![From the Introduction to the First Edition 3
“principal series representations”). More generally, we can replace T with a
group L of block-diagonal matrices, B with the group of corresponding upper
block-triangular matrices P, and we have a semi-direct product decomposition
(called a Levi decomposition) P = V L, where V is the subgroup of P whose
diagonal blocks are identity matrices; we may as before induce from P to G
representations of L lifted to P. The point of this method is that L is isomorphic
to a direct product of general linear groups of smaller degrees than n. We thus
have an inductive process to get representations of G if we know how to decom-
pose induced representations from P to G. This approach has been developed
in the works of Harish-Chandra, Howlett and Lehrer, and is introduced in the
chapter “Harish-Chandra Theory”.
Cohomological Methods
Let us now consider the example of G = Un, the unitary group. It can be defined
as the subgroup of matrices A ∈ GLn(Fq2 ) such that tA[q] = A−1, where A[q]
denotes the matrix whose entries are those of A raised to the qth power. It is thus
the subgroup of GLn(Fq) consisting of the fixed points of the endomorphism
F : A → (tA[q])−1.
A subgroup L of block-diagonal matrices in Un is again a product of unitary
groups of smaller degree. But this time we cannot construct a bigger group P
having L as a quotient. More precisely, the group V of upper block-triangular
matrices with entries in Fq and whose diagonal blocks are the identity matrix
has no fixed points other than the identity under F.
To get a suitable theory, Harish-Chandra’s construction must be generalised;
instead of inducing from V L to G, we construct a variety attached to V
on which both L and G act with commuting actions, and the cohomology
of that variety with -adic coefficients gives a (virtual) bi-module which de-
fines a “generalised induction” from L to G. This approach, due to Deligne
and Lusztig, will be developed in the chapters from “-adic Cohomology” to
“Geometric Conjugacy and Lusztig Series”.
Gelfand–Graev Representations
Using the above methods, a lot of information can be obtained about the char-
acters of the groups G(Fq), when G has a connected centre. The situation is
not so clear when the centre of G is not connected. In this case one can use
the Gelfand–Graev representations, which are obtained by inducing a linear
character “in general position” of a maximal unipotent subgroup (in GLn the
subgroup of upper triangular matrices with ones on the diagonal is such a sub-
group). These representations are closely tied to the theory of regular unipotent](https://image.slidesharecdn.com/63837-250501123329-7121210f/85/Representations-of-Finite-Groups-of-Lie-Type-2nd-Edition-Francois-Digne-15-320.jpg)
![6 Basic Results on Algebraic Groups
the elements of the group G(k) of k-valued points of G; that is, the morphisms
A → k. Since an algebraic variety over k is determined by the set of its k-points,
we will sometimes identify G with the set of its elements.
Examples 1.1.1 (of affine algebraic groups)
(i) The additive group Ga, defined by the algebra k[t], with comultiplication
t → t ⊗ 1 + 1 ⊗ t. We have Ga(k) k+.
(ii) The multiplicative group Gm, defined by the algebra k[t,t−1], with co-
multiplication t → t ⊗ t. We have Gm(k) k×.
(iii) The general linear group GLn, defined by the algebra k[{ti,j}1≤i,j≤n,
det(ti,j)−1]. The comultiplication is given by ti,j →
k ti,k ⊗ tk,j. The group
GLn(k) identifies with the group of invertible n × n-matrices with entries
in k. As an open subvariety of an affine space, GLn is connected.
(iv) A finite group Γ is algebraic; its algebra is the algebra of k-valued func-
tions on Γ, which can be identified to Homk(kΓ,k). A basis of orthogonal
idempotents is formed of the Dirac functions δg(g ) = δg,g . The comulti-
plication is δg →
{g ,g ∈Γ|g g =g} δg ⊗ δg .
Actually every connected one-dimensional affine algebraic group is iso-
morphic to Gm or Ga – this is surprisingly difficult to prove; see for instance
Springer (1998, 3.4.9).
A morphism of algebraic groups is a morphism of varieties which is also
a group homomorphism. A closed subvariety which is a subgroup is naturally
an algebraic subgroup; that is, the inclusion map is a morphism of algebraic
groups.
Remark 1.1.2 For some problems about algebraic groups, working with
varieties instead of schemes loses information. For instance, in our setting,
when k = Fp, an algebraic closure of the prime field Fp, the kernel of the mor-
phism x → xp : Ga → Ga is trivial. However, in the setting of schemes, it can
be computed as Spec(k[t]/tp), which is not allowed in the setting of varieties
since it is a non-reduced algebra.
Proposition 1.1.3 Let {Vi}i∈I be a family of irreducible subvarieties all con-
taining the identity element of an algebraic group G; then the smallest closed
subgroup H of G containing the Vi is equal to the product Wi1 . . . Wik for
some finite sequence (i1,. . . ,ik) of elements of I, where either Wi = Vi or
Wi = {x−1 | x ∈ Vi}.
Reference See Borel (1991, I, 2.2).](https://image.slidesharecdn.com/63837-250501123329-7121210f/85/Representations-of-Finite-Groups-of-Lie-Type-2nd-Edition-Francois-Digne-18-320.jpg)
![8 Basic Results on Algebraic Groups
1.2 Diagonalisable Groups, Tori, X(T),Y(T)
Definition 1.2.1
(i) A torus is an algebraic group T isomorphic to Gr
m for some r called the
rank of T.
(ii) A diagonalisable group is an algebraic group isomorphic to a closed sub-
group of a torus.
(iii) A rational character of an algebraic group G is a morphism of algebraic
groups G → Gm.
(iv) The character group X(G) of an algebraic group G is the group of ratio-
nal characters of G.
Proposition 1.2.2 A diagonalisable group D is equal to Spec(kX(D)).
Proof We first prove the result for a torus T of rank r; its algebra is
A = k[t1,. . . ,tr,t−1
1 ,. . . ,t−1
r ] k[t,t−1]⊗r. An element of X(T) is given by a
morphism of both algebras and coalgebras k[t,t−1] → A, which is defined
by giving the image of t which must be invertible, thus must be a monomial;
this monomial must be unitary for the morphism to be a coalgebra morphism.
The group law on X(T), given by pointwise multiplication, corresponds to the
multiplication of monomials; thus X(T) is isomorphic to the group of unitary
monomials, and A kX(T), the group algebra of the group X(T).
Let now D be a closed subgroup of a torus T; its algebra is a quotient of the
algebra kX(T) of T, hence is spanned by the images of the rational characters
of T; that is, by the restrictions to D of these characters. By the linear indepen-
dence of the characters of a group as functions to k – see, for example, Lang
(2002, Chapter VI, Theorem 4.1) – these restrictions form a basis of the algebra
of D, whence the result.
Proposition 1.2.3
(i) A diagonalisable group is the direct product of a torus by a finite abelian
p -group, where p is the characteristic of k.
(ii) The quotient of a torus by a closed subgroup is a torus.
References See Springer (1998, 3.2.7(i)) and Borel (1991, Corollary of 8.4).
Proposition 1.2.4 Seen as a functor from diagonalisable groups to Z-modules,
X is an exact functor.
Proof Left exactness is a general fact for Hom functors. Now, a similar ar-
gument to that in the proof of 1.2.2 shows that if D ⊂ D are diagonalisable
groups, then the restriction X(D ) → X(D) is surjective.](https://image.slidesharecdn.com/63837-250501123329-7121210f/85/Representations-of-Finite-Groups-of-Lie-Type-2nd-Edition-Francois-Digne-20-320.jpg)
![1.2 Diagonalisable Groups, Tori, X(T),Y(T) 9
Exercise 1.2.5 Show that X is an antiequivalence of abelian categories from
diagonalisable groups over k to Z-modules of finite type without p-torsion,
where p is the characteristic of k, and that X restricts to an antiequivalence
of additive categories from tori to finite rank lattices (here “lattice” means free
Z-module). Use the fact that the inverse antiequivalence maps a Z-module M
to Spec k[M].
Proposition 1.2.6 Any exact sequence of tori is split; in particular, a subtorus
of a torus is a direct factor.
Proof By the antiequivalence of categories X (see 1.2.5) an exact sequence
of tori corresponds to an exact sequence of lattices. Such a sequence is always
split.
Proposition 1.2.7 Given an algebraic action of a torus T on an affine variety
X, there exists an element t ∈ T such that Xt = XT.
Proof It is known that an algebraic group action can be linearised; that is,
there exists an embedding of X into a finite-dimensional k-vector space V and
an embedding of T into GL(V) such that the action of T factors through this
embedding; see Slodowy (1980, I, 1.3). The space V has a decomposition into
irreducible T-submodules, each defining some χ ∈ X(T). The kernel of a non-
trivial such χ has codimension 1 in T, since its algebra is the quotient of the
algebra of T by the ideal generated by χ (identifying the algebra of T with the
group algebra of X(T)). It follows that there is some element t ∈ T which lies
outside the finitely many kernels of the non-trivial characters of T occurring
in V; the fixed points of t on V (and so on X) are the same as those of T,
whence the result.
Proposition 1.2.8
(i) A torus consists of semi-simple elements.
(ii) A connected linear algebraic group containing only semi-simple elements
is a torus.
Proof Proposition 1.2.8 (i) comes from the fact that Gr
m can be embedded as
the group of diagonal matrices in GLr. For (ii) see Springer (1998, 6.3.6).
Definition 1.2.9
(i) A one-parameter subgroup of an algebraic group G is a morphism of
algebraic groups Gm → G.
(ii) The abelian group of one-parameter subgroups of a torus T is denoted
by Y(T).](https://image.slidesharecdn.com/63837-250501123329-7121210f/85/Representations-of-Finite-Groups-of-Lie-Type-2nd-Edition-Francois-Digne-21-320.jpg)
![10 Basic Results on Algebraic Groups
An element of Y(T) is given by a morphism k[t1,. . . ,tr,t−1
1 ,. . . ,t−1
r ] →
k[t,t−1], determined by the images of the ti which must be invertible, thus
monomials. These monomials must be unitary, thus powers of t, for the mor-
phism to be a coalgebra morphism. The group law on one-parameter subgroups
corresponds to multiplying these powers, so Y(T) is isomorphic to a product of
r copies of the group of powers of t.
There is an exact pairing between X(T) and Y(T) (that is, a bilinear map
X(T) × Y(T) → Z making each one the Z-dual of the other) obtained as
follows: given χ ∈ X(T) and ψ ∈ Y(T), the composite map χ ◦ ψ is a homo-
morphism from Gm to itself, so is the map t → tn for some n ∈ Z; the pairing is
defined by ( χ,ψ) → n.
Proposition 1.2.10 The map y ⊗ x → y(x) is a group isomorphism:
Y(T) ⊗Z k× ∼
−
→ T(k).
Proof Let (xi)i=1,...,n and (yi)i=1,...,n be two dual bases of X(T) and Y(T),
respectively. It is easy to check that t →
i=n
i=1 yi ⊗ xi(t) : T(k) → Y(T) ⊗Z k× is
the inverse of the map of the statement, using the fact that
x∈X(T) Ker x = {1}.
(This follows from the isomorphism between the algebra of the variety T and
kX(T); see 1.2.2.)
We now study the relationship between closed subgroups of T and sub-
groups of X(T).
Lemma 1.2.11 Given a torus T, let x1,. . . ,xn be linearly independent ele-
ments of X(T) and λ1,. . . ,λn arbitrary elements of k×; then there exists s ∈ T
such that xi(s) = λi for i = 1,. . . ,n.
Reference See, for example, Humphreys (1975, 16.2, Lemma C).
Definition 1.2.12 Given a torus T and a closed subgroup S of T, we define
S⊥
X(T) = {x ∈ X(T) | ∀s ∈ S, x(s) = 1}; conversely, given a subgroup A of
X(T), we define a subgroup of T by A⊥
T = {s ∈ T | ∀x ∈ A, x(s) = 1} (which is
closed, since A is finitely generated).
Proposition 1.2.13 If k has characteristic p, given a torus T and a subgroup
A of X(T), the group (A⊥
T)⊥
X(T)/A is the p-torsion subgroup of X(T)/A.
Proof First notice that, for any closed subgroup S of T, the group X(T)/S⊥
X(T)
has no p-torsion. Indeed,
pn
x ∈ S⊥
X(T) ⇔ ∀s ∈ S,x(s)pn
= 1 ⇔ ∀s ∈ S, x(s) = 1 ⇔ x ∈ S⊥
X(T),
where the middle equivalence holds, since x → xpn
is an automorphism of k.](https://image.slidesharecdn.com/63837-250501123329-7121210f/85/Representations-of-Finite-Groups-of-Lie-Type-2nd-Edition-Francois-Digne-22-320.jpg)
![1.3 Solvable Groups, Borel Subgroups 11
Thus it is enough to see that (A⊥
T)⊥
X(T)/A is a p-group. Let x ∈ (A⊥
T)⊥
X(T) − A.
It is a standard result on submodules of free Z-modules that there is a basis
(x,x1,. . . ,xr) of A,x such that (mx,x1,. . . ,xr) is a basis of A, with m ∈ Z,
possibly m = 0 (which means omit mx). The result follows if we can prove that
m is a power of p. Let us assume otherwise: then there exists λ ∈ k× such that
λ 1 and λm = 1 (even if m = 0). By 1.2.11, there exists some s ∈ T such that
x(s) = λ,x1(s) = 1,. . . ,xr(s) = 1. Thus mx(s) = 1, so s ∈ A⊥
T, but x(s) 1,
which contradicts x ∈ (A⊥
T)⊥
X(T).
1.3 Solvable Groups, Borel Subgroups
We will denote by Gu the set of unipotent elements of an algebraic
group G.
Proposition 1.3.1 Let G be a connected solvable algebraic group; then:
(i) Gu is a normal connected subgroup of G.
(ii) For every maximal torus T of G, there is a semi-direct product decompo-
sition G = Gu T.
(iii) Let S be a subtorus of G – then NG(S) = CG(S).
Proof For (i) and (ii) see Springer (1998, 6.3.3(ii) and 6.3.5(iv)); these asser-
tions follow from the theorem of Lie–Kolchin, which states that every closed
solvable subgroup of GLn(k) is conjugate to a subgroup of the group of upper
triangular matrices.
For (iii), if n ∈ NG(S), s ∈ S, then [n,s] ∈ [G,G] ∩ S ⊂ Gu ∩ S = 1, where
the first inclusion comes from (ii).
Let us note that this proposition implies that every connected solvable alge-
braic group containing no unipotent elements is a torus.
Definition 1.3.2 Maximal closed connected solvable subgroups of an alge-
braic group are called Borel subgroups.
These groups have paramount importance in the theory. The next theorem
states their basic properties.
Theorem 1.3.3 Let G be a connected algebraic group; then:
(i) All Borel subgroups of G are conjugate.
(ii) Every element of G is in some Borel subgroup.
(iii) The centraliser in G of any torus is connected.
(iv) A Borel subgroup is equal to its normaliser in G.
(v) Every maximal torus of G is in some Borel subgroup.](https://image.slidesharecdn.com/63837-250501123329-7121210f/85/Representations-of-Finite-Groups-of-Lie-Type-2nd-Edition-Francois-Digne-23-320.jpg)
![1.5 Examples of Reductive Groups 15
To see the second assertion, we may embed G in some GL(V); the space
V is a direct sum of isotypic spaces Vχ for some χ ∈ X(Z(G)). The action
of G preserves this decomposition, so the image of G is in
χ GL(Vχ) and
that of Der(G) in
SL(Vχ), while that of Z(G) consists of products of scalar
matrices in each Vχ, whence the result.
1.5 Examples of Reductive Groups
Example 1.5.1 The general linear group GLn; see 1.1.1(iii). The theorem of
Lie–Kolchin (see the proof of 1.3.1) shows that the upper triangular matrices in
GLn, defined by the algebra k[{ti,j}1≤i≤j≤n,{t−1
i,i }1≤i≤n], form a Borel subgroup
B – this group is connected as open in an affine space. The group of diagonal
matrices is a torus T. It is maximal in B since it is equal to the quotient of B by
the upper unitriangular matrices, which are unipotent, and a torus contains only
semi-simple elements; see 1.2.8(i). It is thus maximal in G by 1.3.3(v) and (i).
The lower triangular matrices – conjugate to the upper triangular by the
matrix of the permutation (1,n)(2,n − 1) . . . – form another Borel subgroup,
whose intersection with B is T. Thus GLn is reductive by 1.4.5 and 1.4.4(iii).
However, GLn is not semi-simple: the centre is the group of scalar matrices,
which is connected, thus equal to R(G) by 1.4.8.
The Borel subgroups are the conjugates of the upper triangular matrices,
hence identify as the stabilisers of the complete flags: that is, the increasing
sequences of vector subspaces 0 = F0 F1 · · · Fn = kn. This allows to
identify the variety G/B with the flag variety.
Example 1.5.2 The special linear group SLn = Spec k[ti,j]/(det(ti,j) − 1).
Since a Borel subgroup of SLn is a subgroup of a Borel subgroup of GLn by
maximality, and since the upper triangular matrices of SLn form a connected
group, by the same argument as in GLn they form a Borel subgroup. A similar
argument shows that the diagonal matrices of SLn form a maximal torus. The
group SLn is reductive by the same argument as GLn, thus is semi-simple, as
it has a finite centre (see 1.4.8).
Example 1.5.3 PGLn is the quotient of GLn by Gm embedded as the cen-
tre of GLn. To see it is an affine variety, we can either refer to Proposition
1.1.4 or identify it with the subgroup of g ∈ GL(Mn(k)), which are algebra
automorphisms; that is, such that g(Ei,j)g(Ek,l) = δj,kg(Ei,l) where Ei,j is the
elementary matrix defined by {Ei,j}k,l = δi,jδk,l. The image of a maximal torus
(resp. a Borel subgroup) of GLn is a maximal torus (resp. a Borel subgroup)
of PGLn, since the centre of GLn is contained in all Borel subgroups and all
maximal tori.](https://image.slidesharecdn.com/63837-250501123329-7121210f/85/Representations-of-Finite-Groups-of-Lie-Type-2nd-Edition-Francois-Digne-27-320.jpg)
![16 Basic Results on Algebraic Groups
If char k = p, the centre Z(SLp) is Spec k[t]/(tp −1) = Spec k[t]/(t−1)p,
which as a variety has a single point and thus is the trivial group, but it is not
trivial as a scheme! In characteristic p the natural morphism SLp → PGLp is
not separable; it is a bijection on the points over k but is not an isomorphism of
group schemes, an instance of the problem we mentioned in 1.1.2.
Example 1.5.4 The symplectic group Sp2n. On V = k2n with ordered ba-
sis (e1,. . . ,en,en,. . . ,e1), we define the symplectic bilinear form ei,ej =
ei,ej = 0, ei,ej = −ej,ei = δi,j. The group Sp2n is the (clearly closed)
subgroup of GL2n which preserves this form. If J is the n × n matrix J =
1
...
1
and J =
J
−J
, we have v,v = tvJv ; thus g is symplectic if
and only if tgJg = J. The torus T = {diag(t1,. . . ,tn,t−1
n ,. . . ,t−1
1 )} is a maximal
torus: it is in a unique maximal torus T1 of GL2n, the torus of diagonal matri-
ces, since T1 = CGL2n (T) and T consists of all symplectic diagonal matrices.
The symplectic upper triangular matrices form a subgroup B consisting of the
matrices
B BJ S
0 J tB−1J
, where B is upper triangular and S is symmetric. The
group B is connected since it is the product of the connected varieties of the up-
per triangular and the symmetric matrices. It is solvable since it is a subgroup
of the upper triangular matrices of GL2n. Thus it is contained in some Borel
subgroup of GL2n; that is, in the stabiliser of a complete flag. But the only line
stable by B is e1, and by induction, considering e⊥
1 /e1, the only flag sta-
bilised by B is the flag e1 ⊂ e1,e2 ⊂ · · · ⊂ e1,. . . ,en ⊂ e1,. . . ,en,en ⊂
· · · ⊂ e1,. . . ,en,en,. . . ,e1. Thus the only Borel subgroup of GL2n containing
B is the group B1 of (invertible) upper triangular matrices; thus B is a Borel
subgroup of Sp2n, since any larger solvable connected subgroup would be in
B1, and B = B1 ∩ Sp2n.
The group B is the stabiliser of a complete (that is, maximal) flag of isotropic
subspaces e1 ⊂ e1,e2 ⊂ · · · ⊂ e1,. . . ,en. By conjugation in Sp2n, we see
that any stabiliser of a complete isotropic flag is a Borel subgroup.
We deduce that Sp2n is connected: it is enough to see that every element is
in a Borel subgroup. Any g ∈ Sp2n has at least one eigenvector x ∈ V; as g is
symplectic, it induces a symplectic automorphism h of V1 = x⊥/x (which
is naturally endowed with a symplectic form induced by the initial form on V).
By induction on dim V, we may assume that Sp(V1) is connected, and thus h
is in some Borel subgroup of Sp(V1); that is, it stabilises a complete isotropic
flag of V1. The inverse image in V of this flag, completed by x, is a complete
isotropic flag of V stabilised by g, whence the result. The induction starts with
the group Sp2, which is equal to SL2, hence connected.](https://image.slidesharecdn.com/63837-250501123329-7121210f/85/Representations-of-Finite-Groups-of-Lie-Type-2nd-Edition-Francois-Digne-28-320.jpg)
Representations of Finite Groups of Lie Type 2nd Edition François Digne
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- 5. LONDON MATHEMATICAL SOCIETY STUDENT TEXTS Managing Editor: Ian J. Leary, Mathematical Sciences, University of Southampton, UK 57 Introduction to Banach algebras, operators and harmonic analysis, GARTH DALES et al 58 Computational algebraic geometry, HAL SCHENCK 59 Frobenius algebras and 2-D topological quantum field theories, JOACHIM KOCK 60 Linear operators and linear systems, JONATHAN R. PARTINGTON 61 An introduction to noncommutative Noetherian rings (2nd Edition), K. R. GOODEARL & R. B. WARFIELD, JR 62 Topics from one-dimensional dynamics, KAREN M. BRUCKS & HENK BRUIN 63 Singular points of plane curves, C. T. C. WALL 64 A short course on Banach space theory, N. L. CAROTHERS 65 Elements of the representation theory of associative algebras I, IBRAHIM ASSEM, DANIEL SIMSON & ANDRZEJ SKOWROŃSKI 66 An introduction to sieve methods and their applications, ALINA CARMEN COJOCARU & M. RAM MURTY 67 Elliptic functions, J. V. ARMITAGE & W. F. EBERLEIN 68 Hyperbolic geometry from a local viewpoint, LINDA KEEN & NIKOLA LAKIC 69 Lectures on Kähler geometry, ANDREI MOROIANU 70 Dependence logic, JOUKU VÄÄNÄNEN 71 Elements of the representation theory of associative algebras II, DANIEL SIMSON & ANDRZEJ SKOWROŃSKI 72 Elements of the representation theory of associative algebras III, DANIEL SIMSON & ANDRZEJ SKOWROŃSKI 73 Groups, graphs and trees, JOHN MEIER 74 Representation theorems in Hardy spaces, JAVAD MASHREGHI 75 An introduction to the theory of graph spectra, DRAGOŠ CVETKOVIĆ, PETER ROWLINSON & SLOBODAN SIMIĆ 76 Number theory in the spirit of Liouville, KENNETH S. WILLIAMS 77 Lectures on profinite topics in group theory, BENJAMIN KLOPSCH, NIKOLAY NIKOLOV & CHRISTOPHER VOLL 78 Clifford algebras: An introduction, D. J. H. GARLING 79 Introduction to compact Riemann surfaces and dessins d’enfants, ERNESTO GIRONDO & GABINO GONZÁLEZ-DIEZ 80 The Riemann hypothesis for function fields, MACHIEL VAN FRANKENHUIJSEN 81 Number theory, Fourier analysis and geometric discrepancy, GIANCARLO TRAVAGLINI 82 Finite geometry and combinatorial applications, SIMEON BALL 83 The geometry of celestial mechanics, HANSJÖRG GEIGES 84 Random graphs, geometry and asymptotic structure, MICHAEL KRIVELEVICH et al 85 Fourier analysis: Part I - Theory, ADRIAN CONSTANTIN 86 Dispersive partial differential equations, M. BURAK ERDOǦAN & NIKOLAOS TZIRAKIS 87 Riemann surfaces and algebraic curves, R. CAVALIERI & E. MILES 88 Groups, languages and automata, DEREK F. HOLT, SARAH REES & CLAAS E. RÖVER 89 Analysis on Polish spaces and an introduction to optimal transportation, D. J. H. GARLING 90 The homotopy theory of (∞,1)-categories, JULIA E. BERGNER 91 The block theory of finite group algebras I, M. LINCKELMANN 92 The block theory of finite group algebras II, M. LINCKELMANN 93 Semigroups of linear operators, D. APPLEBAUM 94 Introduction to approximate groups, M. C. H. TOINTON 95 Representations of finite groups of Lie type (2nd Edition), F. DIGNE & J. MICHEL 96 Tensor products of C*-algebras and operator spaces, G. PISIER
- 7. London Mathematical Society Student Texts 95 Representations of Finite Groups of Lie Type second edition FRANÇOIS DIGNE Université de Picardie Jules Verne, Amiens JEAN MICHEL Centre National de la Recherche Scientifique (CNRS), Paris
- 8. University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108481489 DOI: 10.1017/9781108673655 First edition © Cambridge University Press 1991 Second edition © François Digne and Jean Michel 2020 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1991 Second edition 2020 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Digne, François, author. | Michel, Jean, author. Title: Representations of finite groups of Lie type / François Digne, Université de Picardie Jules Verne, Amiens, Jean Michel, Centre National de la Recherche Scientifique (CNRS), Paris. Description: Second edition. | Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2020. | Series: London mathematical society student texts ; 95 | Includes bibliographical references and index. Identifiers: LCCN 2019035714 | ISBN 9781108481489 (hardback) | ISBN 9781108673655 (ebook) Subjects: LCSH: Lie groups. | Representations of groups. Classification: LCC QA387 .D54 2020 | DDC 512/.482--dc23 LC record available at https://lccn.loc.gov/2019035714 ISBN 978-1-108-48148-9 Hardback ISBN 978-1-108-72262-9 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 9. Contents Introduction to the Second Edition page 1 From the Introduction to the First Edition 2 1 Basic Results on Algebraic Groups 5 1.1 Basic Results on Algebraic Groups 5 1.2 Diagonalisable Groups, Tori, X(T),Y(T) 8 1.3 Solvable Groups, Borel Subgroups 11 1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups 13 1.5 Examples of Reductive Groups 15 2 Structure Theorems for Reductive Groups 19 2.1 Coxeter Groups 19 2.2 Finite Root Systems 24 2.3 Structure of Reductive Groups 29 2.4 Root Data, Isogenies, Presentation of G 35 3 (B,N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-Simple Elements 39 3.1 (B,N)-Pairs 39 3.2 Parabolic Subgroups of Coxeter Groups and of (B,N)-Pairs 42 3.3 Closed Subsets of a Crystallographic Root System 45 3.4 Parabolic Subgroups and Levi Subgroups 51 3.5 Centralisers of Semi-Simple Elements 56 v
- 10. vi Contents 4 Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem 59 4.1 k0-Varieties, Frobenius Endomorphisms 59 4.2 The Lang–Steinberg Theorem; Galois Cohomology 63 4.3 Classification of Finite Groups of Lie Type 70 4.4 The Relative (B,N)-Pair 75 5 Harish-Chandra Theory 79 5.1 Harish-Chandra Induction and Restriction 79 5.2 The Mackey Formula 83 5.3 Harish-Chandra Theory 86 6 Iwahori–Hecke Algebras 91 6.1 Endomorphism Algebras 91 6.2 Iwahori–Hecke Algebras 97 6.3 Schur Elements and Generic Degrees 104 6.4 The Example of G2 108 7 The Duality Functor and the Steinberg Character 113 7.1 F-rank 113 7.2 The Duality Functor 116 7.3 Restriction to Centralisers of Semi-Simple Elements 123 7.4 The Steinberg Character 126 8 -Adic Cohomology 130 8.1 -Adic Cohomology 130 9 Deligne–Lusztig Induction: The Mackey Formula 137 9.1 Deligne–Lusztig Induction 137 9.2 Mackey Formula for Lusztig Functors 140 9.3 Consequences: Scalar Products 146 10 The Character Formula and Other Results on Deligne–Lusztig Induction 148 10.1 The Character Formula 148 10.2 Uniform Functions 153 10.3 The Characteristic Function of a Semi-Simple Class 157 11 Geometric Conjugacy and the Lusztig Series 161 11.1 Geometric Conjugacy 161 11.2 More on Centralisers of Semi-Simple Elements 167 11.3 The Lusztig Series 170
- 11. Contents vii 11.4 Lusztig’s Jordan Decomposition of Characters: The Levi Case 175 11.5 Lusztig’s Jordan Decomposition of Characters: The General Case 183 11.6 More about Unipotent Characters 188 11.7 The Irreducible Characters of GLF n and UF n 191 12 Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters 196 12.1 Regular Elements 196 12.2 Regular Unipotent Elements 201 12.3 Gelfand–Graev Representations 207 12.4 Regular and Semi-Simple Characters 214 12.5 The Character Table of SL2(Fq) 220 13 Green Functions 225 13.1 Invariants 225 13.2 Green Functions and the Springer Correspondence 231 13.3 The Lusztig–Shoji Algorithm 236 14 The Decomposition of Deligne–Lusztig Characters 242 14.1 Lusztig Families and Special Unipotent Classes 242 14.2 Split Groups 244 14.3 Twisted Groups 247 References 249 Index 255
- 13. Introduction to the Second Edition We had two main aims in writing this edition: • Be more self-contained where possible. For instance, we have added brief overviews of Coxeter groups and root systems, and given some more details about the theory of algebraic groups. • While retaining the same level of exposition as in the first edition, we have given a more complete account of the representation theory of finite groups of Lie type. In view of the second aim, we have added the following topics to our expo- sition: • We cover Ree and Suzuki groups extending our exposition of Frobenius mor- phisms to the more general case of Frobenius roots. • We have added to Harish-Chandra theory the topic of Hecke algebras and given as many results as we could easily do for fields of arbitrary character- istic prime to q, in view of applications to modular representations. • We have added a chapter on the computation of Green functions, with a brief review of invariant theory of reflection groups, and a chapter on the decomposition of unipotent Deligne–Lusztig characters. Acknowledgements In addition to the people we thank in the introduction to the first edition, we have benefitted from the input of younger colleagues such as Cédric Bonnafé, Olivier Dudas, Meinolf Geck, Daniel Juteau, and Raphaël Rouquier, and we give special thanks to Gunter Malle for a thorough proofreading of this edition. We also thank the many people who pointed out to us misprints and other errors in the first edition. 1
- 14. From the Introduction to the First Edition These notes follow a course given at the Paris VII university during the spring semester of academic year 1987–88. Their purpose is to expound basic results in the representation theory of finite groups of Lie type (a precise definition of this concept will be given in the chapter “Rationality, Frobenius”). Let us start with some notations. We denote by Fq a finite field of charac- teristic p with q elements (q is a power of p). The typical groups we will look at are the linear, unitary, symplectic, orthogonal, . . . , groups over Fq. We will consider these groups as the subgroups of points with coefficients in Fq of the corresponding groups over an algebraic closure Fq (which are algebraic reduc- tive groups). More precisely, the group over Fq is the set of fixed points of the group over Fq under an endomorphism F called the Frobenius endomorphism; this will be explained in the chapter “Rationality, Frobenius”. In the follow- ing paragraphs of this introduction we will try to describe, by some examples, a sample of the methods used to study the complex representations of these groups. Induction from Subgroups Let us start with the example where G = GLn(Fq) is the general linear group over Fq. Let T be the subgroup of diagonal matrices; it is a subgroup of the group B of upper triangular matrices, and there is a semidirect product decom- position B = U T, where U is the subgroup of the upper triangular matrices which have all their diagonal entries equal to 1. The representation theory of T is easy, since it is a commutative group (actually isomorphic to a product of n copies of the multiplicative group F× q ). Composition with the natural homomor- phism from B to T (quotient by U) lifts representations of T to representations of B. Inducing these representations from B to the whole of the general lin- ear group gives representations of G (whose irreducible constituents are called 2
- 15. From the Introduction to the First Edition 3 “principal series representations”). More generally, we can replace T with a group L of block-diagonal matrices, B with the group of corresponding upper block-triangular matrices P, and we have a semi-direct product decomposition (called a Levi decomposition) P = V L, where V is the subgroup of P whose diagonal blocks are identity matrices; we may as before induce from P to G representations of L lifted to P. The point of this method is that L is isomorphic to a direct product of general linear groups of smaller degrees than n. We thus have an inductive process to get representations of G if we know how to decom- pose induced representations from P to G. This approach has been developed in the works of Harish-Chandra, Howlett and Lehrer, and is introduced in the chapter “Harish-Chandra Theory”. Cohomological Methods Let us now consider the example of G = Un, the unitary group. It can be defined as the subgroup of matrices A ∈ GLn(Fq2 ) such that tA[q] = A−1, where A[q] denotes the matrix whose entries are those of A raised to the qth power. It is thus the subgroup of GLn(Fq) consisting of the fixed points of the endomorphism F : A → (tA[q])−1. A subgroup L of block-diagonal matrices in Un is again a product of unitary groups of smaller degree. But this time we cannot construct a bigger group P having L as a quotient. More precisely, the group V of upper block-triangular matrices with entries in Fq and whose diagonal blocks are the identity matrix has no fixed points other than the identity under F. To get a suitable theory, Harish-Chandra’s construction must be generalised; instead of inducing from V L to G, we construct a variety attached to V on which both L and G act with commuting actions, and the cohomology of that variety with -adic coefficients gives a (virtual) bi-module which de- fines a “generalised induction” from L to G. This approach, due to Deligne and Lusztig, will be developed in the chapters from “-adic Cohomology” to “Geometric Conjugacy and Lusztig Series”. Gelfand–Graev Representations Using the above methods, a lot of information can be obtained about the char- acters of the groups G(Fq), when G has a connected centre. The situation is not so clear when the centre of G is not connected. In this case one can use the Gelfand–Graev representations, which are obtained by inducing a linear character “in general position” of a maximal unipotent subgroup (in GLn the subgroup of upper triangular matrices with ones on the diagonal is such a sub- group). These representations are closely tied to the theory of regular unipotent
- 16. 4 From the Introduction to the First Edition elements. They are multiplicity-free and contain rather large cross-sections of the set of irreducible characters, and so give useful additional information in the nonconnected centre case. (In the connected centre case, they are linear combinations of Deligne–Lusztig characters.) For instance, in SL2(Fq) they are obtained by inducing a nontrivial linear character of the group of matrices of the form 1 u 0 1 : such a character cor- responds to a nontrivial additive character of Fq; there are two classes of such characters under SL2(Fq), which corresponds to the fact that the centre of SL2 has two connected components (its two elements). The theory of regular elements and Gelfand–Graev representations is ex- pounded in chapter “Regular Elements; Gelfand–Graev Representations”, with, as an application, the computation of the values of all irreducible characters on regular unipotent elements. Acknowledgements We would like to thank the “équipe des groupes finis” and the mathematics department of the École Normale Supérieure, who provided us with a stimulat- ing working environment, and adequate facilities for composing this book. We thank also the Paris VII university, which gave us the opportunity to give the course which started this book. We thank all those who carefully read the ear- lier drafts and suggested improvements, particularly Michel Enguehard, Guy Rousseau, Jean-Yves Hée and the editor. We thank Michel Broué and Jacques Tits, who provided us with various ideas and information, and above all George Lusztig, who invented most of the theory.
- 17. 1 Basic Results on Algebraic Groups A finite group of Lie type – sometimes also called a finite reductive group – is (in a first approximation) the group of points over a finite field Fq of a (usually connected) reductive algebraic group over an algebraic closure Fq. We begin by recalling the definition of these terms. See 4.2.6 for a precise definition of finite groups of Lie type. Let us first establish some notations and conventions we use throughout. If g is an automorphism of a set (resp. variety, group, . . . ) X, we will denote by Xg the set of fixed points of g, and by gx or g(x) the image of the element x ∈ X by g. A group G acts naturally on itself by conjugation, and we will write gh for ghg−1, where g and h are elements of G and will write ad g for the morphism h → gh. We will write Z(G) for the centre of G; if X is a subset of a set on which G acts, we put NG(X) = {g ∈ G | gX = X} = {g ∈ G | ∀x ∈ X, gx ∈ X} and CG(X) = {g ∈ G | ∀x ∈ X, gx = x}. If A is a ring and X a set (resp. a group), we denote by AX the free A-module with basis X (resp. the group algebra of X with coefficients in A). We write H G for a semi-direct product of groups where H is normal in the product. We will generally use bold letters for algebraic groups and varieties. 1.1 Basic Results on Algebraic Groups An algebraic group is an algebraic variety G endowed with a group struc- ture such that the multiplication and inverse maps are algebraic. In this book, we will need only affine algebraic groups over an algebraically closed field k (which will be taken to be Fp from Chapter 4 onwards): that is, affine algebraic varieties G = Spec A, where A is a reduced k-algebra; that is, without non-zero nilpotent elements. The group structure gives a coalgebra structure on A, or even a Hopf algebra structure. For such a group G, we will call elements of G 5
- 18. 6 Basic Results on Algebraic Groups the elements of the group G(k) of k-valued points of G; that is, the morphisms A → k. Since an algebraic variety over k is determined by the set of its k-points, we will sometimes identify G with the set of its elements. Examples 1.1.1 (of affine algebraic groups) (i) The additive group Ga, defined by the algebra k[t], with comultiplication t → t ⊗ 1 + 1 ⊗ t. We have Ga(k) k+. (ii) The multiplicative group Gm, defined by the algebra k[t,t−1], with co- multiplication t → t ⊗ t. We have Gm(k) k×. (iii) The general linear group GLn, defined by the algebra k[{ti,j}1≤i,j≤n, det(ti,j)−1]. The comultiplication is given by ti,j → k ti,k ⊗ tk,j. The group GLn(k) identifies with the group of invertible n × n-matrices with entries in k. As an open subvariety of an affine space, GLn is connected. (iv) A finite group Γ is algebraic; its algebra is the algebra of k-valued func- tions on Γ, which can be identified to Homk(kΓ,k). A basis of orthogonal idempotents is formed of the Dirac functions δg(g ) = δg,g . The comulti- plication is δg → {g ,g ∈Γ|g g =g} δg ⊗ δg . Actually every connected one-dimensional affine algebraic group is iso- morphic to Gm or Ga – this is surprisingly difficult to prove; see for instance Springer (1998, 3.4.9). A morphism of algebraic groups is a morphism of varieties which is also a group homomorphism. A closed subvariety which is a subgroup is naturally an algebraic subgroup; that is, the inclusion map is a morphism of algebraic groups. Remark 1.1.2 For some problems about algebraic groups, working with varieties instead of schemes loses information. For instance, in our setting, when k = Fp, an algebraic closure of the prime field Fp, the kernel of the mor- phism x → xp : Ga → Ga is trivial. However, in the setting of schemes, it can be computed as Spec(k[t]/tp), which is not allowed in the setting of varieties since it is a non-reduced algebra. Proposition 1.1.3 Let {Vi}i∈I be a family of irreducible subvarieties all con- taining the identity element of an algebraic group G; then the smallest closed subgroup H of G containing the Vi is equal to the product Wi1 . . . Wik for some finite sequence (i1,. . . ,ik) of elements of I, where either Wi = Vi or Wi = {x−1 | x ∈ Vi}. Reference See Borel (1991, I, 2.2).
- 19. 1.1 Basic Results on Algebraic Groups 7 H is called the subgroup generated by the Vi. It is clear from 1.1.3 that the subgroup generated by a family of connected subvarieties is connected. An algebraic group is called linear if it is isomorphic to a closed subgroup of GLn. It is clear from the definition that a linear algebraic group is affine; the converse is also true; see Springer (1998, 2.3.7(i)). All algebraic groups con- sidered in the sequel will be linear; an example of non-linear algebraic groups are the elliptic curves. Proposition 1.1.4 Let G be a linear algebraic group and H a closed sub- group; then: (i) The quotient G/H exists and is a quasi-projective variety (that is, an open subvariety of projective variety). (ii) If H is a normal subgroup, then G/H is an affine variety and a linear algebraic group for the induced group structure. References See Springer (1998, 5.5.5 and 5.5.10). The connected components of an algebraic group G are finite in number and coincide with its irreducible components; the component containing the identity element of G is called the identity component and denoted by G0. It is a characteristic subgroup of G, and G/G0 identifies with the set of con- nected components of G. Conversely, every normal closed subgroup of finite index contains G0 – these properties are elementary; see, for example, Springer (1998, 2.2.1). An element of a linear algebraic group G is called semi-simple (resp. unipo- tent) if its image in some embedding of G as a closed subgroup of some GLn is semi-simple; that is, conjugate to a diagonal matrix (resp. unipotent, that is, conjugate to an upper unitriangular matrix). This property does not depend on the embedding. Every element has a unique decomposition, its Jordan decomposition, as the product of two commuting semi-simple and unipotent elements; see Springer (1998, 2.4.8). Proposition 1.1.5 Let G be a linear algebraic group over Fp, where p is a prime. Then every element of G has finite order: the semi-simple elements are the p -elements, and the unipotent elements are the p-elements. Proof This results from the fact that the above result holds for GLn(Fp). Indeed, the diagonal entries of a diagonal matrix are in F × p , thus of order prime to p, and a matrix X is unipotent if and only if N = X − In is nilpotent, equiva- lently Xpa = Npa + In = In for a large enough.
- 20. 8 Basic Results on Algebraic Groups 1.2 Diagonalisable Groups, Tori, X(T),Y(T) Definition 1.2.1 (i) A torus is an algebraic group T isomorphic to Gr m for some r called the rank of T. (ii) A diagonalisable group is an algebraic group isomorphic to a closed sub- group of a torus. (iii) A rational character of an algebraic group G is a morphism of algebraic groups G → Gm. (iv) The character group X(G) of an algebraic group G is the group of ratio- nal characters of G. Proposition 1.2.2 A diagonalisable group D is equal to Spec(kX(D)). Proof We first prove the result for a torus T of rank r; its algebra is A = k[t1,. . . ,tr,t−1 1 ,. . . ,t−1 r ] k[t,t−1]⊗r. An element of X(T) is given by a morphism of both algebras and coalgebras k[t,t−1] → A, which is defined by giving the image of t which must be invertible, thus must be a monomial; this monomial must be unitary for the morphism to be a coalgebra morphism. The group law on X(T), given by pointwise multiplication, corresponds to the multiplication of monomials; thus X(T) is isomorphic to the group of unitary monomials, and A kX(T), the group algebra of the group X(T). Let now D be a closed subgroup of a torus T; its algebra is a quotient of the algebra kX(T) of T, hence is spanned by the images of the rational characters of T; that is, by the restrictions to D of these characters. By the linear indepen- dence of the characters of a group as functions to k – see, for example, Lang (2002, Chapter VI, Theorem 4.1) – these restrictions form a basis of the algebra of D, whence the result. Proposition 1.2.3 (i) A diagonalisable group is the direct product of a torus by a finite abelian p -group, where p is the characteristic of k. (ii) The quotient of a torus by a closed subgroup is a torus. References See Springer (1998, 3.2.7(i)) and Borel (1991, Corollary of 8.4). Proposition 1.2.4 Seen as a functor from diagonalisable groups to Z-modules, X is an exact functor. Proof Left exactness is a general fact for Hom functors. Now, a similar ar- gument to that in the proof of 1.2.2 shows that if D ⊂ D are diagonalisable groups, then the restriction X(D ) → X(D) is surjective.
- 21. 1.2 Diagonalisable Groups, Tori, X(T),Y(T) 9 Exercise 1.2.5 Show that X is an antiequivalence of abelian categories from diagonalisable groups over k to Z-modules of finite type without p-torsion, where p is the characteristic of k, and that X restricts to an antiequivalence of additive categories from tori to finite rank lattices (here “lattice” means free Z-module). Use the fact that the inverse antiequivalence maps a Z-module M to Spec k[M]. Proposition 1.2.6 Any exact sequence of tori is split; in particular, a subtorus of a torus is a direct factor. Proof By the antiequivalence of categories X (see 1.2.5) an exact sequence of tori corresponds to an exact sequence of lattices. Such a sequence is always split. Proposition 1.2.7 Given an algebraic action of a torus T on an affine variety X, there exists an element t ∈ T such that Xt = XT. Proof It is known that an algebraic group action can be linearised; that is, there exists an embedding of X into a finite-dimensional k-vector space V and an embedding of T into GL(V) such that the action of T factors through this embedding; see Slodowy (1980, I, 1.3). The space V has a decomposition into irreducible T-submodules, each defining some χ ∈ X(T). The kernel of a non- trivial such χ has codimension 1 in T, since its algebra is the quotient of the algebra of T by the ideal generated by χ (identifying the algebra of T with the group algebra of X(T)). It follows that there is some element t ∈ T which lies outside the finitely many kernels of the non-trivial characters of T occurring in V; the fixed points of t on V (and so on X) are the same as those of T, whence the result. Proposition 1.2.8 (i) A torus consists of semi-simple elements. (ii) A connected linear algebraic group containing only semi-simple elements is a torus. Proof Proposition 1.2.8 (i) comes from the fact that Gr m can be embedded as the group of diagonal matrices in GLr. For (ii) see Springer (1998, 6.3.6). Definition 1.2.9 (i) A one-parameter subgroup of an algebraic group G is a morphism of algebraic groups Gm → G. (ii) The abelian group of one-parameter subgroups of a torus T is denoted by Y(T).
- 22. 10 Basic Results on Algebraic Groups An element of Y(T) is given by a morphism k[t1,. . . ,tr,t−1 1 ,. . . ,t−1 r ] → k[t,t−1], determined by the images of the ti which must be invertible, thus monomials. These monomials must be unitary, thus powers of t, for the mor- phism to be a coalgebra morphism. The group law on one-parameter subgroups corresponds to multiplying these powers, so Y(T) is isomorphic to a product of r copies of the group of powers of t. There is an exact pairing between X(T) and Y(T) (that is, a bilinear map X(T) × Y(T) → Z making each one the Z-dual of the other) obtained as follows: given χ ∈ X(T) and ψ ∈ Y(T), the composite map χ ◦ ψ is a homo- morphism from Gm to itself, so is the map t → tn for some n ∈ Z; the pairing is defined by ( χ,ψ) → n. Proposition 1.2.10 The map y ⊗ x → y(x) is a group isomorphism: Y(T) ⊗Z k× ∼ − → T(k). Proof Let (xi)i=1,...,n and (yi)i=1,...,n be two dual bases of X(T) and Y(T), respectively. It is easy to check that t → i=n i=1 yi ⊗ xi(t) : T(k) → Y(T) ⊗Z k× is the inverse of the map of the statement, using the fact that x∈X(T) Ker x = {1}. (This follows from the isomorphism between the algebra of the variety T and kX(T); see 1.2.2.) We now study the relationship between closed subgroups of T and sub- groups of X(T). Lemma 1.2.11 Given a torus T, let x1,. . . ,xn be linearly independent ele- ments of X(T) and λ1,. . . ,λn arbitrary elements of k×; then there exists s ∈ T such that xi(s) = λi for i = 1,. . . ,n. Reference See, for example, Humphreys (1975, 16.2, Lemma C). Definition 1.2.12 Given a torus T and a closed subgroup S of T, we define S⊥ X(T) = {x ∈ X(T) | ∀s ∈ S, x(s) = 1}; conversely, given a subgroup A of X(T), we define a subgroup of T by A⊥ T = {s ∈ T | ∀x ∈ A, x(s) = 1} (which is closed, since A is finitely generated). Proposition 1.2.13 If k has characteristic p, given a torus T and a subgroup A of X(T), the group (A⊥ T)⊥ X(T)/A is the p-torsion subgroup of X(T)/A. Proof First notice that, for any closed subgroup S of T, the group X(T)/S⊥ X(T) has no p-torsion. Indeed, pn x ∈ S⊥ X(T) ⇔ ∀s ∈ S,x(s)pn = 1 ⇔ ∀s ∈ S, x(s) = 1 ⇔ x ∈ S⊥ X(T), where the middle equivalence holds, since x → xpn is an automorphism of k.
- 23. 1.3 Solvable Groups, Borel Subgroups 11 Thus it is enough to see that (A⊥ T)⊥ X(T)/A is a p-group. Let x ∈ (A⊥ T)⊥ X(T) − A. It is a standard result on submodules of free Z-modules that there is a basis (x,x1,. . . ,xr) of A,x such that (mx,x1,. . . ,xr) is a basis of A, with m ∈ Z, possibly m = 0 (which means omit mx). The result follows if we can prove that m is a power of p. Let us assume otherwise: then there exists λ ∈ k× such that λ 1 and λm = 1 (even if m = 0). By 1.2.11, there exists some s ∈ T such that x(s) = λ,x1(s) = 1,. . . ,xr(s) = 1. Thus mx(s) = 1, so s ∈ A⊥ T, but x(s) 1, which contradicts x ∈ (A⊥ T)⊥ X(T). 1.3 Solvable Groups, Borel Subgroups We will denote by Gu the set of unipotent elements of an algebraic group G. Proposition 1.3.1 Let G be a connected solvable algebraic group; then: (i) Gu is a normal connected subgroup of G. (ii) For every maximal torus T of G, there is a semi-direct product decompo- sition G = Gu T. (iii) Let S be a subtorus of G – then NG(S) = CG(S). Proof For (i) and (ii) see Springer (1998, 6.3.3(ii) and 6.3.5(iv)); these asser- tions follow from the theorem of Lie–Kolchin, which states that every closed solvable subgroup of GLn(k) is conjugate to a subgroup of the group of upper triangular matrices. For (iii), if n ∈ NG(S), s ∈ S, then [n,s] ∈ [G,G] ∩ S ⊂ Gu ∩ S = 1, where the first inclusion comes from (ii). Let us note that this proposition implies that every connected solvable alge- braic group containing no unipotent elements is a torus. Definition 1.3.2 Maximal closed connected solvable subgroups of an alge- braic group are called Borel subgroups. These groups have paramount importance in the theory. The next theorem states their basic properties. Theorem 1.3.3 Let G be a connected algebraic group; then: (i) All Borel subgroups of G are conjugate. (ii) Every element of G is in some Borel subgroup. (iii) The centraliser in G of any torus is connected. (iv) A Borel subgroup is equal to its normaliser in G. (v) Every maximal torus of G is in some Borel subgroup.
- 24. 12 Basic Results on Algebraic Groups Outline of proof To show (i) – see Springer (1998, 6.2.7(iii)) – one first shows that G/B is a complete variety; a complete variety has the property that a con- nected solvable group acting on it always has a fixed point. Thus another Borel subgroup B acting on G/B by left translation has a fixed point; that is, there exists g ∈ G such that B gB = gB, so g−1B g ⊂ B, whence the result. Property (ii) similarly results from properties of complete varieties; see Springer (1998, 6.4.5(i)). For (iii) see Springer (1998, 6.4.7(i)). For (iv) see Springer (1998, 6.4.9). Property (v) results from the definition of a Borel subgroup. It follows from (iv) above and the remark that the closure of a solvable group is solvable – see Borel (1991, I, 2.4) – that the words “closed connected” can be omitted from the definition of a Borel subgroup when G is connected. Theorem 1.3.4 Let T be a torus of a connected algebraic group G, then: (i) NG(T)0 = CG(T) = CG(T)0. (ii) The Borel subgroups of CG(T) are the groups CG(T) ∩ B, where B runs over the Borel subgroups of G containing T. References See, for example Springer (1998, 3.2.9 and 6.4.7). It follows from (i) that the quotient NG(T)/CG(T) is finite. Definition 1.3.5 A closed subgroup of a connected algebraic group G which contains a Borel subgroup is called a parabolic subgroup. Corollary 1.3.6 Let G be a connected algebraic group: (i) Any parabolic subgroup is equal to its normaliser in G and is connected. (ii) Two parabolic subgroups containing the same Borel subgroup and G-conjugate are equal. (iii) All maximal tori of G are conjugate; every semi-simple element of G is in some maximal torus. (iv) Two elements of a maximal torus T are G-conjugate if and only if they are NG(T)-conjugate. Proof Let us prove (i). As the Borel subgroups are connected, P0 contains a Borel subgroup B. As another Borel subgroup of G in P0 is P0-conjugate to B, we have NG(P0) = P0NG(B) = P0B = P0. As P ⊂ NG(P0) we have P = P0. Let us prove (ii). Using again that Borel subgroups of P are P-conjugate, we get that two conjugate parabolic subgroups containing the same Borel subgroup are NG(B)-conjugate, thus are equal, since NG(B) = B. Since any maximal torus is in some Borel subgroup, the first assertion in (iii) results from 1.3.3 (i) and the same property for connected solvable groups;
- 25. 1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups 13 see Springer (1998, 6.3.5 (iii)). The second assertion of (iii) comes from 1.3.3 (ii) and the same property for connected solvable groups; see Springer (1998, 6.3.5 (i)). Let us prove (iv). If s and gs both lie in T, then T and g−1 T are two maxi- mal tori containing s, thus are two maximal tori of the group CG(s), and since they are connected they lie in the identity component CG(s)0. By (iii) they are conjugate by some element x ∈ CG(s)0; that is, xT = g−1 T so gx ∈ NG(T). As gxs = gs, we get the result. Statement 1.3.6(iii) allows to give the following definition. Definition 1.3.7 The rank of an algebraic group is the rank of its maximal tori. Exercise 1.3.8 If a closed subgroup H of the connected algebraic group G contains a maximal torus T of G, then NG(H) ⊂ H0 · NG(T). 1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups A unipotent algebraic group is a group containing only unipotent elements. Proposition 1.4.1 Every unipotent subgroup of an affine algebraic group G is nilpotent. Proof By embedding G into GLn, we can reduce to the case G = GLn; any unipotent subgroup of GLn(k) is conjugate to a subgroup of the group of up- per triangular matrices which have all their diagonal entries equal to 1 – see Springer (1998, 2.4.12) for a proof – whence the result. Proposition 1.4.2 As an algebraic variety, a connected unipotent group is iso- morphic to an affine space; in characteristic 0 a unipotent group is necessarily connected. Proof For the first part, see Springer (1998, 14.2.7 and 14.3.10). If the charac- teristic is 0 and U is a unipotent algebraic group, then U/U0 is a finite unipotent group. But a unipotent element in characteristic 0 has infinite order, since this holds in GLn, whence the second part. Corollary 1.4.3 The maximal connected unipotent subgroups of an algebraic group G are the groups Bu where B is a Borel subgroup of G. Proof By 1.4.1 such a subgroup is nilpotent, thus in a Borel subgroup, whence the result by 1.3.1(i).
- 26. 14 Basic Results on Algebraic Groups Proposition 1.4.4 Let G be an algebraic group. (i) The product of the closed connected normal solvable subgroups of G is also a closed connected normal solvable subgroup of G, called the radical of G and denoted by R(G). (ii) Similarly, the set of all closed connected normal unipotent subgroups of G has a unique maximal element called the unipotent radical of G and denoted by Ru(G). (iii) Ru(G) = R(G)u (where R(G)u is defined as in 1.3.1(i)). Proof Using 1.1.3 it follows that the product in (i) is actually finite. This im- plies (i), since the product of two solvable groups normalising each other is still solvable. To see (ii) and (iii), we first remark that a closed connected normal unipotent subgroup is in R(G), thus in R(G)u. We then observe that R(G)u is normal in G, being characteristic in R(G), and is connected by 1.3.1(i). Corollary 1.4.5 R(G) is the identity component of the intersection of all Borel subgroups. Proof Indeed, R(G) is contained in at least one Borel subgroup. Since it is normal and all Borel subgroups are conjugate, it is contained in their inter- section. Since it is connected, it is contained in the identity component of this intersection. Conversely, this component is solvable and normal. Definition 1.4.6 An algebraic group is called reductive if its unipotent radical is trivial, and semi-simple if its radical is trivial. Exercise 1.4.7 For an algebraic group G, the group G/R(G) is semi-simple and the group G/Ru(G) is reductive. Proposition 1.4.8 If G is connected and reductive, then R(G) = Z(G)0, the identity component of the centre of G. Proof By 1.3.1(iii), since Ru(G) is trivial, R(G) is a torus, normal in G. Since G is connected and for any torus T we have NG(T)0 = CG(T)0 (see 1.3.4), R(G) is central in G, and, being connected, is in Z(G)0. Conversely, Z(G)0, being normal solvable and connected, is contained in R(G). Proposition 1.4.9 If G is reductive and connected, its derived group Der(G) is semi-simple and has a finite intersection with Z(G). Proof The first assertion results from the second one, since the radical R(Der(G)), being characteristic by 1.4.8, is in R(G), thus in Z(G) ∩ Der(G); it is trivial, since it is connected and this last group is finite.
- 27. 1.5 Examples of Reductive Groups 15 To see the second assertion, we may embed G in some GL(V); the space V is a direct sum of isotypic spaces Vχ for some χ ∈ X(Z(G)). The action of G preserves this decomposition, so the image of G is in χ GL(Vχ) and that of Der(G) in SL(Vχ), while that of Z(G) consists of products of scalar matrices in each Vχ, whence the result. 1.5 Examples of Reductive Groups Example 1.5.1 The general linear group GLn; see 1.1.1(iii). The theorem of Lie–Kolchin (see the proof of 1.3.1) shows that the upper triangular matrices in GLn, defined by the algebra k[{ti,j}1≤i≤j≤n,{t−1 i,i }1≤i≤n], form a Borel subgroup B – this group is connected as open in an affine space. The group of diagonal matrices is a torus T. It is maximal in B since it is equal to the quotient of B by the upper unitriangular matrices, which are unipotent, and a torus contains only semi-simple elements; see 1.2.8(i). It is thus maximal in G by 1.3.3(v) and (i). The lower triangular matrices – conjugate to the upper triangular by the matrix of the permutation (1,n)(2,n − 1) . . . – form another Borel subgroup, whose intersection with B is T. Thus GLn is reductive by 1.4.5 and 1.4.4(iii). However, GLn is not semi-simple: the centre is the group of scalar matrices, which is connected, thus equal to R(G) by 1.4.8. The Borel subgroups are the conjugates of the upper triangular matrices, hence identify as the stabilisers of the complete flags: that is, the increasing sequences of vector subspaces 0 = F0 F1 · · · Fn = kn. This allows to identify the variety G/B with the flag variety. Example 1.5.2 The special linear group SLn = Spec k[ti,j]/(det(ti,j) − 1). Since a Borel subgroup of SLn is a subgroup of a Borel subgroup of GLn by maximality, and since the upper triangular matrices of SLn form a connected group, by the same argument as in GLn they form a Borel subgroup. A similar argument shows that the diagonal matrices of SLn form a maximal torus. The group SLn is reductive by the same argument as GLn, thus is semi-simple, as it has a finite centre (see 1.4.8). Example 1.5.3 PGLn is the quotient of GLn by Gm embedded as the cen- tre of GLn. To see it is an affine variety, we can either refer to Proposition 1.1.4 or identify it with the subgroup of g ∈ GL(Mn(k)), which are algebra automorphisms; that is, such that g(Ei,j)g(Ek,l) = δj,kg(Ei,l) where Ei,j is the elementary matrix defined by {Ei,j}k,l = δi,jδk,l. The image of a maximal torus (resp. a Borel subgroup) of GLn is a maximal torus (resp. a Borel subgroup) of PGLn, since the centre of GLn is contained in all Borel subgroups and all maximal tori.
- 28. 16 Basic Results on Algebraic Groups If char k = p, the centre Z(SLp) is Spec k[t]/(tp −1) = Spec k[t]/(t−1)p, which as a variety has a single point and thus is the trivial group, but it is not trivial as a scheme! In characteristic p the natural morphism SLp → PGLp is not separable; it is a bijection on the points over k but is not an isomorphism of group schemes, an instance of the problem we mentioned in 1.1.2. Example 1.5.4 The symplectic group Sp2n. On V = k2n with ordered ba- sis (e1,. . . ,en,en,. . . ,e1), we define the symplectic bilinear form ei,ej = ei,ej = 0, ei,ej = −ej,ei = δi,j. The group Sp2n is the (clearly closed) subgroup of GL2n which preserves this form. If J is the n × n matrix J = 1 ... 1 and J = J −J , we have v,v = tvJv ; thus g is symplectic if and only if tgJg = J. The torus T = {diag(t1,. . . ,tn,t−1 n ,. . . ,t−1 1 )} is a maximal torus: it is in a unique maximal torus T1 of GL2n, the torus of diagonal matri- ces, since T1 = CGL2n (T) and T consists of all symplectic diagonal matrices. The symplectic upper triangular matrices form a subgroup B consisting of the matrices B BJ S 0 J tB−1J , where B is upper triangular and S is symmetric. The group B is connected since it is the product of the connected varieties of the up- per triangular and the symmetric matrices. It is solvable since it is a subgroup of the upper triangular matrices of GL2n. Thus it is contained in some Borel subgroup of GL2n; that is, in the stabiliser of a complete flag. But the only line stable by B is e1, and by induction, considering e⊥ 1 /e1, the only flag sta- bilised by B is the flag e1 ⊂ e1,e2 ⊂ · · · ⊂ e1,. . . ,en ⊂ e1,. . . ,en,en ⊂ · · · ⊂ e1,. . . ,en,en,. . . ,e1. Thus the only Borel subgroup of GL2n containing B is the group B1 of (invertible) upper triangular matrices; thus B is a Borel subgroup of Sp2n, since any larger solvable connected subgroup would be in B1, and B = B1 ∩ Sp2n. The group B is the stabiliser of a complete (that is, maximal) flag of isotropic subspaces e1 ⊂ e1,e2 ⊂ · · · ⊂ e1,. . . ,en. By conjugation in Sp2n, we see that any stabiliser of a complete isotropic flag is a Borel subgroup. We deduce that Sp2n is connected: it is enough to see that every element is in a Borel subgroup. Any g ∈ Sp2n has at least one eigenvector x ∈ V; as g is symplectic, it induces a symplectic automorphism h of V1 = x⊥/x (which is naturally endowed with a symplectic form induced by the initial form on V). By induction on dim V, we may assume that Sp(V1) is connected, and thus h is in some Borel subgroup of Sp(V1); that is, it stabilises a complete isotropic flag of V1. The inverse image in V of this flag, completed by x, is a complete isotropic flag of V stabilised by g, whence the result. The induction starts with the group Sp2, which is equal to SL2, hence connected.
- 29. 1.5 Examples of Reductive Groups 17 The permutation (1,2n)(2,2n − 1) . . . is symplectic and conjugates B to the symplectic lower triangular matrices; thus by the same argument as for GLn, we find that Sp2n is reductive. The centre of Sp2n is formed of the scalar symplectic matrices, which are only I2n and − I2n; thus Sp2n is semi-simple (see also the end of Example 2.3.14 for a computation with roots). Example 1.5.5 The orthogonal and special orthogonal groups. We will assume the characteristic of k different from 2. On V = kn (n ≥ 1) with or- dered basis (e1,. . . ,en), we define the symmetric bilinear form ei,ej = ⎧ ⎪ ⎨ ⎪ ⎩ 1 if i + j = n + 1, 0 otherwise. The group On is the (closed) subgroup of GLn which preserves this form. Simi- larly as for the symplectic group, we consider the n × n matrix J = 1 ... 1 ; we have v,v = tvJv , and g is orthogonal if tgJg = J. A diagonal matrix diag(t1,. . . ,tn) is orthogonal if and only if titn+1−i = 1 for all i. These matrices form a torus if n is even. If n is odd, they verify t(n+1)/2 = ±1 and form a non-connected group whose identity component is defined by t(n+1)/2 = 1. Using the same argument as for the symplectic group, we get that this identity component is a maximal torus of On. The orthogonal upper triangular matrices form a subgroup B; if n = 2m is even, it consists of the matrices B BJ A 0 J tB−1J , where A and B are matrices of size m × m with B upper triangular and A antisymmetric, and J is the matrix defined in Example 1.5.4. If n = 2m + 1 is odd, B consists of the matrices B −BJ tw BJ A 0 1 w 0 0 J tB−1J , where A, B, and J are the same as in the even case and w is a row vector of size m. The group B is connected since it is the product of the connected varieties of the upper triangular matrices and of antisymmetric matrices, and in addition, in the odd case, the product by Gm a . By the same argument as for the symplectic group, we get that B is a Borel subgroup of On. In the even case, the group B is also the stabiliser of the maximal flag of isotropic subspaces e1 ⊂ e1,e2 ⊂ · · · ⊂ e1,. . . ,en/2. By conjugating in On, we see that any stabiliser of a complete isotropic flag is a Borel subgroup. A similar characterisation can be made in the odd case, since B is then the
- 30. 18 Basic Results on Algebraic Groups identity component of the stabiliser of the maximal flag of isotropic subspaces e1 ⊂ e1,e2 ⊂ · · · ⊂ e1,. . . ,e(n−1)/2. The matrix equation defining On shows that the determinant of an orthogo- nal matrix has to be ±1. There are orthogonal elements of determinant −1, for example, if n ≥ 2, 1 In−2 1 . Hence the group On is not connected. We define the special orthogonal group SOn as On ∩ SLn. The proof that SOn is connected goes along the same lines as for Sp2n; see Exercise 1.5.6 below. In the even case, the induction starts with SO2, which is the one dimensional torus consisting of the matrices a 0 0 a−1 . In the odd case, the induction starts with SO1 = {1}. The permutation (1,n)(2,n − 1) . . . is orthogonal and conjugates B to orthogonal lower triangular matrices; thus, as in the above examples, we deduce that SOn is reductive. The centre of SOn is formed of the scalar orthogonal ma- trices, which are {± In} if n is even and In if n is odd; thus SOn is semi-simple (see the end of Examples 2.3.15 and 2.3.16 for a computation with roots). Exercise 1.5.6 Show that any g ∈ SOn has an isotropic eigenvector x and that the restriction of g to V1 = x⊥/x is in SO(V1). (Hint: decompose V = x ⊕ V1 ⊕ y with y ∈ V − x⊥ and x⊥ = x ⊕ V1.) Deduce the connectedness of SOn by induction on dim V. Notes There are a number of books we recommend that provide a good exposition of the basic theory of algebraic groups; for instance, the “classical” books of Borel (1991), Springer (1998), or Milne (2017), or the less complete but more pedagogical approach of Geck (2003).
- 31. 2 Structure Theorems for Reductive Groups We give in this chapter the main structure theorems for reductive groups. In order to do that, we first recall the definition and some properties of Cox- eter groups and root systems (refer to 2.3.1(i) and (iii) to see why root systems appear in this context). 2.1 Coxeter Groups Let W be a group generated by a set S of elements left stable by taking inverses. Let S∗ be the free monoid on S; that is, the set of words on S (finite sequences of elements of S). Let w ∈ W be the image of s1 . . . sk ∈ S∗. The word s1 . . . sk ∈ S∗ is called a reduced expression for w if it has minimal length among the words represent- ing w; we then write l(w) = k. We call l the length on W with respect to S. We assume now the set S which generates W consists of involutions; that is, each element of S is its own inverse (and is different from 1). Notice that reversing words is then equivalent to taking inverses in W. For s,s ∈ S we will denote by Δ(m) s,s the word ss ss . . . m terms . If the product ss has finite order m, we will just write Δs,s for Δ(m) s,s ; then the relation Δs,s = Δs ,s holds in W. Writing the relation (ss )m = 1 this way has the advantage that transforming a word by the use of this relation does not change the length – this will be useful later. Relations of this kind are called braid relations because they define the braid groups, which are groups related to the Coxeter groups but also have a topological definition. Definition 2.1.1 A pair (W,S) where S is a set of involutions generating the group W is a Coxeter system if s ∈ S | s2 = 1,Δs,s = Δs ,s for s,s ∈ S with ss of finite order is a presentation of W. 19
- 32. 20 Structure Theorems for Reductive Groups We may ask if a presentation of the above kind always defines a Coxeter system. That is, given a presentation with relations Δ(m) s,s = Δ(m) s ,s , is m the order of ss in the defined group and is the order of s equal to 2? This is always the case – see Bourbaki (1968, V, §4.3); but it is not obvious. If (W,S) is a Coxeter system, we say that W is a Coxeter group and that S is a Coxeter generating set of W. According to, for example, Bourbaki (1968, V, §4.4 corollaire 2) a Coxeter group has a faithful representation in which the elements of S act as reflections (see 2.2.1), so we also call the elements of S the generating reflections of W, and the W-conjugates of elements of S the reflections of W, their set being denoted by Ref(W). Characterisations of Coxeter Groups Theorem 2.1.2 Let W be a group generated by a set S of involutions. The following are equivalent: (i) (W,S) is a Coxeter system. (ii) There exists a (unique) map N from W to the set of subsets of Ref(W), such that N(s) = {s} for s ∈ S and for x,y ∈ W we have N(xy) = N(y)+̇y−1N(x)y, where +̇ denotes the symmetric difference of two sets (the sum modulo 2 of the characteristic functions). (iii) (Exchange condition) If s1 . . . sl(w) is a reduced expression for w ∈ W and s ∈ S is such that l(sw) ≤ l(w), then there exists i such that sw = s1 . . . ŝi . . . sl(w), where ŝi denotes a missing term. (iv) W satisfies l(sw) l(w) for s ∈ S, w ∈ W, and (Matsumoto’s lemma) two reduced expressions of the same word can be transformed into each other by using just the braid relations. Formally, given any monoid M and any morphism f : S∗ → M such that f (Δs,s ) = f (Δs ,s) when ss has finite order, then f is constant on the reduced expressions of any w ∈ W. We will see along the way that |N(w)| = l(w). Note that (iii) could be called the “left exchange condition”. By symmetry there is a right exchange condition where sw is replaced by ws. Proof We first show that (i)⇒(ii). The definition of N may look technical and mysterious, but the intuition is – see 2.2.11(i) below – that W has a reflection representation permuting a set of “root vectors” (there are two opposite roots attached to each reflection), that these roots are divided into positive and nega- tive by a linear form which does not vanish on any root, and that N(w) records the reflections whose roots change sign by the action of w. Lemma 2.1.3 If N is as in (ii), then N(s1 . . . sk) = {sk}+̇{sk sk−1}+̇ · · · +̇{sksk−1...s2 s1}.
- 33. Discovering Diverse Content Through Random Scribd Documents
- 34. ELIZABETH CROWNED Elizabeth of Hungary, a widow at the age of twenty, was sought in marriage by Frederick II., Emperor of Germany. She, having taken a vow never to marry again, declined his offer, and devoted her life to deeds of kindness and charity. She died at the age of twenty-four, and was canonized as a saint by Gregory IX. At this ceremony Frederick placed upon her head a golden crown, saying, Since thou wouldst not be crowned as my Empress, I crown thee to-day as an immortal Queen in the kingdom of God. When once I saw thee, fair, yet sad and lone,— Tho wealth and beauty waited at thy hand— I would have crowned thee, saintly one, mine own; Glad would have had thee share with me my throne, Bride of my heart, and Empress of my land! But thou wert wedded to thy valiant dead, And to the service of a Christ-like love; So by thy hand the suffering poor were led, And from thy bounty were the hungry fed, Till came thy summons to the Court Above. Now hast thou passed from tears and pain away, Thine ear hath caught the heavenly melodies;— So be it mine, with reverent touch, to-day, On thy fair head this diadem to lay, And crown thee Queen immortal for the skies!
- 35. WHO IS SUFFICIENT? Six-and-thirty little mortals Coming to be taught; And mine that most delightful task To rear the tender thought. Merry, mischief-loving children, Thoughtless, glad and gay, Loving lessons—just a little, Dearly loving play. Six-and-thirty souls immortal, Coming to be fed; Needing food convenient for them, As their daily bread. Bright and happy little children, Innocent and free, Coming here their life-long lessons Now to learn of me. Listen to the toilsome routine, List, and answer them, For these things who is sufficient 'Mong the sons of men? Now they, at the well-known summons, Cease their busy hum; And, some with pleasure, some reluctant, To the school-room come. Comes a cunning little urchin With defiant eye, Making music with his marbles As he passes by.
- 36. But, alas! the pretty toys are Taken from him soon, And the music-loving Willie Strikes another tune! Comes a lisping little beauty, Scarce five summers old; Baby voice and blue eyes pleading, Please, misth, I'm stho cold! Little one, the world is chilly, All too cold for thee; From its storms Our Father shield thee, And thy refuge be. While I turn to caution Johnny Not to make such noise; Mary parses: Earth's an adverb, In the passive voice. Well, indeed, it must be passive, Else it is not clear How such open language-murder, Goes unpunished here. Second Reader Class reciting— Lesson verse or prose? None in all the class is certain; Each one thinks he knows. Well, is queried then, the difference Who can now define? Answers Rob: In verse they never Finish out the line! Boy, thy thought doth strangely thrill me, And as hours roll on, Hears my heart a solemn query: Is my day's work done?
- 37. Do I make of this my life-task Prose or idle rhyme? Do I in the sight of Heaven Finish out the line? Oh, it is too fine a knowledge For our mortal sight, All these restless little creatures How to lead aright. He who prayeth while he worketh, Taking lessons still Of the Friend of little children, Learning all His will; He alone can walk before them Worthily and well; He alone of life's strange language Can the meaning tell. May I then with heart as tender As a little child Lead my flock; and Father, keep them Pure and undefiled.
- 38. PEACE O blessed peace, that floweth like a river, Unstayed, unwearied, ever on and on; That hath its fount and spring in Christ the giver, And finds its ocean round the great white Throne. O peace of God, that passeth understanding, Thou art the answer to my soul's long quest; Doubts, fears and sins, their serried hosts disbanding, I leave, launch on thy wave, and anchored, rest.
- 39. BOYS AND GIRLS We were seven in all, as the dear rustic maid To the poet so sweetly protested; And together we rambled and studied and played, Each imbibing a share of the sunshine and shade Wherewith our young life was invested. And black eyes and blue eyes and brown eyes and gray Looked up to the face of our mother, As she led us in study in labor or play, Or told of Our Father, and taught us to pray, And to cherish and love one another. O, the rapture of being when life is a-tune With the song-life and beauty of morning; When the roseate dawn brightens into the noon, And the year hastens on to the splendor of June, In her fragrance and matchless adorning. So our years flitted by and the youngest of all— Our dark-eyed and fun-loving brother— Was grown to be manly and lithesome and tall, And to couteous titles we answered the call, But were still boys and girls to each other. O, the joy of endeavor, endurance and toil On thro' summer-time vigor and sweetness, Of triumph o'er that which would hinder or foil, Of the patience of hope after tears and turmoil, In the glory of autumn's completeness. And the toil and the turmoil and tears have been ours—
- 40. From our ranks we have missed a loved brother We've encountered the thorns, but we've cherished the flowers; We've passed under the clouds on to sunnier hours, And we're still boys and girls to each other.
- 41. A SMILE The gliding of a fairy form And rosy lips that knew no guile, With wonder parted, came to ask, Papa, what is a smile? A smile, whate'er it is, then stole That gentle parent's features o'er; For ne'er to him had been proposed Query so strange before. But while he pondered in his heart How he should to his child reply, A new, triumphant joy lit up Her loving, lustrous eye;— And with this gladsome, new-found thought, She answered in her own behalf: Oh, now, I know; a smile must be The whisper to a laugh!
- 42. A SPARROW ALONE ON THE HOUSETOP Sing, little sparrow, sing thy song. No peril neareth thee; Tho night be dark or day be long, Or clouds hang low, sing on, sing on, The dear God heareth thee. Sing, little bird, whate'er befall— Trill out thine utmost need; Thou canst not soar, thou canst not fall But He will note who knoweth all, And He thy plaint will heed. O little sparrow, far and high Thy soft notes God-ward go, And I with thee send up my cry, And both shall somewhere find reply, God careth for us so.
- 43. TO MOTHER O mother, from thy home beyond the stars Hast thou not known the yearning of thy child For thy sweet love? Hast thou not heard her wild And piteous moaning for thy soft caress? Felt her heart's aching for the tenderness And the low patience of thy loving voice? Hast thou not seen her 'mid life's toils and jars, Pant as a bird behind its prison bars, For freedom to fly forth and be with thee? And canst thou not, sweet mother, send reply? Oh, thro' the depths of glory, thro' the sky, Look for one moment down and say to me That all of loss on earth thou findest to be Great gain in heaven; that thou dost rejoice In all that was, and is, and shall betide At last to all; and that, in Him who died, Yet liveth evermore, I, too, shall see All discord blended into harmony; And that I, too, shall be, as thou art, satisfied.
- 44. PSALM CXXI INSCRIBED TO MY SISTER, R. S. B. Lift up thine eyes unto the hills; A pure and fragrant breath Is wafted from their purple tops,— The Heaven-sent breath of Faith. Lift up thine eyes unto the hills; Beyond their shadowy slope The Sun of Righteousness doth rise In roseate dawn of Hope. Lift up thine eyes unto the hills; Around, below, above, The holy sky is all aglow With the warm light of Love. Lift up thine eyes unto the hills;— Faith, Hope and Love are given To point from fading joys of earth, To endless joy of Heaven.
- 45. TO R. T. B. ON HER MARRIAGE DAY Sister, we know That God is good, and He hath led us on By pleasant ways or painful to this day. Our lives went on together until now. In childhood and in youth the same fond home Hath been our earthly refuge; the same Rock Our shelter when earth had no rest or shade. At the same fancy we have often smiled, For the same sorrow wept; and oft our souls, In mingling aspirations, have sent up The same thanksgiving, the same burning prayer. Yes, we have lived together; we have known The visible blending of the outward life Made real by the holier unison Of loving spirit and aspiring mind. The spells of joy have bound us—and of hope, And tears—which are the diamond links of love— Have made the chain of our affection strong. It may be thus no more; yet—God is good— I hush the moaning of my riven heart, And smile that thou art happy; and give thanks That thy sweet life, rejoicing, hath put on Its richest diadem, its crown of love. May the kind Father grant that crown to be All worthy of the wearer; may His smile Lend brightness to it ever; and at last, When it is laid with earthly robes away,
- 46. O may the infinite and eternal Love Rest like a glory on thy radiant brow.
- 47. ON NEW YEAR, 1897 TO G. D. AND S. F. B. God bless you thro' this bright new year, The first you spend together; Give peace and trust thro' cloudy days, Joy in its sunny weather. And may the days as days go by, Still richer seem and sweeter, And passing seasons make your lives In every good completer. There are not words to tell the love In which I could caress you; Your dear united names I breathe, And once more pray, God bless you.
- 48. TO ANNA ON HER SIXTEENTH BIRTHDAY Sixteen! and life to thee looks bright and fair;— A book unread, rose-tinted, golden edged, Encased in binding curious, costly, rare;— And all the years to be thou holdest pledged To give thee from its pages, day by day, Readings to cheer and bless the blithesome way. And life is such a volume, only thou, From garnered storage of the heart and mind, Must fill unwritten pages, and allow Fair pictures—of pure thought, of self resigned, Of kindly deeds—each new-made page to grace;— How blest if none thou, later, woulds't efface! Sixteen! A May-day in the path of life, A marvelous puzzle on the finger twirled; Sixteen again; a stir of earnest strife And toil and tumult in a restless world; Repeated still,—a patient, steadfast hold On good attained,—ripe fruit, and grain of gold. Sixteen once more! Serene in shade or sun, A brighter outlook now; existence grand! Content in hopes fulfilled, in victories won, Mingling with holier yearnings for that land, Whose o'er-flown radiance and whose surplus bliss Have been the glory and the joy of this.
- 50. A SONG OF TENS TO MARY At the tenth birthday all the world looks fair; The twentieth scarcely shades it with a care; At the third decade life soars grand and high; But with the fourth its heyday passes by. The fifth comes on,—a century's half is told; The sixth,—our little girl is growing old. Another half-score milestone passed, and then We've reached the allotted three-score years and ten. Years may be added; should they come to thee May Faith and Wisdom their companion be; Hope thy sure anchor; Peace with thee abide, And Love still be thy light at eventide.
- 51. JESSICA A gentleman once wrote of Elizabeth Fry: Her name has long been a word of beauty in our household. Make thy name a word of beauty, Like the lily pure and fair, From its perfumed cup exhaling Sweetest fragrance on the air. Make thy name a word of beauty Lustrous as the ocean pearl; Constant in life's loving service, Guileless through youth's mazy whirl. Make thy name a word of beauty, Radiant, steadfast, like a star; Shedding from a glowing center Love's effulgence near and far. Aye, we greet thee, rare-sweet maiden, (Make it evermore thy right), Jessica—our word of beauty, Lily, pearl, and star of light.
- 52. TRANSITION Out of the blindness and the night Into clear and constant light. Out of the weariness and pain Into everlasting gain. Out of the toil and durance hard Into rest and rich reward. Out of the doubting and distress Into certain blessedness. Out of the dusty lanes of care Into pastures green and fair. Out of the glaring desert sun To shades where cooling waters run. Out of the din of woe and wrong Into choral waves of song Out of the dwelling, worn and old, Into the city of pearl and gold. Where now, O Death, where is thy sting? Thou art the summons to the King. O Grave, where is thy victory? Thou art the gateway to the free!
- 53. TO A. H. B. A COMMENCEMENT GREETING With Portraits of Eminent Authors Dear Hallam, with this trifling gift Best wishes now I send thee; Through all thy future life may joy And grace and peace attend thee. May this the bright beginning be Of days love-crowned and royal; May griefs and faults and foes be few, Friends manifold and loyal. May gems from authors such as these Store well thy mental coffer, But for thy heart's enrichment please Accept the love I offer. 1882
- 54. TO WINNIE ON HER WEDDING DAY Stars will shine on, tho thou art gone, But we shall miss the gleaming Of one bright eye's responsive smile, And love-light softly beaming. And flowers will bloom,—but we shall miss A fragrance and a beauty That brightened for us here and there The sombre path of duty. And friends will greet us on our way, But we shall miss the sweetness Of a fair presence that hath made So much of life's completeness. And yet 'tis well; we give thee joy, And pray with this caressing; That love and peace without alloy May be thy bridal blessing.
- 55. A LIFE WORK IN MEMORY OF DANIEL HILL He heard the cry of man enslaved In bonds and servile toil; And gave his voice for freedom till The Freedman tilled free-soil. He saw his weaker brother reel, Pierced by Drink's poisoned dart, And wrought and wrote with fervent zeal To stay the Tempter's art. He heard the clash of sword and gun In deadly battle-strife; And pleaded till his day was done For Love's sweet rule in life. He rests in peace. Who now shall wear The mantle he let fall? Who teach as he the Father-love, The brotherhood of all?
- 56. VISIONS I saw when Israel toiled and groaned beneath the Pharoah's rod, And in his hopeless bondage moaned his helpless prayer to God. I saw when from the river's brink the infant leader rose, Who, reared in Egypt's royal court, still felt his brothers' woes. I heard him at the burning bush his swift excuses bring: Who, who am I, that I should stand before the Egyptian king? And who am I that I should lead the people of thy choice? My warning word they will not heed, nor hearken to my voice. And who am I that I should move a monarch to relent? I, but a man, and slow of speech, nor wise, nor eloquent. I marked the answer: Plead no more thy vain excuse to me; I am the Lord; my servant thou; my glory thou shalt see. I am the Lord; the power is mine; 'tis thine to hear and do; The Lord almighty is to save, by many or by few. The man of doubt exchanged his fears for faith in God and right, While meek obedience on his brow sat like a crown of light. The slow of speech grew eloquent, till Israel gladly heard; And bolder waxed the Leader, till the king's hard heart was stirred, And he in fierce displeasure drove the captives from his land; Not knowing their deliverance was all divinely planned. Down the long line of two-score years I looked and saw at last, The blissful view from Pisgah's height; the Jordan safely passed;
- 57. And heard—as Memnon's harp had caught the sweet enchanting strain, And sent adown the waves of time brave Miriam's glad refrain— Sing, for the Lord hath triumphed; sing, great wonders can he do; The Lord is mighty and can save by many or by few. I saw again, when sin-enslaved, by Jabin's hand oppressed, A people's cry went up to God for rescue and for rest. Then up rose Deborah, judge and seer, with all her valiant band, And drove the oppressor from her gates, his chariots from her land. And Jael, wife of Heber, slew his captain with the sword; So woman's hand achieved that day the victory for the Lord. And woman's voice extolled in song the great Deliverer's name:— Praise God! He hath avenged His own, for willingly they came. The mountains melt before His face, the tribes their strength renew; The Lord is mighty and doth save by many or by few. I saw when Gideon led his band down to the water's bank To prove and set them in array, as man by man they drank, And with the handful chosen thus went forth against the foe, And vanquished all the Midian host, and laid their princes low. Not with the thousands called from far, who pitched by Harod's well; Nor yet the undismayed who stood when the faint-hearted fell; But Now, with these three hundred men, go forward, said the Lord; Do thou thy part, let them do theirs, trust, and obey my word. Their torches flashed like dancing flames, their trumpets loudly blew;
- 58. Strange warfare! but the Lord can save by many or by few. Once more I saw when Israel quailed before Philistia's pride; While great Goliath, day by day, Jehovah's power defied. The weak and timid fled away, the valiant shrank with fear;— 'Twas threatened death or dire defeat, and life and fame are dear. Even Saul, their chosen king, forgot (admiring Israel's boast!) That he stood head and shoulders high above his martial host. And are there none, he cried, who dare to meet this vaunting foe? And must the banner of our God trail in dishonor low? Then forth there came a ruddy youth: That banner I'll defend; Be it not said our God hath none on whom He may depend. Let no heart fail to-day because of this Philistine's boast; The battle is the Lord's and He will vanquish this proud host. Then spake he to the giant foe: A loyal servant I Of Israel's God, whose holy name thou darest to defy. In that dread name I charge thee stand, and shield thee as thou may; The fowls of air, the beasts of earth shall feast on thee to-day. 'Twas but a pebble from the brook, sent by a loyal will; But sword and spear not mightier were God's purpose to fulfil. For one may chase a thousand, and ten thousand flee from two; The God of right is strong to save by many or by few. * * * * * Years, ages pass and now I see a land beloved and fair; And lo! a cruel enemy hath gained possession there.
- 59. The riches of this goodly land into his coffers pour; Insatiate and unscrupulous, his constant cry is More! More money clinking in my till, more men—my licensed prey; More boys to feed my traffic when these men have passed away. Thus man is robbed of purse and soul, home of its peace and joy; The wife of husband is bereft, the mother of her boy. The land doth mourn. On every side the spoiler hath his way; No past oppression hath surpassed this vision of to-day. And who, like Moses, will exchange his self-distrust and fear For faith to meet the encroaching foe and check his bold career? And who, like Deborah, will arise and lead a valiant band To drive the Tyrant from her gates, the Traffic from her land? Who will, like Gideon and his men, the light of truth dare throw On darkest evil, and the trump of coming victory blow? Or who, like David, will come forth in God's great name, alone, And lay the boastful giant low, as once with sling and stone? When Avarice and unholy Pride against the good contend, The battle is the Lord's and He His people will defend. The great Red Sea of wrong, while He doth pass, shall stand aside; Mountains shall bow before Him, and proud Jordan's waves divide. Each epoch hath its burning bush, and each its palm-tree shade; And each its oak of Ophrah, where the pledge of peace is made. And each its fold, where kingly soul in shepherd guise is found; And when the Master calleth there the place is holy ground. Holy the place; but whose the hour? perchance He calleth thee, Or thee; who, who will answer now, Lord, here am I; send me?
- 60. O, for the love of land and home, make answer brave and true; Our God is mighty still to save, by many or by few.
- 61. BE YE ALSO READY Let us be still before Him. Yet once more That voice hath spoken to our startled souls Which fell in solemn cadence on the ear Of the hushed listeners on Mt. Olive's hill: At eventide, at midnight, or at morn, The Son of Man shall come, shall surely come; Be ready, for ye may not know the hour. And if at eventide, when Nature folds Her toil-spent hands and sinks into repose; Or if at midnight hour of gloom Thou come, Or when the morning spreads her wings of light, Oh make us ready for the solemn call. Supply our need, of knowledge, wisdom, grace, Dear Lord, that with confiding joy our souls, Made pure of sin and strong in faith, may go To meet Thee at Thy coming. If the sound Of sweet home-voices follow to the brink Of death's dark river, as they fainter grow, Then let us hear Thy still small voice of love; Say to us, It is I—be not afraid. Or if the angel of the icy hand Should find us when no human friend is near And summon us away, then as we lose Our hold of earth and fall away from life, O wilt Thou grant our parting spirits may Go out in silence and be found with Thee.
- 62. MIMOSA A modest plant; soft shades of green In leaflets poised on slender stem; And all outspread to catch the glow Of morning sun or dew-drop gem. But, lo, what change! When finger-tips But touch the leaflets' fringe, the charm Of life is gone—Mimosa shrinks, As conscious of some present harm. So would I have my soul recoil From touch of wrong or thought of sin; So throw its portals wide again, To let the dew and sunshine in.
- 63. AT THE CRISIS I.—THE STEAMBOAT BELLS When steamboats approach Mt. Vernon their bells begin to toll, and continue the mournful service until the sacred spot is again left in the distance. Mt. Vernon's shade sweet vigil keeps Where on her breast her hero sleeps; O passing bells, soft be your tone, Toll gently for our Washington. Toll, the great Warrior's strife is o'er; Toll, for the Statesman pleads no more; Toll—for a Man is fallen—on, Peal out your dirge for Washington. Toll for a people's wounded heart, Toll for a bleeding Nation's smart, Toll for a World!—toll sadly on— The world hath lost a Washington. Ring out your wailing on the air, And let it be a voice of prayer; He whom we greatly need is gone;— God give another Washington. 1863 Thus while she listened to the mournful knell That woke sad echoes on Potomac's shore; Saw how from Sumter's height her banner fell,
- 64. And heard, not distant far, loud battle's roar;— Thus, while she heard the impatient bondman's moan, Knew her own power defied, her trust betrayed; While Treason rose to hurl her from her throne— The Spirit of the Union mused and prayed. II.—THE EMANCIPATOR God gave another; while we stood Aghast before the coming flood Of war, and its attending woes, The one for whom she prayed arose. Blinded and deaf, we knew him not; Yet saw him wipe out slavery's blot; Heard him proclaim his people free, From lake to gulf, from sea to sea. Saw this and heard, but deaf and blind, We failed to recognize the Mind, Which, going on from strength to strength, From grace to grace, had grown at length, Thro the stern lessons of the hour, Of danger, censure, praise and power, To be the Man among us, one, Whom now we hail, since he is gone, Lincoln, our more than Washington. 1866
- 65. ON THE DEATH OF DR. JAMES E. RHOADS Fallen? No; his part was finished In the earthly toil and strife; He hath but lain his armor by, And entered into life. Silent? No; tho' hushed forever Tones that did like music thrill, Through example, helpful, holy, Lo, he speaketh still. Vanished? Lost to those that loved him? No; his spirit lingering near Still doth woo them, onward, upward, Whispering, Be of cheer. Crowned? Aye, crowned in earth and heaven; Here with laurels fairly won; There with star-lit diadem, Inscribed Well done! well done!
- 66. ETERNAL YOUTH Looking in thine eyes of azure, Looking on thy hair of gold, Once I wished, Evangelina, That there were no growing old. For I thought of how thy sweet eyes Would grow dim with tears and care; How the years would turn to silver All thy wealth of golden hair. How the lines of life would gather O'er the face so placid now; Traces of its toil and struggle Touching lip and cheek and brow. This I thought, and wished the shadows Might not lengthen o'er thy way; Wished there were no time but spring-time, Were no evening of the day. Now I fear, Evangelina, That my wish was half a prayer, That the listening Father heard me, That thou liest, an answer, there. For thou liest in thy beauty,— Eyes of blue and hair of gold, Lip and cheek and brow of marble, Folded fingers, still and cold;— O my angel, God hath called thee Where there is no growing old.
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