this, let 302242>(g) = 1,b(g) = <31/dgt(g)). Then q5(GL(n;lR))consistsof all real (71 + 1) X (n + 1) matrices m that satisfy the equations mn+1,j = Ofor 1 g j 302247 n, ml-,n+1 = 0 for 1 302247 7} 302247 n, and det(m) = 1. Note that 1/1(GL(n;C)) consists of all complex (n + 1)X (n + 1) matrices that satisfy the samesystem of equations. That proves (2.8.2) for GL. The sameargument, with q5(g) = 1/;(g) = g, provesit for SL.)
- 48. 2.8. COMPLEXIFICATION OF REAL GROUPS 31) Now let342200231s run through some cases that we met earlier. (2.8.3) GL(n;ll-ll)C = GL(2n;(C) and SL(n;lHl)C = SL(2n; (C). When we view H342200235 as a 4n342200224dimensional real vector space R4342200235, we obtain a map qb : GL(n;lHI) 342200224+ GL(4n;lR), and the scalar multiplications by the pure imaginary unit quaternions i, j and k are transformed to anticommuting linear transformations I,J,K E GL(4n;IR)of square 342200224I. Note that) 302242>(GL(n;1H1)) = {9 E GL(4n;R) I 91 = Ig,gJ = J9 and 9K = K9}-) When we View C2342200235 as R4" we obtain a map 1/; : GL(2n;(C) 342200224> GL(4n;]R), such that scalar multiplication by i is transformed to the samelineartransformation I E GL(4n;]R), and 1/1(GL(2n;(C))= {gE GL(4n;R) l gI = Ig}. Now we view the spaceof 4n x 471. real matrices that commute with I as a complex vector space V, with complex conjugation given by an 302273342200224342200224> J;2:J3422002351. So U = {cc 6 V 1 :rJ = Jae} is a real vector space with complexi357254201cation V. Making use of the map that realizes (2.82), we obtain the 357254201rst statement of (2.8.3). The second statement follows.) (2-84) 0(p, q)c = 0(1) + q;(C) and 30(1). q).; = 50(1) + <1; C)-) For the full orthogonal groups we use the system of equations on the matrix entries given by 342200230gI,,,qg 342200224 Ipyq = 0, and for the special orthogonal groups we use the additional equations that come from det(g) 342200224 1 = 0. Then (2.8.4) is immediate.) (2.8.5) S'p(n;lR)C = S'p(n; C). The proof of (2.8.5)is the sameas the proof of (2.84), with J in place of IN}.) (2-8-5) U(p,q)c = GL(:v+ M3) and 5U(p,q)c = SL(p+ q; C)- Write 2 E U(p, q) as 2 = :1: + iy with a:,y real, and de357254201ne 302242(z) = ( Z Then) my Z <z5(U(p,q)) = {9 6 0(2p, 291) I N = J9} Where J = (P15)~ Thus 302242(U(1>,q)) is the set of all 2n >< 2n rea.l matrices, n = p + q, that satisfy certain real polynomial equations. As affine variety or as differentiable manifold it has real dimension n2, so its complexi357254201cation has real dimension 2722. That complexi357254201cation still consists of matrices that commute with J, in other words of matrices in q$(GL(n; (C)),which) has real dimension 2n2. That proves (2.8.6).) (2-8-7) 511(1). q)c = Sp(p + q; C). The proof of (2.8.7) is the same as the proof of (2.8.6).) (2.8.8) S'0*(2n)C = S'O(2n;(C). The proof of (2.8.8) is the same as the proof of (2.8.6).) (2.8.9) GL(n;<C)c = GL(n;(C) X GL(n;(C) and S'L(n;C)C = SL(n;(C) >< SL(n;(C).)
- 49. 32 2. SOME EXAMPLES) De357254201ne g = (g 1/d357254202tw) e SL(n + 1;<C) for g e GL(n; C). De357254201ne 302242(g) = (3; 3) where 302247 = 3: 342200224|342200224 iy with :c,y real. Then q5(GL(n;(C))complexi357254201es to the group of all (fw 2342200231 where z and w are of the form (3 1/ dgtw) with g E GL(n;(C). Thus G'L(n; C)342200230: = GL(n; (C) X GL(n; C). The statement on SL follows.) (28.10) O(n;<C)C = O(n;(C) >< O(n;(C) and SO(n;(C)C = S'O(n;<C) >< SO(n;(C). The proof of (28.10) is essentiallythe sameasthe proof of (2.8.9).) (2.8.11) S'p(n;(C)C = Sp(n;(C) >< Sp(n; (C). The proof of (2.8.11)is essentially the sameasthe proof of (2.8.9). LEMMA 2.8.12. IfA is an algebra over R then Aut(.A)C = Aut(AC).) PROOF. Choose a basis of A over R; then Aut(.A) consists of all invertible real linear transformations of A that preserve the multiplication table (relative to that basis). Now both Aut(./1)):and Aut(./lg) consist of all invertible complex linear transformations of A: that preserve the multiplication table. D) 2.9. p342200224adic Groups) The p342200224adic and adele groups are related more with number theory than with harmonic or geometricanalysis,but we mention them because they are important examplesof locally compact topological groups.) A topological 357254201eld is a 357254201eld (in the sense of algebra) in which the 357254201eld operations are continuous. VVhen a topological 357254201eld is locally compact one obtains somelocally compact topological groups that are of importance in number theory. Formally,) DEFINITION 2.9.1. Let lF = (IF, +, X) be both a 357254201eld and a topological space, then F is a topological 357254201eld if) the additive structure (IF, +) and the topology form a commutative topological group,) the multiplicative structureIFX = lB342200230 {0} is a topological group, and) the map F)342200230 X IF 342200224> IF, given by (a, an) e342200224> am, is continuous. 0) Of course R, (C and H are locally compact topological 357254201elds. Any 357254201eld with the discrete topology is a locally compacttopological 357254201eld, but somehow that is not very interesting. There are, however, a number of other interesting ones, for example the p342200224adic number 357254201elds Q1, .) We recall the de357254201nition of Q1, . Let p be a prime number. The gradicvaluation on the rational number 357254201eld Q is given by: M1, = 0, and if 0 75 as E Q then lmlp = p'342200235 where at = p"u/ 12 in such a way that n, u and 11 are integers (note: n can be negative) with u and 21 not divisible by p. That givesthe metric dp(x,y) = (2-3/1,, on Q. Its completion (asa 357254201eld and as a metric space) is the locally compact 357254201eld)
- 50. 342200231..uzw.342200234A<i'&342200230R*2L*?302253X?.?302247%"}F) r :..302273:,; . .. . .1. ..) 2.10. HEISENBERG GROUPS 33) Q1, of p342200224adic numbers. Sometimes it is convenient to denoteR as Q00 . The point of this, for us in the present setting, is) PROPOSITION 2.9.2. Let l357254201342200230 be a nondiscrete locally compact 357254201eld of character- istic zero. Then T342200230 is a division algebra of 357254201nite dimension over a p342200224adic number 357254201elds Qp or over R = Q00. Conversely every such division algebra is a nondiscrete locally compact 357254201eld.) Nondiscrete locally compact 357254201eld of 357254201nite characteristic have an analogous de- scription. Let lF;,, denote the pure transcendental extension of degree1of the 357254201eld of p elements. Then the nondiscrete locally compact 357254201elds of characteristic p are just the division algebras of 357254201nite dimension over F}, .) We omit the proof of Proposition 2.9.2. That, and related results described below in Section 3.8, are found in Andr303251 Weil342200231s book 342200234Basic Number Theory.342200235) Let IF be a nondiscrete locally compact 357254201eld of characteristic zero. Then GL(n;lF') consists of the invertible elements of lF"X342200235. Any closed subgroup de- 357254201ned by polynomial equations on the matrix entries is a locally compact group. Our orthogonal groups, symplectic groups, and other linear Lie groups, were de- 357254201ned by equations With integer coe357254202icients, so they make sense over any 357254201eld, in particular over F. This gives a rich supplyof locally compact groups that are ex- tremely interesting in number theory. In Section 3.8, using the notions of Haar measure and the module of an automorphism, we will see how the integer matrices give maximal compact subgroups for linear Lie groupsover p342200224adic 357254201elds, and how the various gradic groups 357254201t together to form the adele groups.) 2.10. Heisenberg Groups) The 342200234Heisenberg groups342200235 are ubiquitous in mathematics, playing important rolesin number theory, harmonic analysis, and the theory of homogeneous riemann- ian manifolds. In Part 2 we will use them to illustrate an important construction of Mackey for unitary representations. They are in the background in Part 3 when we discuss uncertainty principles on commutative spaces. In Part 4 we will see how they enter into several of the basicconstructions of commutative spaces. For applications in those considerations of group structure and harmonic analysis we give the de357254201nitions in greater than usual generality.) Let IF denote a real division algebra R (real numbers),(C (complex numbers), ll-ll (quaternions) or <0 (octonions). Then IF has a standard decomposition IF = lR+I1nlF where ImlF, the space of pure imaginary elements of IE342200230, has real dimension 0, 1, 3 or 7. This is consistent with (2.64), and we View Re and Im as the projection lF~>lRandlF342200224>ImlF.) We View the space IF" of n342200224tuplesover F as a right vector space, so scalars act on the right and linear transformations act on the left. When p and q are non342200224negative integers, p + q = n, we have the hermitian vector space) IFM342200231 : IF" with hermitian form h(a:,y) : Zfxgy342200230 342200224 ::xp+302243@p+e)
- 51. 34 2. SOME EXAMPLES) and its unitary group U(p,q; In the octonion case one has to be careful: U(p,q;302251) = SO(p, q) >< Spin(7). We now have a group) Hp,q;]F : real vector space Im]F + lF342200235342200230q with group composition) (2101) (2, w)(z', w342200231) = (2 + 2' + Im h(w, w342200231), w + 11)342200231).) Groups Hp7q;]1: % Hp/,q/;F/ if and only if F = IF342200231 and (ii) p -1-q = p342200231 + q342200231. Finally, g(z,w) = (2, de357254201nes an action of the unitary group U(p, q; F) by automor- phisms on Hp1q;]5342200230. The semidirect product group G,,7q;1; = Hpg357254202p ><1 U(p,q;lF) has group composition) (2.102) (2,w,g)(z342200231, 111', g342200231) = (2 + 2' -1-Im h(w, g(w342200231)), w + g(w342200231), gg342200231). We will mostly be concerned with the case q = 0, leading to the groups (2.103) Hn;};.342200230 '.= Hn,0;][r and Gm357254201w 2: Gn;0;]p.) The usual Heisenberg group is H := Hmc. Sothe groups H,,,q;;p, and even the groups Hn;}F, form a slight generalization. Of coursethe HWER are just a real vector groups. The othersare 2342200224step nilpotent.) Occasionally one wants more of a generalization of the usual Heisenberg group.) Given an integer 3 g 1 we decompose IFSXS = Im IFSXS + Re F5342200234, direct sum of real vector spaces, where the projectionsare given by) 1) Imz = 302247(z 342200224 2*) and Rez 2 + z*) where 2* is conjugate transpose.) Given integers t, u 3 O, we have a hermitian map H I F342200235(342200230+342200234) >< F342200235(t+342200234) 342200224> F3342200234 given by 7'l((13422002301,rv2),($11,312)) = $131? - $22/3 where a:1,y1 E IFS342200235 and .7cg,y2 E FSX342200234. We write lE3422002305>342200230("342200234) for lF3422002303"(342200234*'342200234) with the hermitian map H. Putting these together we have a (very) generalized Heisenberg group Hs,t7u;]302247342200230 : real vector space Im IFS342200234 + lF5X("342200234) with group composition) (2.104) (2, w)(z', w342200231) = (2 + z342200231 + Im'H(w, w342200231), w + 10').) Since the (t,u) pertainsto rows in F5"(t*342200234) the automorphism action of U(t,u;lF) on H3yt,u;]14342200230 is g(z, w) = (z, wg*). The semidirect product group G_;342200231t,302273u_;]]4342200230 = H57t342200231u;]p342200230 >4 U (t, n; F) has group composition) (2.105) (2, w, g)(z342200231, w',g') 2 (2 + 2' + ImH(w,w342200231g*), w + u/g*,gg342200231).) In particular Glsytyu357254202p342200230 has center Im]F3"5, H1yt,u;];.~ 342200234=342200231 Ht342200231u;]F and G3422002311,tyu;][302253342200230 % Gt,u;]F, and Hsytwp is commutativejust when either 3 =: 1 with F = R, or t = u = 0.) The groups H satyudlr and Crlsvtj-u;]F appear in the study of maximal parabolic subgroupsof unitary groups U(p, q; See [W9], [W10] and [W11]. We will meet a few of them in Chapter 13 when we look at the classi357254201cation of commutative nilmanifolds, and in Chapter 14 we will see that harmonic analysison thosegroups, based on a theory of square integrable representations, is particularly elegant.)
- 52. 273 73m302273.. 273.. 273.,g..~) CHAPTER 3) Integration and Convolution) Let G be a locally compact topological group. In this chapter weconstruct the basictoolfor analysis on G: a left translation invariant Radon measurel known as left Haar measure MG on G. As always, there is a certain emphasisonexamples. We then study properties of the associated left Haar integral fa f(g) dp,G These include structure and properties of the LebesguespacesLP(G),the convolution product, the modular function, the group algebraL1(G), and the measure algebra M eas(G). Finally, we use the notion of modulus of an automorphism to construct adele groups of linear algebraic groups.) 3.1. De357254201nition and Examples) The invariant integral on G will be left invariant, in other words invariant under left translations. That means fa f(a:g)dug (g) = fa f(g) d,uG In other words, the associated measure is left invariant in the sense that ac (xA) = as (A) whenever an E G and A C G is a Borelset. Formally,) DEFINITION 3.1.1. A left Haar measure on G is a left invariant Radon mea- sure on G that is not identically zero. A left Haar integral is the left invariant integral de357254201ned by a left Haar measure. <)) Before going on we look at some examples.) EXAMPLE 3.1.2. Let G be the additive group R342200235. Then Lebesgue measure is a left Haar measureon G, and the Lebesgue integral is a left Haar integral on G. 0) EXAMPLE 3.1.3. The circle group G = {zE (C I = 1} has left Haar measure 357254201d357254202 where z = 62"342200234). It has total mass 1 and the corresponding invariant integral) is f I342200224> .217 02" f(e342200234342200231)d6. <> EXAMPLE 3.1.4. Let G be a discretegroup. Thencounting measure is a left Haar measure on G and f 302273342200224+ Zgea f (g) is a left Haar integral. 0) EXAMPLE 3.1.5. Let G = GL(n;]R) and let dx denote the volume element of Lebesgue measure on lR"X342200234. The left action g : av r342200224> gm scales Lebesgue measure on each column of :1: by {det(g)|, so d(ga:) = |det(g)|"d:r. Thus ldet(x)|""d:c is a left Haar measure on GL(n;lR). This same argument shows that |det(x)|3422002302"d;zt is a left Haar measure on GL(n;(C).) 1Let 8 denote the a342200224a.lgebra of Borel sets in G. It is the cr342200224algebra generated by the open sets. A R/adon measure on G is a positive inner342200224regular Borel measure ,u : B 342200224> [0, 00] that is 357254201nite on compact sets. Inner regularity means that ,u(A) = sup{p(F) I F C A and F compact} for all A E B.) 35)
- 53. 36 3. INTEGRATION AND CONVOLUTION) EXAMPLE 3.1.6. Let G be the group of all n X n invertible upper triangular real matrices) 901,1 961,2 501,3 --- 961,7; 0 (E212 .TI23422002313 . . . $2)342200235 (31.7) It : 0 0 56373 . .. 333,7, 0 0 O ... $7,) The left a.ction g : x 302273342200224> gm scales Lebesgue measure on the 357254201rst column of 1: by |g171[,scalesLebesgue measureonthe secondcolumn of :c by |g1,1g2,2f, etc., 357254201nally scales Lebesgue measure on the last columnlof m by |g1Y1g2,2...gn,,,|, so d(g:c) = |9i342200231,19Z302247 ---93022422z3422002241,n34220022419n,nld$- Thus |9l342200230,19302243342200235,E '--9i3422002241,n34220022419n,nl_1d3342200231/' 13 3 left Haar measure on the group of all n X n invertible upper triangular real matrices..This same argument shows that fg357254202lggigl . .. g3,_1,,,_1gn,n|3422002342dw is a left Haar measure on the group of all n X n invertible upper triangular complex matricesand that |g{34220023071g3022433022471...g3,_1,,,_1g,,,,,|3422002304d;c is a left Haar measure on the group of all n x n invertible upper triangular quaternion matrices. 0) EXAMPLE 3.1.8. Let G be as in Example 3.1.6 except that (317) is a block form matrix. In other words, G342200231 consists of the 7" X 7342200230 real matrices (317) Where mm- is 7342200234, X rj and r1 + ---+ r,, = 7". Then, combining the considerations of the last two examples,) det g1 1 342200234+'342200235+T" det ggg 342200231"2+342200230342200235+T" ...det ,,_1n_1 T342200235"+342200230"+342200231" det gnn 342200231" 342200230Ida: , , 9 , ,) is a left Haar measure on G. As before, in the complex version of this, left Haar measure is) |det(.q1,1)342200234+"'+T" d342202254t(92.2)342200235+342200230342200234+T" ---d342202254t(9n3022531.n3422002241)3422002353422002351+"342200230+T" d342202254t(9n,n)T"|T2d342202254342202254~) These examples are, in fact, all the 342200234parabolic subgroups342200235 of the real and complex general linear groups. ()) EXAMPLE 3.1.9. Let G and H be locally compact groups with respective left Haar measures pa and ,u.H. Then ,uG >< pH is a left Haar measure on the product group G x H. Semidirect products in general are a bit more subtle, and they342200231ll have to wait for Proposition 3.3.10, which requires some discussion of the module of an automorphism. 0) 3.1.10. One can also considerright Haar measures and right Haar integrals:) x/(Aw) = A/(A) and fa f(9w)dx/(9) = fa f(9)du342200231(9). The map 9 H 9* inter342200224 changes left and right Haar measures.) 3.2. Existenceand Uniqueness ofHaar Measure) In this Section we prove Haar342200231s famous result2) 2This result has a long history, and the theorem as stated is due to Andr303251 Weil. For more details, seethe notes to 30224715 in the book of Hewitt and Ross. Here are a few of the high points. Many special cases were worked out by various mathematicians, often quite explicitly. Peter and Weyl proved the existence for compact Liegroups, and von Neumann proved existence,uniqueness and unimodularity for arbitrary compact topological groups. A. Haar proved existence in 1933 for separable locally compact groups. Later A. Weil reformulated it in terms of linear functionals and)
- 54. 3.2. EXISTENCE AND UNIQUENESSOF HAAR MEASURE 37) THEOREM 3.2.1. Let G be a locally compact topological group. Then G has a left Haar integral, and any two are proportional.) 3.2.2. Existence is easy for Lie groups using differential forms. If G is a Lie group of real dimension n, if {51,...{,,} is a basis of the (Lie algebra of) left translation invariant real vector 357254201elds on G, and if {(421, . . .,w"}is the dual basis of left translation invariant linear differential forms on G, then (2 2 wl / - - - / cu342200235 is a left translation invariant volume element of G, and the Riemann integral against 0 de357254201nes a left Haar integral on G.) 3.2.3. Let Gc(G)denotethe spaceof compactly supported continuous functions f : G 342200224342200224> (C with norm f|f||OO = supgea We will construct our integral as a (continuous) positive linear functional I on GC(G). Then we apply the Riesz342200224 Markov Theorem, which says that there is a unique Radon measure ,aG on G such) that 10342200235) = fa f(g) duG(9) for all f 6 00(0)-) PROOF. If I : f 302273342200224> I (f) is a left Haar integral, then its restriction I + to (32.4) C'C+(G) = {f E Cc(G) | f(G) C R and f(g) 2 0 for all g E G} satis357254201es (i) I+ : G: (G) 342200224> lR+ = {r 6 IR | r 2 0,} and is not identically zero, (325) (ii)I+(rf) = rI+(f)for r E lR+ and f E Gc+(G), )I+(f1 + f2) = I+(f1) + I+(f2) 30224301342200235 .f1vf2 E C:(G)aa11d (iv) I+(342202254mf) : I+(f) for 3:E G and f E C:(G)) (iii) Conversely, if I + satis357254201es (32.5), then it extends by linearity to a nontrivial positive linear functional I : GC(G)342200224> C, and that gives a left Haar integral. Sothe proof of existence comes down to the construction of an I + that satis357254201es (3.2.5).) Existence. Suppose f, h E C342200231: (G) with h 75 0. Then there is a non342200224empty open set U C G and a constant c > 0such that h(g) > c for g E U. The supportSupp (f) is compact, so it is covered by a 357254201nite union U133342200235 x,-U of left translates of U.) Now [342202254302242,(h)](a:,-g) = h(g) implies f 302247 Z?:1(|[f|lm/c)Zx, We have exhibited) (3.26) n > 0 integer,'y,- 2 0, {$1, . . . ,x,,} C G such that f g Zn:*y,342202254,,,(h). i=1 De357254201ne a ratio (32.7) (f 2 h) = inf | f satis357254201es (32.6) for some {$1, . . .,ac,,}C G}. 2'21 Notice that f g ||h|[oO(f : h) and that (f : h) > 0 iff a303251 0.) Ifk7303251Other1(f:k)302247(f:h)(h:k). Iff7E0then(h:k)302247(h:f)(f:k). Combine these: if f 72$ 0 then 1/(h : 302247 : : k) 302247 (f 2 h). In other words,) extended the result to arbitrary locally Compact groups. After that, Kakutani noted that Haar342200231s) construction is valid for arbitrary locally compact groups. The proof here is due to H. Cartan. It has the feature that uniqueness comes along with the existence.)
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- 56. Macedonia, and when Alexander invaded Asia Ptolemy attended him. After Alexander’s death Ptolemy obtained the government of Egypt, where he gained the esteem of the people by acts of kindness. He made himself master of Phœnicia and Syria, and rendered assistance to the people of Rhodes against their enemies, for which he received the name of Soter. He laid the foundation of a library, which became the most celebrated in the world. He died in his eighty-fourth year, about 284 years B.C. He was succeeded by his son, Ptolemy Philadelphus, who showed himself to be a worthy successor of his father. His palace was an asylum for learned men, and he greatly increased the library his father had founded. Ptolemy Third succeeded his father Philadelphus on the Egyptian throne. He conquered Syria and Cilicia, and returned home laden with spoils. He was, like his predecessors, a patron of learning and the arts. Ptolemy Fourth, called Philopater, succeeded to the throne, his reign being marked by acts of cruelty and oppression. He died in his thirty-seventh year, after a reign of seventeen years, 204 years B.C. Numerous members of this celebrated family in succession occupied the throne, not, however, approaching to the greatness of the founders of the family. Ptolemæ´us. A celebrated geographer and astronomer in the reign of Adrian and Antoninus. He was a native of Alexandria, or, as some say, of Pelusium. In his system of the world, designated the Ptolemaic system, he places the earth in the center of the universe, which was generally received as correct till it was confuted by Copernicus. Pyr´rhus. A famous king of Epirus, son of Æacides and Phthia. He is celebrated for his military talents; and not only his friends, but his enemies, have been warm in extolling him. He is said to have had no superior in the art of war. He wrote several books on encampments and the ways of training an army. He fought against the Romans with much valor, and they passed encomiums on his great military skill. He was killed in an attack on Argos, by a tile thrown on his head from a housetop. Pyr´rhus. See Neoptolemus. Pythag´oras. A celebrated philosopher born at Samos. He first made himself known in Greece at the Olympic games, where, when he was eighteen years old, he obtained the prize for wrestling. He also distinguished himself by his discoveries in geometry, astronomy, and mathematics. He was the first who supported the doctrine of
- 57. metempsychosis, or transmigration of the soul into different bodies. He believed that the universe was created from a shapeless mass of passive matter by the hands of a powerful Being, who was the mover and soul of the world, and of whose substance the souls of mankind were a portion. The time and place of death of this great philosopher are unknown, but some suppose that he died at Metapontum about 497 years B.C. Py´thon. A celebrated serpent sprung from the mud and stagnated waters which remained on the surface of the earth after the deluge of Deucalion. Apollo killed the monster. Quintilia´nus, Mar´cus Fa´bius, a celebrated rhetorician, born in Spain. He opened a school of rhetoric at Rome, and was the first who obtained a salary from the State as a public teacher. He died A.D. 95. Quin´tus Cur´tius Ru´fus. A Latin historian supposed to have lived in the reign of Vespasian. He wrote a history of the reign of Alexander the Great. This work is admired for the elegance of its diction. Regil´lus. A small lake in Latium, famous as being the scene of a great Roman victory, which forms the subject of a fine poem by Lord Macaulay, called “The Battle of the Lake Regillus,” included in his “Lays of Ancient Rome.” Reg´ulus, M. Attil´ius. A consul during the first Punic war. He reduced Brundusium, and in his second consulship he captured a great portion of the Carthaginian fleet. After further successes he was taken prisoner by the Carthaginians, who put him to death with refined tortures. Rhadaman´thus. A son of Jupiter and Europa. He reigned in the Cyclades, where his rule was characterized by marked justice and impartiality. Rom´ulus. According to tradition the founder of Rome. He was a son of Mars and Ilia, and was twin brother of Remus. The twins were thrown into the Tiber, but were saved and suckled by a she-wolf till they were found by Fautulus, a shepherd, who brought them up. Disputes arising between the brothers in reference to the building of the city, Romulus caused Remus to be slain. Ros´cius. A celebrated Roman actor. He died about 60 years B.C. Ru´bicon. A small river in Italy. By crossing it, and thus transgressing the boundaries of his province, Cæsar declared war against the senate and
- 58. Pompey. “Passing the Rubicon” has become a proverbial expression, indicating an irrevocable step taken in any weighty matter. Sa´cra, Vi´a. An important street in Rome, where a treaty of peace was made between Romulus and Tatius. Sal´amis. An island of Attica celebrated for a battle fought there between the fleets of the Greeks and the Persians, in which the latter suffered defeat. Sallus´tius Cris´pus. A celebrated Latin historian. He wrote a history of the Catalinian conspiracy, and died thirty-five years before the Christian era. Sanchoni´athon. A Phœnician historian born at Berytus, or, as some say, at Tyre. He lived a few years before the Trojan war; and wrote on the antiquities of Phœnicia. Sa´por. A king of Persia, who succeeded to the throne about the 238th year of the Christian era. He wished to increase his dominions by conquest, but was defeated by Odenatus, who defeated his army with great slaughter. He was assassinated A.D. 273. Sa´por. The second king of Persia of that name. He fought against the Romans, and obtained several victories over them. Died A.D. 380. Sap´pho, celebrated for her beauty and poetical talents, was born at Lesbos about 600 years before Christ. She became enamored with Phaon, a youth of Mitylene, but he not reciprocating her passion, she threw herself into the sea from the rock of Leucadia. Sardanapa´lus. The last king of Assyria, celebrated for his luxury and indolence. His effeminacy induced his subjects to conspire against him with success, on which he set fire to his palace and perished in the flames, B.C. 820. Satur´nus. The son of Cœlus, or Uranus, by Terra. It was customary to offer human victims on his altars till this custom was abolished by Hercules. He is generally represented as an old man bent with age, and holding a scythe in his right hand. Sat´yri. Demigods whose origin is unknown. They had the feet and legs of a goat, their body bearing the human form. Scævola, Mu´tius, surnamed Cordus, was famous for his courage. He attempted to assassinate Porsenna, but was seized; and to show his
- 59. fortitude when confronted with Porsenna, he thrust his hand into the fire, on which the king pardoned him. Scip´io. The name of a celebrated family at Rome, the most conspicuous of which was Publius Cornelius, afterwards called Africanus. He was the son of Publius Scipio, and commanded an army against the Carthaginians. After obtaining some victories he encountered Hannibal at the famous battle of Zama, in which he obtained a decisive victory. He died about 184 years before Christ, in his forty-eighth year. Scip´io, Lu´cius Corne´lius, surnamed Asiaticus, accompanied his brother Africanus in his expedition in Africa. He was made consul A.U.C. 562, and sent to attack Antiochus, king of Syria, whom he completely routed. He was accused of receiving bribes of Antiochus, and was condemned to pay large fines which reduced him to poverty. Scip´io, P. Æmilia´nus. Called Africanus the younger. He finished the war with Carthage, the total submission of which occurred B.C. 147. The captive city was set on fire, and Scipio was said to have wept bitterly over the melancholy scene. On his return to Rome he was appointed to conclude the war against Numantia, the fall of which soon occurred, and Scipio had Numantinus added to his name. He was found dead in his bed, and was presumed to have been strangled, B.C. 128. Sem´ele. A daughter of Cadmus and Hermione, the daughter of Mars and Venus. She was the mother of Bacchus. After death she was made immortal under the name of Thyone. Semir´amis. A celebrated queen of Assyria, who married the governor of Nineveh, and at his death she became the wife of King Ninus. She caused many improvements to be effected in her kingdom, as well as distinguishing herself as a warrior. She is supposed to have lived 1965 years before the Christian era. Sen´eca, L. Annæ´us, at an early period of his life, was distinguished by his talents. He became preceptor to Nero, in which capacity he gained general approbation. The tyrant, however, determined to put him to death, and he chose to have his veins opened in a hot bath, but death not ensuing, he swallowed poison, and was eventually suffocated by the soldiers who were in attendance. This occurred in his fifty-third year, and in the sixty- fifth of the Christian era. His works, which were numerous, were chiefly on moral subjects.
- 60. Sera´pis. One of the Egyptian deities, supposed to be the same as Osiris. He had a magnificent temple at Memphis, another at Alexandria, and a third at Canopus. Sesos´tris. A celebrated king of Egypt, who lived long prior to the Trojan war. He was ambitious of military fame, and achieved many conquests. On his return from his victories he employed himself in encouraging the fine arts. He destroyed himself after a reign of forty-four years. Seve´rus, Lu´cius Septim´ius. A Roman emperor, born in Africa, noticeable for his ambition. He invaded Britain, and built a wall in the north as a check to the incursions of the Caledonians. He died at York in the 211th year of the Christian era. Sile´nus. A demigod, who is represented generally as a fat old man riding on an ass, with flowers crowning his head. Sil´ius Ital´icus, C. A Latin poet who retired from the bar to consecrate his time to study. He imitated Virgil, but with little success. His poetry, however, is commended for its purity. Simon´ides. A celebrated poet of Cos, who lived B.C. 538 years. He wrote elegies, epigrams, and dramatic pieces, esteemed for their beauty. Sire´nes. The Sirens. They lured to destruction those who listened to their songs. When Ulysses sailed past their island he stopped the ears of his companions with wax, and had himself tied to the mast of his ship. Thus he passed with safety, and the Sirens, disappointed of their prey, drowned themselves. Sis´yphus. Son of Æolus and Enaretta. After death he was condemned, in the infernal regions, to roll a stone to the summit of a hill, which always rolled back, and rendered his punishment eternal. Soc´rates. The most celebrated philosopher of antiquity, born near Athens, whose virtues rendered his name venerated. His independence of spirit created for him many enemies, and he was accused of making innovations in the religion of the Greeks. He was condemned to death by drinking hemlock, and expired a few moments after imbibing the poison, in his seventieth year, B.C. 400. His wife was Xanthippe, remarkable for her shrewish disposition, for which her name has become proverbial. So´lon, one of the wise men of Greece, was born at Salamis, and educated at Athens. After traveling over Greece he returned, and was elected archon and sovereign legislator, in which capacity he effected numerous reforms
- 61. in the state, binding the Athenians by a solemn oath to observe the laws he enacted for one hundred years. After this he visited Egypt, and on returning to Athens after ten years’ absence, he found most of his regulations disregarded by his countrymen. On this he retired to Cyprus, where he died in his eightieth year, 558 years before the Christian era. Som´nus, son of Nox and Erebus, was one of the infernal deities and presided over sleep. Soph´ocles. A celebrated tragic poet of Athens. He was distinguished also as a statesman, and exercised the office of archon with credit and honor. He wrote for the stage, and obtained the poetical prize on twenty different occasions. He was the rival of Euripides for public applause, each having his admirers. He died at the age of ninety-one, 406 years before Christ. Sophonis´ba. A daughter of Hasdrubal, the Carthaginian, celebrated for her beauty. She married Syphax, prince of Numidia, and when he was conquered by the Romans she became a captive to their ally, the Numidian general Masinissa, whom she married. This displeased the Romans, and Scipio ordered Masinissa to separate from Sophonisba, and she, urged to this by Masinissa, took poison, about 203 years before Christ. Soz´omen. A historian who died 450 A.D. He wrote an important work on ecclesiastical history. Sphinx. A monster, having the head and breasts of a woman, the body of a dog, the tail of a serpent, the wings of a bird, and the paws of a lion. The Sphinx was sent into the neighborhood of Thebes by Juno, where she propounded enigmas, devouring those who were unable to solve them. One of the riddles proposed was—What animal walked on four legs in the morning, two at noon, and three in the evening? Œdipus solved it, giving as the meaning—A man, who when an infant crawled on his hands and feet, walking erect in manhood, and in the evening of life supporting himself with a stick. On hearing the solution the Sphinx destroyed herself. Stagi´ra. A town on the borders of Macedonia, where Aristotle was born; hence he is called the Stagirite. Sta´tius, P. Papin´us. A poet, born at Naples in the reign of Domitian. He was the author of two epic poems, the Thebais, in twelve books, and the Achilleis, in two books. Sten´tor. One of the Greeks who went to the Trojan war. He was noted for the loudness of his voice, and from him the term “stentorian” has become
- 62. proverbial. Sto´ici. A celebrated sect of philosophers founded by Zeno. They preferred virtue to all other things, and regarded everything opposed to it as an evil. Stra´bo. A celebrated geographer, born at Amasia, on the borders of Cappadocia. He flourished in the age of Augustus. His work on geography consists of seventeen books, and is admired for its purity of diction. Styx. A celebrated river of the infernal regions. The gods held it in such veneration that they always swore by it, the oath being inviolable. Sueto´nius, C. Tranquil´lus. A Latin historian who became secretary to Adrian. His best known work is his Lives of the Cæsars. Sul´la. See Sylla. Syb´aris. A town on the bay of Tarentum. Its inhabitants were distinguished by their love of ease and pleasure, hence the term “Sybarite.” Syl´la (or Sulla), L. Corne´lius. A celebrated Roman, of a noble family, who rendered himself conspicuous in military affairs; and became antagonistic to Marius. In the zenith of his power he was guilty of the greatest cruelty. His character is that of an ambitious, tyrannical, and resolute commander. He died about seventy years before Christ, aged sixty. Sy´phax. A king of the Masæsyllii in Numidia, who married Sophonisba, the daughter of Hasdrubal. He joined the Carthaginians against the Romans, and was taken by Scipio as a prisoner to Rome, where he died in prison. Tac´itus, C. Corne´lius. A celebrated Latin historian, born in the reign of Nero. Of all his works the “Annals” is the most extensive and complete. His style is marked by force, precision, and dignity, and his Latin is remarkable for being pure and classical. Tac´itus, M. Clau´dius. A Roman, elected emperor by the Senate when he was seventy years of age. He displayed military vigor, and as a ruler was a pattern of economy and moderation. He died in the 276th year of the Christian era. Tan´talus. A king of Lydia, father of Niobe and Pelops. He is represented by the poets as being, in the infernal regions, placed in a pool of water which flowed from him whenever he attempted to drink, thus causing him to suffer perpetual thirst; hence the origin of the term “tantalizing.”
- 63. Tarquin´ius Pris´cus, the fifth king of Rome, was son of Demaratus, a native of Greece. He exhibited military talents in the victories he gained over the Sabines. During peace he devoted attention to the improvement of the capital. He was assassinated in his eightieth year, 578 years B.C. Tarquin´ius Super´bus. He ascended the throne of Rome after Servius Tullius, whom he murdered, and married his daughter Tullia. His reign was characterized by tyranny, and eventually he was expelled from Rome, surviving his disgrace for fourteen years, and dying in his ninetieth year. Tar´tarus. One of the regions of hell, where, according to Virgil, the souls of those who were exceptionally depraved were punished. Telem´achus. Son of Penelope and Ulysses. At the end of the Trojan war he went in search of his father, whom, with the aid of Minerva, he found. Aided by Ulysses he delivered his mother from the suitors that beset her. Tem´pe. A valley in Thessaly through which the river Peneus flows into the Ægean. It is described by the poets as one of the most delightful places in the world. Teren´tius Pub´lius (Terence). A native of Africa, celebrated for the comedies he wrote. He was twenty-five years old when his first play was produced on the Roman stage. Terence is admired for the purity of his language and the elegance of his diction. He is supposed to have been drowned in a storm about 159 B.C. Te´reus. A king of Thrace who married Procne, daughter of Pandion, king of Athens. He aided Pandion in a war against Megara. Terpsich´ore. One of the Muses, daughter of Jupiter and Mnemosyne. She presided over dancing. Tertullia´nus, J. Septim´ius Flor´ens. A celebrated Christian writer of Carthage, who lived A.D. 196. He was originally a Pagan, but embraced Christianity, of which faith he became an able advocate. Tha´is. A celebrated woman of Athens, who accompanied Alexander the Great in his Asiatic conquests. Tha´les. One of the seven wise men of Greece, born at Miletus in Ionia. His discoveries in astronomy were great, and he was the first who calculated with accuracy a solar eclipse. He died about 548 years before the Christian era. Thali´a. One of the Muses. She presided over festivals and comic poetry.
- 64. Themis´tocles. A celebrated general born at Athens. When Xerxes invaded Greece, Themistocles was intrusted with the care of the fleet, and at the famous battle of Salamis, fought B.C. 480, the Greeks, instigated to fight by Themistocles, obtained a complete victory over the formidable navy of Xerxes. He died in the sixty-fifth year of his age, having, as some writers affirm, poisoned himself by drinking bull’s blood. Theoc´ritus. A Greek poet who lived at Syracuse in Sicily, 282 B.C. He distinguished himself by his poetical compositions, of which some are extant. Theodo´sius, Fla´vius. A Roman emperor surnamed Magnus from the greatness of his exploits. The first years of his reign were marked by conquests over the Barbarians. In his private character Theodosius was an example of temperance. He died in his sixtieth year, A.D. 395, after a reign of sixteen years. Theodo´sius Second became emperor of the Western Roman empire at an early age. His territories were invaded by the Persians, but on his appearance at the head of a large force they fled, losing a great number of their army in the Euphrates. Theodosius was a warm advocate of the Christian religion. He died, aged forty-nine, A.D. 450. Theophras´tus. A native of Lesbos. Diogenes enumerates the titles of more than 200 treatises which he wrote. He died in his 107th year, B.C. 288. Thermop´ylæ. A narrow Pass leading from Thessaly into Locris and Phocis, celebrated for a battle fought there, B.C. 480, between Xerxes and the Greeks, in which three hundred Spartans, commanded by Leonidas, resisted for three successive days an enormous Persian army. Thersi´tes. A deformed Greek, in the Trojan war, who indulged in ridicule against Ulysses and others. Achilles killed him because he laughed at his grief for the death of Penthesilea. Shakspeare, who introduced Thersites in his play of “Troilus and Cressida,” describes him as “a deformed and scurrilous Grecian.” The´seus, king of Athens and son of Ægeus by Æthra, was one of the most celebrated heroes of antiquity. He caught the bull of Marathon and sacrificed it to Minerva. After this he went to Crete amongst the seven youths sent yearly by the Athenians to be devoured by the Minotaur, and by the aid of Ariadne he slew the monster. He ascended his father’s throne B.C. 1235. Pirithous, king of the Lapithæ, invaded his territories, but the
- 65. two became firm friends. They descended into the infernal regions to carry off Proserpine, but their intentions were frustrated by Pluto. After remaining for some time in the infernal regions, Theseus returned to his kingdom to find the throne filled by an usurper, whom he vainly tried to eject. He retired to Scyros, where he was killed by a fall from a precipice. Thes´pis. A Greek poet of Attica, supposed to be the inventor of tragedy, B.C. 536. He went from place to place upon a cart, on which he gave performances. Hence the term “Thespians” as applied to wandering actors. The´tis. A sea deity, daughter of Nereus and Doris. She married Peleus, their son being Achilles, whom she plunged into the Styx, thus rendering him invulnerable in every part of his body except the heel by which she held him. This´be. A beautiful girl of Babylon, beloved by Pyramus. Thrasybu´lus. A famous general of Athens, who, with the help of a few associates, expelled the Thirty Tyrants, B.C. 401. He was sent with a powerful fleet to recover the Athenian power on the coast of Asia, and after gaining many advantages, was killed by the people of Aspendos. Thucid´ydes. A celebrated Greek historian born at Athens. He wrote a history of the events connected with the Peloponnesian war. He died at Athens in his eightieth year, B.C. 391. Tibe´rius, Clau´dius Ne´ro. A Roman emperor descended from the Claudii. In his early years he entertained the people with magnificent shows and gladiatorial exhibitions, which made him popular. At a later period of his life he retired to the island of Capreæ, where he indulged in vice and debauchery. He died aged seventy-eight, after a reign of twenty- two years. Tibul´lus, Au´lus Al´bius. A Roman knight celebrated for his poetical compositions. His favorite occupation was writing love poems. Four books of elegies are all that remain of his compositions. Timo´leon. A celebrated Corinthian, son of Timodemus and Demariste. When the Syracusans, oppressed with the tyranny of Dionysius the Younger, solicited aid from the Corinthians, Timoleon sailed for Syracuse with a small fleet. He was successful in the expedition, and Dionysius gave himself up as a prisoner. Timoleon died at Syracuse, amidst universal regret.
- 66. Ti´mons. A native of Athens, called the Misanthrope from his aversion to mankind. He is the hero of Shakspeare’s play of “Timon of Athens” in which his churlish character is powerfully delineated. Timo´theus. A famous musician in the time of Alexander the Great. Dryden names him in his well-known ode, “Alexander’s Feast.” Tire´sias. A celebrated prophet of Thebes. Juno deprived him of sight, and, to recompense him for the loss, Jupiter bestowed on him the gift of prophecy. Tisiph´one. One of the Furies, daughter of Nox and Acheron. Tita´nes. The Titans. A name given to the gigantic sons of Cœlus and Terra. The most conspicuous of them are Saturn, Hyperion, Oceanus, Iapetus, Cottus, and Briareus. Ti´tus Vespasia´nus. Son of Vespasian and Flavia Domitilla, known by his valor, particularly at the siege of Jerusalem. He had been distinguished for profligacy, but on assuming the purple, he became a model of virtue. His death, which occasioned great lamentations, occurred A.D. 81, in the forty-first year of his age. Traja´nus, M. Ul´pius Crini´tus. A Roman emperor born at Ithaca. His services to the empire recommended him to the notice of the emperor Nerva, who adopted him as his son, and invested him with the purple. The actions of Trajan were those of a benevolent prince. He died in Cilicia, in August A.D. 117, in his sixty-fourth year, and his ashes were taken to Rome and deposited under a stately column which he had erected. Tribu´ni Ple´bi. Magistrates at Rome created in the year, U.C. 261. The office of Tribune to the people was one of the first steps which led to more honorable employments. Triptol´emus. Son of Oceanus and Terra, or, according to some authorities, son of Celeus, king of Attica, and Neæra. He was in his youth cured of a severe illness by Ceres, with whom he became a great favorite. She taught him agriculture, and gave him her chariot drawn by dragons, in which he traveled over the earth, distributing corn to the inhabitants. Tri´ton. A sea deity, son of Neptune and Amphitrite. He was very powerful, and could calm the sea and abate storms at his pleasure. Trium´viri. Three magistrates appointed to govern the Roman state with absolute power.
- 67. Tul´lus Hostil´ius succeeded Numa as king of Rome. He was of a warlike disposition, and distinguished himself by his expedition against the people of Alba, whom he conquered. Typhœ´us, or Ty´phon. A famous giant, son of Tartarus and Terra, who had a hundred heads. He made war against the gods, and was put to flight by the thunderbolts of Jupiter, who crushed him under Mount Ætna. Tyrtæ´us. A Greek elegiac poet born in Attica. Of his compositions none are extant except a few fragments. Ulys´ses. The famous king of Ithaca, son of Anticlea and Laertes (or, according to some, of Sisyphus). He married Penelope, daughter of Icarius, on which his father resigned to him the crown. He went to the Trojan war, where he was esteemed for his sagacity. On the conclusion of the war he embarked for Greece, but was exposed to numerous misfortunes on his journey. In his wanderings, he, with some of his companions, was seized by the Cyclops, Polyphemus, from whom he made his escape. Afterwards he was thrown on the island of Æea, where he was exposed to the wiles of the enchantress Circe. Eventually he was restored to his own country, after an absence of twenty years. The adventures of Ulysses on his return from the Trojan war form the subject of Homer’s Odyssey. Ura´nia. One of the Muses, daughter of Jupiter and Mnemosyne. She presided over astronomy. Valentinia´nus the First. Son of Gratian, raised to the throne by his merit and valor. He obtained victories over the Barbarians in Gaul and in Africa, and punished the Quadi with severity. He broke a blood-vessel and died, A.D. 375. Immediately after his death, his son, Valentinian the Second, was proclaimed emperor. He was robbed of his throne by Maximus, but regained it by the aid of Theodosius, emperor of the East. He was strangled by one of his officers. He was remarkable for benevolence and clemency. The third Valentinian was made emperor in his youth, and on coming to maturer age he disgraced himself by violence and oppression. He was murdered A.D. 454. Valeria´nus, Pub´lius Licin´ius. A celebrated Roman emperor, who, on ascending the throne, lost the virtues he had previously possessed. He made his son Gallienus his colleague in the empire. He made war against
- 68. the Goths and Scythians. He was defeated in battle and made prisoner by Tapor, king of Persia, who put him to death by torture. Var´ro. A Latin author, celebrated for his great learning. He wrote no less than five hundred volumes, but all his works are lost except a treatise De Re Rusticâ, and another De Linguâ Latinâ He died B.C. 28, in his eighty- eighth year. Ve´nus. One of the most celebrated deities of the ancients; the goddess of beauty, and mother of love. She sprang from the foam of the sea, and was carried to heaven, where all the gods admired her beauty. Jupiter gave her in marriage to Vulcan, but she intrigued with some of the gods, and, notably, with Mars, their offspring being Hermione, Cupid, and Anteros. She became enamored of Adonis, which caused her to abandon Olympus. Her contest for the golden apple, which she gained against her opponents Juno and Minerva, is a prominent episode in mythology. She had numerous names applied to her, conspicuous amongst which may be named Anadyomene, under which cognomen she is distinguished by the picture, representing her as rising from the ocean, by Apelles. She was known under the Grecian name of Aphrodite. Vespasia´nus Ti´tus Fla´vius. A Roman emperor of obscure descent. He began the siege of Jerusalem, which was continued by his son Titus. He died A.D. 79, in his seventieth year. Ves´ta. A goddess, daughter of Rhea and Saturn. The Palladium, a celebrated statue of Pallas, was supposed to be preserved within her sanctuary, where a fire was kept continually burning. Vesta´les. The Vestals, priestesses consecrated to the service of Vesta. They were required to be of good families and free from blemish and deformity. One of their chief duties was to see that the sacred fire of Vesta was not extinguished. Virgil´ius, Pub´lius Ma´ro, called the prince of the Latin poets, was born at Andes, near Mantua, about seventy years before Christ. He went to Rome, where he formed an acquaintance with Mæcenas, and recommended himself to Augustus. His Bucolics were written in about three years, and subsequently he commenced the Georgics, which is considered one of the most perfect of all Latin compositions. The Æneid is supposed to have been undertaken at the request of Augustus. Virgil died in his fifty-first year B.C. 19.
- 69. Virgin´ia. Daughter of the centurion L. Virginius. She was slain by her father to save her from the violence of the decemvir, Appius Claudius. Virgin´ius. A valiant Roman, father of Virginia. (See Virginia.) The story of Virginius and his ill-fated daughter is the subject of the well-known tragedy of “Virginius,” one of the early productions of J. Sheridan Knowles. It is rarely performed in the present day. Vulca´nus. The god who presided over fire, and who was the patron of those who worked in iron. According to Homer, he was the son of Jupiter and Juno, and was so deformed, that at his birth his mother threw him into the sea, where he remained nine years; but other writers differ from this opinion. He married Venus at the instigation of Jupiter. He is known by the name of Mulciber. The Cyclopes were his attendants, and with them he forged the thunderbolts of Jupiter. Xanthip´pe or Xantip´pe. The wife of Socrates, remarkable for her ill- humor and fretful disposition. She was a constant torment to her husband, and on one occasion, after bitterly reviling him, she emptied a vessel of dirty water on him, on which the philosopher coolly remarked, “After thunder rain generally falls.” Xenoc´rates. An ancient philosopher born at Calcedonia, and educated in the school of Plato, whose friendship he gained. Died B.C. 314. Xen´ophon. A celebrated Athenian, son of Gryllus, famous as a general, philosopher, and historian. He joined Cyrus the Younger in an expedition against Artaxerxes, king of Persia, and after the decisive battle of Cunaxa, in which Cyrus was defeated and killed, the skill and bravery of Xenophon became conspicuous. He had to direct an army of ten thousand Greeks, who were now more than six hundred leagues from home, and in a country surrounded by an active enemy. He rose superior to all difficulties till the celebrated “Retreat of the Ten Thousand” was effected; the Greeks returning home after a march of two hundred and fifteen days. Xenophon employed his pen in describing the expedition of Cyrus, in his work the “Anabasis.” He also wrote the “Cyropædia,” “Memorabilia,” “Hellenica,” etc. He died at Corinth in his ninetieth year, about 360 years before the Christian era. Xer´xes succeeded his father Darius on the throne of Persia. He entered Greece with an immense army, which was checked at Thermopylæ by the
- 70. valor of three hundred Spartans under king Leonidas, who, for three successive days, successfully opposed the enormous forces of Xerxes, and were at last slaughtered. From this period the fortunes of Xerxes waned. His fleet being defeated at Salamis, and mortified with ill-success, he hastened to Persia, where he gave himself up to debauchery, and was murdered in the twenty-first year of his reign, about 464 years before the Christian era. Za´ma. A town of Numidia, celebrated as the scene of the victory of Scipio over Hannibal, B.C. 202. Ze´no, a celebrated philosopher, the founder of the sect of Stoics, was born at Citium in Cyprus. He opened a school in Athens, and soon became noticed by the great and learned. His life was devoted to sobriety and moderation. He died at the age of ninety-eight, B.C. 264. Ze´no. A philosopher of Elea or Velia, in Italy. He was the disciple, or, according to some, the adopted son of Parmenides. Being tortured to cause him to reveal his confederates in a plot he had engaged in, he bit off his tongue that he might not betray his friends. Zeno´bia. A celebrated princess of Palmyra, the wife of Odenatus. After her husband’s death, the Roman emperor Aurelian declared war against her. She took the field with seven hundred thousand men, and though at first successful, she was eventually conquered. Aurelian, when she became his prisoner, treated her with great humanity and consideration. She was admired for her literary talents as well as her military abilities. Zeux´is. A celebrated painter born at Heraclea. He flourished 468 years before the Christian era. He painted some grapes so naturally that the birds came to peck them on the canvas; but he was disgusted with the picture, because the man painted as carrying the grapes was not natural enough to frighten the birds. Zo´ilus. A sophist and grammarian of Amphipolis, B.C. 259. He became known by his severe criticisms on the works of Isocrates and Homer. Zoroas´ter. A king of Bactria, supposed to have lived in the age of Ninus, king of Assyria, some time before the Trojan war. He rendered himself known by his deep researches in philosophy. He admitted no visible object of devotion except fire, which he considered the proper emblem of a Supreme Being. He was respected by his subjects and contemporaries for
- 71. his abilities as a monarch, a lawgiver, and a philosopher, and though many of his doctrines may be deemed puerile, he had many disciples. The religion of the Parsees of the present day was founded by Zoroaster. Zos´imus. A Greek historian who lived about the year 410 of the Christian era. He wrote a history of some of the Roman emperors, which is characterized by graceful diction, but he indulges in malevolent and vituperative attacks on the Christians in his History of Constantine.
- 72. A LIST OF COMMON ABBREVIATIONS OF WORDS USED IN WRITING AND PRINTING. A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y. A 1, first class. a or aa (Gr. ana), in med., of each the same quantity. A.B., Bachelor of Arts. A.D. (L. anno Domini), in the year of our Lord. ad lib., or ad libit. (L. ad libitum), at pleasure. Æ., Æt. (L. ætatis), of age; aged. A.M. (L. artium magister), Master of Arts. A.M. (L. ante meridiem), before noon. A.M. (L. anno mundi), in the year of the world. anon., anonymous. A.R.A., Associate of the Royal Academy. A.R.S.A., Associate of the Royal Scottish Academy. A.R.S.S. (L. antiquariorum regiæ societatis socius), Fellow of the Royal Society of Antiquaries. AS., Anglo-Saxon. A.U.C. (L. anno urbis conditæ, or anno ab urbe conditâ), in the year of, or from the building of the city, viz., Rome. B.A., Bachelor of Arts. Bart. or Bt., Baronet. B.C., before Christ. B.C.L., Bachelor of Civil Law. B.D., Bachelor of Divinity.
- 73. B.LL., also LL.B., Bachelor of Laws. B.Sc., Bachelor of Science. B.S.L., Botanical Society of London. C. (L. centum), a hundred. Cantab. (L. Cantabrigiensis), of Cambridge. Cantuar., Canterbury. cap. (L. caput, the head), chapter; cap., capital; cap., a capital letter; caps., capital letters. C.B., Companion of the Bath. C.E., Civil Engineer. cent. (L. centum), a hundred. cf. (L. confer), compare. chap., chapter. con. (L. contra), against; in opposition. cos., cosine. cres., crescendo. crim. con., criminal conversation; adultery. ct., cent; also (L. centum), a hundred. curt., current—that is, in this period of time, as month, year, or century. cwt. (c. for L. centum, a hundred; wt. for Eng. weight), a hundred-weight. D.C. (It. da capo), in music, again; from the beginning. D.C.L., Doctor of Civil or Canon Law. D.D. (L. divinitatis doctor), Doctor of Divinity. D.G. (L. Dei gratiâ), by the grace of God; (L. Deo gratias), thanks to God. do. or Do., the same. doz., dozen. Dr., doctor; debtor. D.S. (It. dal segno), from the sign. D.Sc., Doctor of Science. Dunelm., Durham. D.V. (L. Deo volente), God willing.
- 74. dwt. (L. denarius, a silver coin, a penny; and first and last letters of Eng. weight), pennyweight. Ebor. (L. Eborăcum), York. E.C., Established Church. Ed., editor. E.E., errors excepted. e.g., (L. exempli gratiâ, for the sake of example), for example; for instance. E.I., East Indies; East India. E.I.C., East India Company; E.I.C.S., East India Company’s Service. E. long., east longitude. E.N.E., east-north-east. E.S.E., east-south-east. Esq. or Esqr., Esquire. etc. (L. et cætera), &c.; and others; and so forth. et seq. (L. et sequentia), and the following. ex., example; exception; ex, “out of,” as, a cargo ex Maria. exch., exchequer; exchange. Exon. (L. Exonia), Exeter. f., feminine; farthing or farthings; foot or feet. Fahr., Fahrenheit. far., farriery; farthing. F.A.S., Fellow of the Society of Arts. F.A.S.E., Fellow of the Antiquarian Society, Edinburgh. F.B.S.E., Fellow of the Botanical Society of Edinburgh. F.C., Free Church. fcp., foolscap. F.D. (L. fidei defensor), Defender of the Faith. F.E.I.S., Fellow of the Educational Institute of Scotland. F.E.S., Fellow of the Entomological Society; Fellow of the Ethnological Society. F.G.S., Fellow of the Geological Society.
- 75. F.H.S., Fellow of the Horticultural Society. Fl., Flemish; Florida; florin. F.L.S., Fellow of the Linnæan Society. F.M., field-marshal. fo., fol., folio. F.P., fire-plug. F.P.S., Fellow of the Philological Society. Fr., France; French. F.R.A.S., Fellow of the Royal Astronomical Society. F.R.C.P., Fellow of the Royal College of Preceptors, or of Physicians. F.R.C.P.E., Fellow of the Royal College of Physicians, Edinburgh. F.R.C.S., Fellow of the Royal College of Surgeons. F.R.C.S.E., Fellow of the Royal College of Surgeons, Edinburgh. F.R.C.S.I., Fellow of the Royal College of Surgeons, Ireland. F.R.C.S.L., Fellow of the Royal College of Surgeons, London. F.R.G.S., Fellow of the Royal Geographical Society. F.R.S., Fellow of the Royal Society. F.R.S.E., Fellow of the Royal Society, Edinburgh. F.R.S.L., Fellow of the Royal Society of Literature. F.S.A., Fellow of the Society of Arts, or of Antiquaries: F.S.A., Scot., an F.S.A. of Scotland. ft., foot or feet. F.T.C.D., Fellow of Trinity College, Dublin. F.Z.A., Fellow of the Zoological Academy. G.A., General Assembly. G.C.B., Knight Grand Cross of the Bath. G.P.O., General Post-office. gtt. (L. gutta or guttæ), a drop or drops. H.B.C., Hudson Bay Company. H.E.I.C., Honorable East India Company. H.G., Horse Guards.
- 76. hhd., hogshead; hogsheads. H.I.H., His (or Her) Imperial Highness. H.M.S., Her (or His) Majesty’s steamer, ship, or service. H.R.H., His (or Her) Royal Highness. H.S.S. (L. historiæ societatis socius), Fellow of the Historical Society. ib., ibid. (L. ibidem), in the same place. id. (L. idem), the same. i. e. (L. id est), that is. I.H.S. (L. Iesus Hominum Salvator), Jesus the Saviour of Men. incog. (L. incognito), unknown. in lim. (L. in limine), at the outset. in loc. (L. in loco), in its place. inst., instant—that is, the present month. in trans. (L. in transitu), on the passage. I.O.U., three letters being identical in sound with the three words “I owe you,”—written as a simple acknowledgment for money lent, followed by sum and signature. Ir., Ireland; Irish. i.q. (L. idem quod), the same as. J.P., Justice of the Peace. K.C.B., Knight Commander of the Bath (Great Britain). K.G., Knight of the Garter (Great Britain). K.G.C., Knight of the Grand Cross (Great Britain). K.G.C.B., Knight of the Grand Cross of the Bath (Great Britain). Knt., knight. K.P., Knight of St. Patrick (Ireland). Kt. or Knt., knight. K.T., Knight of the Thistle (Scotland). K.S.E., Knight of the Star of the East. L. or lb. (L. libra), a pound in weight.
- 77. lat., latitude, N. or S. lb.—see L. leg. (It. legato), smoothly. L.G., Life Guards. lib. (L. liber), a book. Linn., Linnæus; Linnæan. LL.B., (L. legum, of laws, and baccalaureus, bachelor), Bachelor of Laws, an academic title. LL.D., (L. legum, of laws, and doctor, doctor), Doctor of Laws, an academic title, higher than LL.B. long., longitude, E. or W. L.S.D., or £ s. d. (said to be from L. libra, a balance, a pound in weight; solidus, a coin of the value of 25 denarii, subsequently only a half of that value; and denarius, a silver coin worth about 8½d. Eng.), pounds, shillings, pence—that is, in any written statement of money, L. is put over pounds, S. over shillings, and D. over pence; in printing, £ for L. is put before the sum, as £15, s. and d. in single letter, after, as 4s. 6d. M. (L. mille), a thousand. M.A. (L. magister artium), Master of Arts, an academic title. M.C.S., Madras Civil Service. M.D., (L. medicinæ, of medicine, doctor, doctor), Doctor of Medicine. M.E., Mining Engineer. Mdlle. (F. Mademoiselle), Miss. Mme. (F. Madame), Madam. Mons. (F. Monsieur), Mr.; Sir. M.P., Member of Parliament. M.P.S., Member of the Philological Society; Member of the Pharmaceutical Society. M.R.A.S., Member of the Royal Asiatic Society; Member of the Royal Academy of Science. M.R.C.P., Member of the Royal College of Preceptors, or of Physicians. M.R.C.S., Member of the Royal College of Surgeons.
- 78. M.R.G.S., Member of the Royal Geographical Society. MS., manuscript; MSS., manuscripts. Mus. B., Bachelor of Music; Mus. D., Doctor of Music. N.B., North British; North Britain, that is Scotland; New Brunswick; (L. nota, note, bene, well), note well, or take notice. N.E., north-east; New England. N.N.E., north-north-east. N.N.W., north-north-west. non obst. (L. non, not, obstante, standing over against, withstanding), notwithstanding. non pros. (L. non, not, prosequitur, he follows after, he prosecutes), he does not prosecute—applied to a judgment entered against a plaintiff who does not appear. non seq. (L. non, not, sequitur, it follows), it does not follow. N.P., notary public. N.S., new style; Nova Scotia. N.T., New Testament. N.W., north-west. ob. (L. obiit), he died. obs., obsolete. O.S., old style. Oxon. (L. Oxonia), Oxford. oz., ounce. p., page; pp., pages. P.C., Privy Council or Councillor. P.D. or Ph.D., Doctor of Philosophy. per an. (L. per annum), by the year. per cent. (L. per, by; centum, a hundred,) by the hundred. pinx., pxt. (L. pinxit), he or she painted it. P.M., postmaster; (L. post meridiem), afternoon. P.M.G., postmaster-general.
- 79. P.O., post-office; P.O.O., Post-Office Order. pp., pages. P.P., parish-priest. P.P.C., (F. pour prendre congé, to take leave), put on calling cards to intimate leave-taking. pr. (L. per, by), by the. pres., also preses, prĕs´-ĕs; president. prof., professor. pro tem. (L. pro tempore), for the time being. prox. (L. proximo), next; of the next month. P.S., (L. post scriptum), postscript. p.t., post-town. pxt. (L. pinxit), he or she painted it. Q. or Qu., question; query. Q.C., Queen’s Counsel; Queen’s College. q.e. (L. quod est), which is. Q E.D. (L. quod erat demonstrandum), which was to be demonstrated. Q.E.F. (L. quod erat faciendum), which was to be done. Q.E.I. (L. quod erat inveniendum), which was to be found out. q.l. (L. quantum libet), as much as you please. Q.M.G., quartermaster-general. qr., quarter; quire: qrs., quarters. qt., quart: qts., quarts. q.v. (L. quod vide), which see. R., L. rex, king; regina, queen. R., L. recipe, take. R.A., Royal Academy, or Academician; Royal Artillery; Rear-Admiral; Right Ascension. R.C., Roman Catholic. Ref. Ch., Reformed Church. Reg. Prof., Regius Professor.
- 80. R.I.P. (L.), requiescat in pace. R.Rev., right reverend. R.S.A., Royal Society of Antiquaries; Royal Scottish Academy. R.S.S. (L. regiæ societatis socius), Fellow of the Royal Society. Rt., Right. S., south. S.A., South America; South Africa; South Australia. Sarum, Salisbury. S.A.S. (L. societatis antiquariorum socius), Fellow of the Society of Antiquaries. s. caps., small capital letters. sc. or sculp. (L. sculpsit), he or she engraved it. s. or scil. (L. scilicet), to wit; namely. scr., scruple. sculp. or sculpt. (L. sculpsit), he or she engraved it. S.E., south-east. sec., secretary; second. Sep. or Sept., Septuagint; also LXX. seq. (L. sequentes or sequentia), the following; the next. S.G., solicitor-general. S.H.S. (societatis historiæ socius), Fellow of the Historical Society. S.J., Society of Jesus. S.L., solicitor-at-law. Sol.-Gen., solicitor-general. S.P.C.K., Society for Promoting Christian Knowledge. S.P.G., Society for the Propagation of the Gospel. sq., square: sq. ft., square feet: sq. in., square inches: sq. m., square miles: sq. yds., square yards. S.R.I. (L. sacrum Romanum imperium), the Holy Roman Empire. s.s., steamship. S.S.E., south-south-east.
- 81. S.S.W., south-south-west. S.T.P. (L. sacræ theologia professor), Professor of Theology. super., superfine. supp., supplement. S.W., south-west. syn., synonym; synonymous. T.O., turn over. tr. or trs., transpose. U.C., Upper Canada; (L. urbe condita, the founding of the city), the year of Rome. univ., university. U.P., United Presbyterian. U.S., United States. v.g. (L. verbi gratiâ), for example. vid. (L. vide), see. viz. (a corruption of L. videlicet), namely; to wit. vol., volume: vols., volumes. V.P., vice-president. vul., vulgate. W., west; western. Winton, Winchester. W. long., west longitude. W.M.S., Wesleyan Missionary Society. W.N.W., west-north-west. W.S.W., west-south-west. wt., weight. X. or Xt., Christ: Xm. or Xmas., Christmas: Xn. or Xtian., Christian. yd., yard: yds., yards. THE END.
- 82. 1883.—1883. G. W. Carleton & Co. NEW BOOKS AND NEW EDITIONS, RECENTLY ISSUED BY G. W. Carleton & Co., Publishers, Madison Square, New York. The Publishers, on receipt of price, send any book on this Catalogue by mail, postage free. All handsomely bound in cloth, with gilt backs suitable for libraries. Mary J. Holmes' Works. Tempest and Sunshine $1 50 English Orphans 1 50 Homestead on the Hillside 1 50 'Lena Rivers 1 50 Meadow Brook 1 50 Dora Deane 1 50 Cousin Maude 1 50 Marian Grey 1 50 Edith Lyle 1 50 Daisy Thornton 1 50 Chateau D´Or (New) 1 50 Darkness and Daylight 1 50 Hugh Worthington 1 50 Cameron Pride 1 50 Rose Mather 1 50 Ethelyn's Mistake 1 50 Millbank 1 50 Edna Browning 1 50 West Lawn 1 50 Mildred 1 50 Forrest House 1 50 Madeline (New) 1 50
- 83. Marion Harland's Works. Alone $1 50 Hidden Path 1 50 Moss Side 1 50 Nemesis 1 50 Miriam 1 50 At Last 1 50 Helen Gardner 1 50 True as Steel (New)1 50 Sunnybank 1 50 Husbands and Homes 1 50 Ruby's Husband 1 50 Phemie's Temptation 1 50 The Empty Heart 1 50 Jessamine 1 50 From My Youth Up 1 50 My Little Love 1 50 Charles Dickens—15 Vols.—"Carleton's Edition." Pickwick and Catalogue $1 50 Dombey and Son 1 50 Bleak House 1 50 Martin Chuzzlewit 1 50 Barnaby Rudge—Edwin Drood. 1 50 Child's. England—Miscellaneous 1 50 Christmas Books—Two Cities 1 50 David Copperfield 1 50 Nicholas Nickleby 1 50 Little Dorrit. 1 50 Our Mutual Friend 1 50 Curiosity Shop—Miscellaneous 1 50 Sketches by Boz—Hard Times 1 50 Great Expectations—Italy 1 50 Oliver Twist—Uncommercial 1 50
- 84. Sets of Dickens' Complete Works, in 15 vols.—[elegant half calf bindings] 50 00 Augusta J. Evans' Novels. Beulah $1 75 Macaria 1 75 Ines 1 75 St. Elmo $2 00 Vashti 2 00 Infelice (New) 2 00 May Agnes Fleming's Novels. Guy Earlscourt's Wife $1 50 A Wonderful Woman 1 50 A Terrible Secret 1 50 A Mad Marriage 1 50 Norine's Revenge 1 50 One Night's Mystery 1 50 Kate Danton 1 50 Silent and True 1 50 Heir of Charlton 1 50 Carried by Storm 1 50 Lost for a Woman 1 50 A Wife's Tragedy 1 50 A Changed Heart 1 50 Pride and Passion 1 50 Sharing Her Crime (New) 1 50 Allan Pinkerton's Works. Expressmen and Detectives $1 50 Mollie Maguires and Detectives 1 50 Somnambulists and Detectives 1 50 Claude Melnotte and Detectives 1 50 Criminal Reminiscences, etc. 1 50 Rail-Road Forger, etc. 1 50 Bank Robbers and Detectives 1 50
- 85. Gypsies and Detectives 1 50 Spiritualists and Detectives 1 50 Model Town and Detectives 1 50 Strikers, Communists, etc. 1 50 Mississippi Outlaws, etc. 1 50 Bucholz and Detectives 1 50 Bertha Clay's Novels. Thrown on the World $1 50 A Bitter Atonement 1 50 Love Works Wonders 1 50 Evelyn's Folly 1 50 Under a Shadow (New) 1 50 A Woman's Temptation 1 50 Repented at Leisure 1 50 Between Two Loves 1 50 Lady Damer's Secret 1 50 A Struggle for a Ring (New) 1 50 "New York Weekly" Series. Brownie's Triumph—Sheldon$1 50 The Forsaken Bride. do.1 50 Earle Wayne's Nobility. do. 1 50 A New Book. do.1 50 His Other Wife—Ashleigh 1 50 Curse of Everleigh—Pierce 1 50 Peerless Cathleen—Agnew 1 50 Faithful Margaret—Ashmore 1 50 Nick Whiffles—Robinson 1 50 Grinder Papers—Dallas 1 50 Lady Leonora—Conklin 1 50 Miriam Coles Harris' Novels. Rutledge $1 50 Frank Warrington 1 50 Louie's Last Term, St. Mary's 1 50
- 86. A Perfect Adonis 1 50 Missy (New) 1 50 The Sutherlands 1 50 St. Philips 1 50 Round Hearts for Children 1 50 Richard Vandermarck 1 50 Happy-Go-Lucky 1 50 A. S. Roe's Select Stories. True to the Last $1 50 The Star and the Cloud 1 50 How Could He Help it? 1 50 A Long Look Ahead 1 50 I've Been Thinking 1 50 To Love and to be Loved 1 50 Julie P. Smith's Novels. Widow Goldsmith's Daughter$1 50 Chris and Otho 1 50 Ten Old Maids 1 50 His Young Wife 1 50 Lucy (New) 1 50 The Widower 1 50 The Married Belle 1 50 Courting and Farming 1 50 Kiss and be Friends 1 50 Artemas Ward. Complete Comic Writings—With Biography, Portrait and 50 illustrations $1 50 The Game of Whist. Pole on Whist—The English standard work. With the "Portland Rules" $75 Victor Hugo's Great Novel.
- 87. Les Miserables—Translated from the French. The only complete edition $1 50 Mrs. Hill's Cook Book. Mrs. A. P. Hill's New Southern Cookery Book, and domestic receipts $2 00 Carleton's Popular Quotations. Carleton's New Hand-Book—Familiar Quotations, with their authorship $1 50 Carleton's Classical Dictionary—Condensed Mythology for popular use 75 Celia E. Gardner's Novels. Stolen Waters. (In verse)$1 50 Broken Dreams.—do. 1 50 Compensation.—do. 1 50 A Twisted Skein.—do. 1 50 Tested 1 50 Rich Medway 1 50 A Woman's Wiles 1 50 Terrace Roses 1 50 "New York Weekly" Series. Thrown on the World. $1 50 A Bitter Atonement. 1 50 Love Works Wonders. 1 50 Evelyn's Folly. 1 50 Lady Damer's Secret. 1 50 A Woman's Temptation. 1 50 Repented at Leisure. 1 50 Between Two Loves. 1 50 Peerless Cathleen. 1 50 Brownie's Triumph. 1 50 The Forsaken Bride. 1 50 His Other Wife. 1 50 Nick Whiffles. 1 50
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