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Conference Board of the Mathematical Sciences
Regional Conference Series in Mathematics
Number 118
Hodge Theory,
Complex Geometry, and
Representation Theory
Mark Green
Phillip Griffiths
Matt Kerr
American Mathematical Society
with support from the
National Science Foundation

Digitized by the Internet Archive
in 2023 with funding from
Kahle/Austin Foundation
https://archive.org/details/nodgetheorycompl0000gree

CBMS
Regional Conference Series in Mathematics
Number 118
Hodge Theory,
Complex Geometry, and
Representation Theory
Mark Green
Phillip Griffiths
Matt Kerr
Published for the
by the
American Mathematical Society
Providence, Rhode Island
with support from the
National Science Foundation

NSF/CBMS Regional Conference in Mathematical Sciences: Hodge Theory,
Complex Geometry, and Representation Theory held at Texas Christian
University, Fort Worth, Texas, June 18-22, 2012
Research partially supported by
National Science Foundation Grant DMS-1068974
2010 Mathematics Subject Classification. Primary 14M15, 17B56, 22D10, 32G20, 32M10,
14D07, 14M17, 17B45, 20G99, 2245, 22E46, 22F30, 32N10, 32L25, 32Q28, 53C30.
For additional information and updates on this book, visit
www.ams.org/bookpages/CBMS-118
Library of Congress Cataloging-in-Publication Data
Griffiths, Phillip, 1938—
Hodge theory, complex geometry, and representaion theory / Mark Green, Phillip Griffiths,
Matt Kerr.
p. cm. — (CBMS Regional conference in mathematics ; number 118)
Includes bibliographical references and index.
ISBN 978-1-4704-1012-4
1. Hodge theory. 2. Geometry, Differential. I. Title.
QA564.G6335 2013
516.3/6 2013029739
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Acquisitions Department, American Mathematical Society,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
e-mail to reprint-permission@ams. org.
© 2013 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10987654321 18 17 16 15 14 13

Contents
Introduction ; il
Acknowledgement
Lecture 1. The Classical Theory: Part I 5
Beginnings of representation theory! LS
Lecture 2. The Classical Theory: Part II ie
Lecture 3. Polarized Hodge Structures and Mumford-Tate Groups and
Domains 31
Lecture 4. Hodge Representations and Hodge Domains 51
Lecture 5. Discrete Series and n-Cohomology 69
Introduction 69
Appendix to Lecture 5: The Borel-Weil-Bott (BWB) theorem 91
Lecture 6. Geometry of Flag Domains: Part I ; 95
Appendix to Lecture 6: The Gp- and Kc- orbit structure of D and the
Gp-orbit structure ofU 120
Lecture 7. Geometry of Flag Domains: Part II 147
Appendix to Lecture 7: The Borel-Weil-Bott theorem revisited 161
Lecture 8. Penrose Transforms in the Two Main Examples 165
Appendix to Lecture 8: Proofs of the results on Penrose transforms for D
and D’ 178
Lecture9. Automorphic Cohomology 191
Appendix I to Lecture 9: The K-types of the TDLDS for SU(2, 1)
and Sp(4) 209
Appendix II to Lecture 9: Schmid’s proof of the degeneracy of the HSSS
for TDLDS in the SU/(2,1) and Sp(4) cases 214
Appendix III to Lecture 9: A general result relating TDLDS and
Dolbeault cohomology of Mumford-Tate domains 218
Lecture 10. Miscellaneous Topics and Some Open Questions Dili
Appendix to Lecture 10: Boundary components and Carayol’s result 245
Bibliography 299
Index 303
il

iv CONTENTS
Notations used in the talks 307

Introduction
This monograph is based on ten lectures given by the second author at the
CBMS sponsored conference Hodge Theory, Complex Geometric and Representation
Theory that was held during June, 2012 at Texas Christian University, and on
selected developments that have occured since then in the general areas covered by
those lectures. The original material covered in the lectures and in the appendices
is largely on joint work by the three authors.
This work roughly separates into two parts. One is the lectures themselves,
which appear here largely as they were given at the CBMS conference and which
were circulated at that time. The other part is the appendices to the later lectures.
These cover material that was either related to the lecture, such as selected further
background or proofs of results presented in the lectures, or new topics that are re-
lated to the lecture but have been developed since the conference. We have chosen
to structure this monograph in this way because the lectures give a fairly succinct,
in some places informal, account of the main subject matter. The appendices then
give, in addition to some further developments, further details and proofs of several
of the main results presented in the lectures.
These lectures are centered around the subjects of Hodge theory and represen-
tation theory and their relationship. A unifying theme is the geometry of homoge-
neous complex manifolds.
Finite dimensional representation theory enters in multiple ways, one of which
is the use of Hodge representations to classify the possible realizations of a reduc-
tive, Q-algebraic group as a Mumford-Tate group. The geometry of homogeneous
complex manifolds enters through the study of Mumford-Tate domains and Hodge
domains and their boundaries. It also enters through the cycle and correspondence
spaces associated to Mumford-Tate domains. Running throughout is the analysis of
the Gp-orbit structure of flag varieties and the Gr-orbit structure of the complex-
ifications of symmetric spaces Gp/K where K contains a compact maximal torus.
Infinite dimensional representation theory and the geometry of homogeneous
complex manifolds interact through the realization, due primarily to Schmid, of
the Harish-Chandra modules associated to discrete series representations, especially
their limits, as cohomology groups associated to homogeneous line bundles. It also
enters through the work of Carayol on automorphic cohomology, which involves
the Hodge theory associated to Mumford-Tate domains and to their boundary
components.
Throughout these lectures we have kept the “running examples” of SL2, SU(2,1),
Sp(4) and SO(4, 1). Many of the general results whose proofs are not given in the
lectures are easily verified in the running examples. They also serve to illustrate
and make concrete the general theory.

2 INTRODUCTION
We have attempted to keep the lecture notes as accessible as possible. Both
the subjects of Hodge theory and representation theory are highly developed and
extensive areas of mathematics and we are only able to touch on some aspects where
they are related. When more advanced concepts from another area have been used,
such as local cohomology and Grothendieck duality from algebraic geometry at the
end of Lecture 6, we have illustrated them through the running examples in the
hope that at least the flavor of what is being done will come through.
Lectures 1 and 2 are basically elementary, assuming some standard Riemann
surface theory. In this setting we will introduce essentially all of the basic concepts
that appear later. Their purpose is to present up front the main ideas in the theory,
both for reference and to try to give the reader a sense of what is to come. At the
end of Lecture 2 we have given a more extensive summary of the topics that are
covered in the later lectures and in the appendices. The reader may wish to use
this as a more comprehensive introduction. Lecture 3 is essentially self-contatined,
although some terminology from Lie theory and algebraic groups will be used.
Lecture 4 will draw on the structure and representation theory of complex Lie
algebras and their real forms. Lecture 5 will use some of the basic material about
infinite dimensional representation theory and the theory of homogeneous complex
manifolds. In Lectures 6 and 7 we will draw from complex function theory and, in
the last part of Lecture 6, some topics from algebraic geometry. Lectures 8 and 9
will utilize the material that has gone before; they are mainly devoted to specific
computations in the framework that has been established. The final Lecture 10 is
devoted to issues and questions that arise from the earlier lectures.
We refer to the end of Lecture 2 for a more detailed account of the contents of
the lectures and appendices.
As selected general references to the topics covered in this work we mention
e for a general theory of complex manifolds, [Cat1], [Ba], [De], [GH], [Huy]
and [We];
e for Hodge theory, in addition to the above references, [Cat2], [ET], [PS], [Vol],
[Vo2};
e for period domains and variations of Hodge structure, in addition to the refer-
ences just listed, [CM-SP], [Cal;
e for Mumford-Tate groups and domains and Hodge representation [Mol1]}, [Mo2],
[GGK1]1] and [Rol];
e for general references for Lie groups [Kn1] and for representation theory [Kn2];
specific references for topics covered in Lecture 5 are the expository papers
[Sch2], [Sch3];
e for a general reference for flag varieties and flag domains [FH W)]; [GS1] for an
early treatment of some of the material presented below, and [GGK2], [GG1]
and [GG2] for a more extensive discussion of some of the topics covered in this
monograph;
e for a general reference for Penrose transforms [BE] and [EGW]; [GGK2],
GG1| for the material in this work;
e for mixed Hodge structures [PS] and [ET], for limiting mixed Hodge structures
CKS1], [CKS2], and [KU], [KP1] and [KP2] for boundary components of
Mumford-Tate domains;
e for the classical theory of Shimura varieties from a Hodge-theoretic perspective
Ke2].

ACKNOWLEDGEMENT 3
Acknowledgement
It is a pleasure to thank Sarah Warren for doing a marvelous job of typing this
manuscript.

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LECTURE, 1
The Classical Theory: Part I
The first two lectures will be largely elementary and expository. They will
deal with the upper-half-plane H and Riemann sphere P! from the points of view
of Hodge theory, representation theory and complex geometry. The topics to be
covered will be
compact Riemann surfaces of genus one (= 1-dimensional complex tori)
and polarized Hodge structures (PHS) of weight one;
the space H of PHS’s of weight one and its compact dual P! as homoge-
neous complex manifolds;
the geometry and representation theory associated to H;
equivalence classes of PHS’s of weight one, as parametrized by [H, and
automorphic forms;
the geometric representation theory associated to P', including the real-
ization of higher cohomology by global, holomorphic data;
Penrose transforms in genus g = 1 and g 2 2.
Assumptions.
basic knowledge of complex manifolds (in this lecture mainly Riemann
surfaces);
elementary topology and manifolds, including de Rham’s theorem;
some familiarity with classical modular forms will be helpful but not
essential;!
some familiarity with the basic theory of Lie groups and Lie algebras.
Complex tori of dimension one. We let X = compact, connected complex
manifold of dimension one and genus one. Then X is a complex torus C/A where
A => {nim + 1272 }ny,no€Z Cc C
1The classical theory will be covered in the article [Ke1] by Matt Kerr in the Contemporary
Mathematics volume, published by the AMS and that is associated to the CBMS conference.
5

6 1. THE CLASSICAL THEORY: PART I
is a lattice. The pictures are
SS oan aS SS
Oy
Here 6; © 7 and d2 © 7 give a basis for H,(X, Z).
The complex plane C = {z = x + iy} is oriented by
dx dy = (4) dz Adz > 0.
We choose generators 7,72 for A with 7, A m2 > 0, and then the intersection
number
61-09 = +1.
We set Vz = H!(X,Z), V = Vz ®Q= H}(X,
Q) and denote by
Q:Ve®V>Q
Howh= aleias
the cup-product, which via Poincaré duality H,(X,Q) = H1(X,Q) is the intersec-
tion form.
We have
ym xc wm x closed 1-forms w
> Be modulo exact
( ) pr( ) { l-forms p=d¢ }
al
TER OGM) TNC
and it may be shown that
Hpr(X) = spang {dz, dz}.
The pairing of cohomology and homology is given by periods
| dz
5:au
and Il = (ey) is the period matrix (note the order of the 7;’s).
Using the basis for H!(X,C) dual to the basis 51, 5 for H,(X,C), we have
H*(X,C) = C? =column vectors
U WU
dz ie
I
We may scale C by z > Az, and then II = XII so that the period matrix should be
thought of as point in P? with homogeneous coordinates [2°]. By scaling, we may

1. THE CLASSICAL THEORY: PART I tf
normalize to have 7; = 1, so that setting r = m2 the normalized period matrix is
[1] where Imr > 0.
Differential forms on an n-dimensional complex manifold Y with local holo-
morphic coordinates 21,...,2, are direct sums of those of type (p,q)
yee Nat Ndzy, iNdz5, NEON AZ
——_
P q
Thus the C® forms of degree r on Y are
a
AvP (Y)= APA(Y).
Setting
H*°(X) = span{dz}
H®*(X) = span{dz}
we have
OPO a HVX
This says that the above decomposition of the 1-forms on X induces a similar
decomposition in cohomology. This is true in general for a compact Kahler manifold
(Hodge’s theorem) and is the basic starting point for Hodge theory. A recent source
is [Cat1].
From dz dz = 0 and (4) dz dz > 0, by using that cup-product is given in
de Rham cohomology by wedge product and integration over X we have
eva TC =0
tee C)= H19(X) @ H91(X)
iQ (H*°(X), H19(X)) > 0.
Using the above bases the matrix for @ is
0 -1
e=() 9)
OTL i= WO — 0)
iQ (UU,
11)= eTIQII > 0.
and these relations are
For II = [7] the second is just Im7 > 0.

DEFINITIONS. (i) A Hodge structure of weight one is given by a Q-vector space
V with a line V!° C VW satisfying
Ve = yio ran)youl
Volav.
(ii) A polarized Hodge structure of weight one (PHS) is given by the above together
with a non-degenerate form
Q:VeaV-Q, Q(v,
v')= —Q(v’,v)
satisfying the Hodge-Riemann bilinear relations
OV V2) =
iQ(V19,V'") >0.
In practice we will usually have V = Vz ®Q. The reason for working with Q
will be explained later.
When dimV = 2, we may always choose a basis so that V = Q? = column
vectors and Q is given by the matrix above. Then V!° = C is spanned by a point
[7]€PVe =P’
Identification. The space of PHS’s of weight one (period domain) is given by
the upper-half-plane
pe 8ae inom
The compact dual H given by subspaces V!° C Ve satisfying Q(V1°, V1°) = 0
(this is automatic in this case) is H = PVc & P! where
P! = {r-plane} U oo = lines through the origin in C?.?
It is well known that H and P! are homogeneous complex manifolds; i.e., they
are acted on transitively by Lie groups. Here are the relevant groups. Writing
isis oti)
and using Q to identify A2V with Q we have
Q(z,w) =*wQz=zAw
and the relevant groups are
‘are Q)&SL2(R) for H
Aut(Vc,Q) =SL2(C) for P?.
In terms of the coordinate 7 the action is the familiar one:
Gir =-
cr +d
where Gg 4) € SL. This is because tT = z9/z1 and
a@ 0 f2o fazo+bz _ aT +b
C od) zi Je Net dei) 4 cred
2(CM-SP] is a general reference for period domains and their differential geometric proper-
ties. A recent source is [Ca].

1. THE CLASSICAL THEORY: PART I 9
If we choose for our reference point i € H (= [?] € P'), then we have the identifi-
cations
4 & SLo(R)/ $O(2)
P1& §L9(C)/B
where (this is a little exercise)
ad epedien K
come Uy|
B= {(° f)“(a d) =-b—eh
The Lie algebras are (here t = Q,R or C)
(= {(° * Javed}
so(2) = {
(? ov) izer}
sn {(e ance).
REMARK. From a Hodge-theoretic perspective the above identifications of the
period domain H and its compact dual H are the most convenient. From a group-
theoretic perspective, it is frequently more convenient to set
THU
Cai,
and identify H with the unit disc A c C Cc P!. When this is done, SL2(R) becomes
the other real form
Su(1, Re = {9z (: 1)€ SLo(C) : *gHg = H}
Prag Gaal
of SL2(R), where here H = ¢ Seal Then
ehh eaONS VAN
Beis of
Thus, for the A model SO(2) becomes a “standard” maximal torus and B is a
“standard” Borel subgroup.
We now think of H as the parameter space for the family of PHS’s of weight
one and with dim V = 2. Over H there is the natural Hodge bundle
eee
with fibres
OV aelinesine Vice
Under the inclusion H( < P!, the Hodge bundle is the restriction of the tautological
line bundle Op:(—1). Both V1° and Op:(—1) are examples of homogeneous vector
bundles.

In general, given
e a homogeneous space
Y=A/B
where A is a Lie group and B C A is a closed subgroup, and
e alinear representation r: B + Aut E where E is a complex vector space,
there is an associated homogeneous vector bundle
c= AxpE
a
eA
where A x pF is the trivial vector bundle A x E factored by the equivalence relation
(a,e) ~ (ab, r(b-*)e)
where a € A,e € E,b€ B. The group A acts on E by a: (a’,e) = (aa’,e) and there
is an A-equivariant action on E + Y. There is an evident notion of a morphism of
homogeneous vector bundles; then E > Y is trivial as a homogeneous vector bundle
if, and only if, r: B — Aut(£) is the restriction to B of a representation of A.
EXAMPLE. Let 7 € H C P! be the reference point. For the standard linear
representation of SL2(C) on Vc, the Borel subgroup B is the stability group of the
flag
(OV V7 ee Vex
It follows that there is over P! an exact sequence of SL2(C)-homogeneous vector
bundles
0 > Opi(—1) — V > Opi (1) — 0
where V = P! x Ve with g € SLe(C) acting on V by g- ((z],v) = ((gz], gv). The
restriction to H of this sequence is an exact sequence of SL2(R)-homogeneous bun-
dles
03 V9 V5 Vv! > 0.
The bundle V!° is given by the representation
cos@ —sin@ _, ei
sin@ cos
of SO(2). Using the form Q the quotient bundle V/V! := V°" is identified with
the dual bundle V1.
The canonical line bundle is
wp1 = Opi (—2).
Thus
w9¢ & (V10)8?.
Proof. For the Grassmannian Y = Gr(n, E) of n-planes P in a vector space E
there is the standard GL(F)-equivariant isomorphism
dpy = Homie. b Py
In the case above where E = C? and z = [2°] € P! we have
Te ~ Vee Q Ve/V;°

1. THE CLASSICAL THEORY: PART I 11
where V,)° is the line in Vc corresponding to z. If we use the group SL2(C) that
preserves Q in place of GLo(C), then
Veen Os
Thus the cotangent space
T*pl ~ y20
where in general we set V™° = (V1°)®"_ The above identification wp: © Op: (—2)
is an SLo(C), but not GL2(C), equivalence of homogenous bundles.
Convention. We set
1/2
Wee =.
The Hodge bundle V° + H has an SL2(R)-invariant metric, the Hodge metric,
given fibrewise by the 2"¢ Hodge-Riemann bilinear relation. The basic invariant of
a metric is its curvature, and we have the following
General fact. Let L — Y be an Hermitian line bundle over a complex mani-
fold Y. Then the Chern (or curvature) form is
il =
é1(D) = (=) 00
log ||s||?
271
where s € O(L) is any non-vanishing local holomorphic section and ||s||? is its length
squared.
Basic calculation.
1 dzAdy_ 1 dt
dr
4n oy? = Qa (Imr)?”
a(Vr) =
This has the following
Consequence. The tangent bundle
Tt oe
has a metric
|2 )
Re(dz dz)
iL
(Ima)?
of constant negative Gauss curvature.
Before giving the proof we shall make a couple of observations.
Any SL2(R) invariant Hermitian metric on is conformally equivalent to dx? +
dy”; hence it is of the form
dx? + dy?
h(x,y ( se)
(x,y) a
for a positive function h(z,y). Invariance under translation r + 7 +b, b € R,
corresponding to the subgroup (4°), implies that h(x,y) = h(y) depends only on
: 1/2
y. Then invariance under t — at corresponding to the subgroup ie wadvia)
a > 0, gives that h(y) = constant. A similar argument gives that c,(V'°) is a
constant multiple of the form above.
The all important sign of the curvature K may be determined geometrically
as follows: Let [ Cc SL2(R) be a discrete group such that Y = PH is a compact

Riemann surface of genus g 2 2 with the metric induced from that on H. By the
Gauss-Bonnet theorem
o>2-2=x(Y)=— f Kaa=K (SAC)).
An
PROOF OF BASIC CALCULATION. We define a section s € T(H, V+") by
s(r) = (7)ane:
1
The length squared is given by
Is(@)I? = a's(7)Qa(z) = 2y.
Using for T = x + iy
0,= Os = 10y)
O-= 5(Oz +10,,)
we obtain
ss 1 png 2
—00 = —-— dy.
a 00 ne (0; + 0,)dx A dy
This gives
hss 2_ 1 dxAdy
5 00log|Is(7) I ae ara
REMARK. There is also a SU(2)-invariant metric on Opi(—1) induced from the
standard metric on C?. For this metric
Isr)Ile= 1 + |r?
(the subscript c on || ||? stands for “compact”). Then we have
1 dx dy
An (1 + |r|?)2
a) =
Thus, V»° — H is a positive line bundle whereas V1:° — P? is a negative line
bundle with
deg Op: (—1) = i ce(Ve?) — lle
pl
This sign reversal between the SL2(R)-invariant curvature on the open domain H
and the SU(2) (= compact form of SL2(C))-invariant metric on the compact dual
H = P' will hold in general and is a fundamental phenomenon in Hodge theory.
Above we have holomorphically trivialized Y!}° + H using the section
s(T) = (a),
We have also noted that we have the isomorphism of SL2(R)-homogeneous line
bundles
ORG = yao
Now w3¢ has a section dr and a useful fact is that under this isomorphism
ii Sa)

BEGINNINGS OF REPRESENTATION THEORY 13
The proof is by tracing through the isomorphism. To see why it should be true
we make the following observation: Under the action of Ge 8) € SL2(R), s(r)
transforms to
a b T meen Ce DN arte
(a) ) araghee
i.e., s(7) transforms by (crt + d)~!. On the other hand, using ad — bc = 1 we find
that ;
d art +b 9 dr
‘Nert+d) (er+d)?”
Thus s(7)* and dr transform the same way under SL2(R), and consequently their
ratio is a constant function on H.
Beginnings of representation theory?
In these lectures we shall be primarily concerned with infinite dimensional rep-
resentations of real, semi-simple Lie groups and with finite dimensional representa-
tions of reductive Q-algebraic groups. Leaving aside some matters of terminology
and definitions for the moment we shall briefly describe the basic examples of the
former in the present framework.
Denote by [(H,V™°) the space of global holomorphic sections over H of the
n‘® tensor power of the Hodge bundle, and by du(r) the SL2(R) invariant area form
dx A dy/y? on K.
DEFINITION. For n 2 2 we set
pt = {¥er(av"®): f w(r)lPay(r) < ooh.
KH
There is an obvious natural action of SL2(R) on '(H,V™°) that preserves the
pointwise norms, and it is a basic result [Kn2] that the map
SL2(R) > Aut(D7)
gives an irreducible, unitary representation of SL2(R).
As noted above there is a holomorphic trivialization of V9 + H given by the
non-zero section
Then using the definition of the Hodge norm and ignoring the factor of 2,
llo(r)
IP= y-
Writing
T)o(T)
pr) = fu(
| (7)|?du(r) = 5) (r)|?(Im 7)"~2dr
A df.
[
tweortann =(5) ffifomPann
Thus we may describe D;* as
{feT (,0%): ffLol +iy)PPu" Ade A dy < oof.
we have
4A general reference for this is [Kel].

For n = 1 we define the norm by
sup f _|fule+ iy) Pao.
y>0 —co
The spaces D> are described analogously using the lower half plane.
Fact ({Kn2]). The D* for n 2 2 are the discrete series representations of
SL2(R). For n =1, DF are the limits of discrete series.
The terminology arises from the fact that in the. spectral decomposition of
L?(SLo(R)) the D# for n 2 2 occur discretely.
There is an important duality between the orbits of SL2(R) and of SO(2,C)
acting on P'. Anticipating terminology to be used later in these lectures we set
e P! = flag variety SL2(C)/B where B is the Borel subgroup fixing 7 = [7];
e SL2(R) = real form of SL2(C) relative to the conjugation A > A;
e SO(2) = mazimal compact subgroup of SL2(R) (in this case it is SL2(R)M
B);
e H = flag domain SL2(R)/SO(2);
e SO(2,C) = complezification of SO(2).
We note that SO(2,C) = C*.
Matsuki duality is a one-to-one correspondence of the sets
{SL2(R)-orbits in P*} 4 {SO(2, C)-orbits in P'}
that reverses the relation “in the closure of.” The orbit structures in this case are
nae HH open SL2(R) orbits
Aiesinh
RU {oo} closed SL2(R) orbit
P1
{i,—i} open SO(2,
C) orbit
inte
a =i closed SO(2, C) orbits
The lines mean “in the closure of.”° The correspondence in Matsuki duality is
Hoi
Tis
RU {0} & P’{i, —i}.
Matsuki duality arises in the context of representation theory as follows: A
Harish-Chandra module is a representation space W for slo(C) and for SO(2, C) that
satisfies certain conditions (to be explained in Lecture 5). A Zuckerman module is,
°Matsuki duality for flag varieties is discussed in [FHW] and in [Sch3] where its connection
to representation is taken up.

BEGINNINGS OF REPRESENTATION THEORY 15
for these lectures, a module obtained by taking finite parts of completed unitary
SL2(IR)-modules. For the D+ the modules are formal power series
b=) /ag (7 —1)*dr 8/2.
k20
We think of these as associated to Gp-modules arising from the open orbit H. The
Lie algebra slo(C), thought of as vector fields on P!, operates on w above by the
Lie derivative, and SO(2,C) operates by linear fractional transformations.
Associated to the closed SO(2,C) orbit i are formal Laurent series
"by fa) @n/2
Pir pees
This is also a (slo(C),SO(2,C))-module. The pairing between SO(2, C)-finite vec-
tors, i.e., finite power and Laurent series, is
(db, 7) = Resi (wp,7).
There are also representations associated to the closed SL2(R) orbit and open
SO(2, C) orbit that are in duality (cf. [Sch3]).
There is a similar picture if one takes the other real form SU(1,1)p of SL2(C).
It is a nice exercise to work out the orbit structure and duality in this case.
We shall revisit Matsuki duality in this case, but set in a general context, in
Lecture 2.
Why we work over Q. Setting X, = C/A we say that X, and Xj, are
isomorphic if there is a linear mapping
ACC
with a(A) = A’. This is equivalent to X, and X,, being biholomorphic as compact
Riemann surfaces. Normalizing the lattices as above the condition is
7 Gr
+ 0 Cap
= ; € SL2(Z).
anes i 4 2(2)
Thus the equivalence classes of compact Riemann surfaces of genus one is identified
with the quotient space SL2(Z)K.
For many purposes a weaker notion of equivalence is more useful. We say that
X, and Xj, are isogeneous if the condition a(A) = A’ is replaced by a(A) © A’.
Then A’/a(A) is a finite group and there is an unramified covering map
XA = XN
More generally, we may say that X, ~ Xv, if there is a diagram of isogenies
Identifying each of the universal covers with the same C, we have A Cc A”, A’ c A”
and then
A®Q=A”" Q@Q=AN' OQ.
The converse is true, which suggests one reason for working over Q.

REMARK. Among the important subgroups of SL2(Z) are the congruence sub-
groups ae 46 i - (; i (mod ny
b.
Then P'(1) = SL2(Z). Geometrically the quotient spaces Mpy) := I'(N)H arise
as parameter spaces for complex tori X, plus additional “rigidifying” data. In this
case the additional data is “marking” the N-torsion points
X,(N) := (1/N)A/A & (Z/NZ)?.
When we require that an ismorphism X,(N) = X,(N) take marked points to
marked points the the equivalence classes of X,(V)’s are [(N)H.
Later in these talks we will encounter arithmetic groups [ which have compact
quotients.

LECTURE 2
The Classical Theory: Part II
This lecture is a continuation f the first one. In it we will introduce and
illustrate a number of the basic concepts and terms that will appear in the later
lectures, where also the formal definitions will be given.
Holomorphic automorphic forms. We have seen above that the equivalence
classes of PHS’s of weight one with dim V = 2 may be identified with SL2(Z)H.
More generally, for geometric reasons discussed earlier one wishes to consider con-
gruence subgroups I Cc SL2(Z) and the quotient spaces
Mr = DNL:
We make two important remarks concerning these spaces:
(i) The fixed points of y € T acting on H occur when we have a PHS
Vea eyo
left invariant by y € Aut(Vz,Q). Thus 7¥ is an integral matrix that lies in
the compact subgroup of SL2(R) which preserves the positive Hermitian form
iQ(V»°, Ve) It follows that ¥ is of finite order, so that locally there is a disc
A around 7 with a coordinate t on A such that
SG ee
for some integer m (in fact, m = 2 or 3). The map
3.= 1)"
then gives a local biholomorphism between A modulo the action of the group
{y'™} and the s-disc. In this way Mp is a Riemann surface. We define sections
of the bundle V™® over the quotient space {y*,k € Z}A of the disc modulo
the action of y to be given by y-invariant sections of V™° > A.
REMARK. It will be a general fact, with essentially the same argument as above,
that isotropy group of a general polarized Hodge structure that lies in an arithmetic
group is finite.
(ii) Mp will not be compact but will have cusps, which are biholomorphic to the
punctured disc A*. The model here is the quotient of the region
He mae es ew
by the subgroup Ip = {(4 7) : n € Z} of translations. Setting
2T1iT
Y Ree
we obtain a biholomorphism
Roa Oalalseu net
of the quotient space with a punctured disc.
U7

18 2. THE CLASSICAL THEORY: PART II
DEFINITION. A holomorphic automorphic form of weight n is given by a holo-
morphic section y € '(Mp, V™°) that is finite at the cusps.
These will be referred to simply as modular forms.! rs
en
We recall that wa, = VW, so that ee ~ y”™° and the sections of Wp
around the fixed points of I are defined as above. Thus automorphic forms of
weight n are given by
W(7) = fy(rar?
where fy(7) is holomorphic on 1 and satisfies
ar+6
i (25) = (er +d)" fy(7).
Around a cusp as above one sets g = e?™*7 and expands the resulting well-defined
function Fy,(q) = fy(7) in a Laurent series
Fy (q) = Ne Gnd.
By definition, the finiteness condition at the cusp is a, = 0 for n < 0.
From a Hodge-theoretic perspective there is a canonical extension V}° > A
of the Hodge bundle V!° — A* given by the condition that the Hodge length of
a section have at most logarithmic growth in the Hodge norm as one approaches
the puncture (cf. [Cat2]). Modular forms are then the holomorphic sections of
vy _, P that extend to holomorphic sections of V%° + TH. In this way they
are defined purely Hodge-theoretically.
Among the modular forms are the special class of cusp forms w, defined by the
equivalent conditions
> Srvc Ill?
dps< 00;?
e ap = 0;
e w vanishes at the origin in the canonical extensions at the cusps.
Representation theory associated to P!. It is convenient to represent P!
as the compact dual of A = SU(1,1)r/T. Thus
Blo (C= SU Tele.
At the Lie algebra level we then have
sei) we (3 =e Niece RI
ano ie i” wae cI
where slo(C) = su(1, 1)g + isu(1, 1)p via
a=a-+tia’
b= B+ if"
C=6+ 16.
‘We refer to [Kel] for a general discusssion of classical modular forms, and to [Ke2] for a
treatment of modular forms as they arise in the theory of Shimura varieties.
?This is not the usual condition, which involves the integral of fy over a horizontal path in
+H. We have used it in order to have a purely Hodge-theoretic formulation.

2. THE CLASSICAL THEORY: PART II 19
As basis for slo(C) we take the standard generators
pe Uipitted emilee soe
PSone th
h=CH, nt? =CX, n =CY
h is a Cartan sub-algebra and the structure equations are
Then setting
ep pare
(i y| = oy,
[X,Y] =H.
The weight lattice P are the integral linear forms on ZH C h. Thus P & Z with
(1,H) = 1. The root vectors are the eigenvectors X,Y of § acting on sl2(C), and
the roots are the corresponding eigenvalues +2, —2 viewed in the evident way as
weights. They generate the root lattice R C P with P/R = Z/2Z. The positive root
is +2 and
nt = span of positive root vector X
n =span of negative root vector Y.
For the Borel subgroup B = He 6s )}, which is the stability group of [9] € P?
corresponding to the origin 0 € A, the Lie algebra
b=Hhe@n .
We note that the roots are purely imaginary on the Lie algebra
{(e )+e
of the maximal torus T C SU(1, 1)r.
As is customary notation in representation theory we set
1
p= ri positive roots) = 1.
The Weyl group W acting on 6 is generated by the reflections in the hyperplanes
defined by roots; in this case it is just tid. One usually draws the picture of zt C 5
with the roots and weights identified. In this case it is 27it = R, P = Z, R = 2Z.
=e eles 00" Fd a2
where “2” is the positive root and W is generated by the identity and w where
W(L) = —zZ.
Given a representation
i pedBn (OMe Se
eM
boyah
where F is a complex vector space, the weights are the simultaneous eigenvalues of
r(h). In this case they are the eigenvalues of r(H). The standard representation is
given by E = C?. The weight vectors are the eigenvectors for r(h). For the standard
representation they are
1 0
sa) aaa)
with weights +1.

Any irreducible representation of SL2(C) is isomorphic to S” := Sym” F for
nm = 0, 1520... The picture of S? 1s
a x
a a
e e ® @ e @ e@ @ @
Ss eee Pp
ceeetan eee
Y Y
—n —n+2 n—2 n
where the dots represent the 1-dimensional weight spaces with weights —n, —n +
2,...,n —2,n. The actions on X and Y are as indicated. If we make the identifi-
cations
Zz e+
VAN wee
then
e S” = homogeneous polynomials F'(zo, z1) of degree n;
SOS 16) en Cas
e 2% is the highest weight vector.
As SL2(C)-modules we have
H°(Opi(n)) = S”.
Geometrically, since Op: (n) = Op:(—n)* we see that on each line L in C?, F(z, 20)
restricts to a form that is homogeneous of degree n. Thus
Fl, € Sym” L* = fibre of Opi(n) at L.
As a homogeneous line bundle
Opi (n) = SLo (C) XB Cc
a 0
where (s Bes ) € B acts on C by the character a”. With our convention above, the
differential of this character, viewed as a linear form on h, is the weight n.
With the notation to be used later we have
Opi (n) = 1b,
where the subscript on L denotes the weight, which is the differential of the char-
acter that defines the homogeneous line bundle.
By Kodaira-Serre duality
H' (Opi(—k — 2))” © H°(wp:(k)),
and using the isomorphism of SL2(C)-homogeneous line bundles
pl a Opi (—2)
H' (Op: (—k — 2))” & H°(Opi(k)) = S*.
Penrose transform for P'. One of the main aspects of these lectures will be
to use the method of Eastwood-Gindikin-Wong [EGW] to represent higher degree
sheaf cohomology by global, holomorphic data. We will now illustrate this for
H!(Op:(—k — 2)).
For this we set
W =P! x P!(diagonal).
Using homogeneous coordinates z = |2°] we have
W= {(z,w) € P! x P! : zw, — z1w0 F O}.

For simplicity of notation we identify A?C? = C and then have zAw = zw, —21W9.2
For calculations it is, as usual, convenient to work upstairs in the open set U in
C? x C? lying over W and keep track of the bi-homogeneity of a function defined
in U.
The correspondence space W has the properties
(A) Wis a Stein manifold (it is an affine algebraic variety);
(B) the fibres of the projection W > P! on the first factor are contractible
(they are just copies of C).
Under these conditions [EGW] showed that there is a natural isomorphism
(x) H?(Op:(m)) © Hf, (T(W, 2% (m)); dx).
As we will now explain, the RHS of (x) is a global, holomorphic object. A further
detailed explanation will be given in Lecture 7. We will explain “in coordinates”
what the various terms mean.
e (2 = sheaf of relative differentials on W;
e (Q%,d,) is the complex --- 3 0% “49715 ...;
e OF (m) = OF @o,, T* Opi (m) where 1*Opi(m) is the pullback bundle;
e Ly: Q%(m)); dx) is the de Rham cohomology arising from the
global sections of the above complex.
The relative forms are defined by
— eo Js *Ol q-1 q
02 = 04,/image {m Qn: @ OY, +08},
and d, is induced by the usual exterior differential d. We think of 7*Op:(m) > W
as a vector bundle whose transition functions are constant on the fibres of 7, and
then d, is well defined on sections of 7*Op:(m).
The pullback sheaf 7~!Op:(m) is the sheaf over W whose sections over an open
set Z C W are the sections of Op:(m) over 7(Z). We have an inclusion
a Opi (m) =i 1* Opi (m)
where the subsheaf 7~!Opi(m) is given by the sections of the bundle 7*Op:(m)
that are constant on the fibres of W > P?.
In coordinates (z,w) = (Zo, 21;Wo,w1) on U, 2% means that we mod out by
dz and dz,. Setting
We w dwo = wodwy
we have
. F(z,w) holomorphic in U and homogeneous
=i = ) ;
: D(W, — (m)) - {of degree m in z and of degree zero in w ;
e d, F(z, w) = Fu)(z,w)dwo + Fw, (z,w)du1.*
Using Euler’s relation
WoFw, + wikw, = 90
when F(z, w) is homogenous of degree zero in w we obtain
Fea F,
dyP(2,0)= ( |Wwe (=) w.
Wy Wo
3Thus our symmetry group is SL2(C) and not GL2(C).
4This equation is true for an F(z, w) with any bi-homogeneity in z, w.

For the reasons to be seen below, it is now convenient to set m = —k — 2. Then
G(z,w)Y :
Sree (=e GNuyee where G(z,w) is homogeneous of
degree zero in z and of degree k in w
THEOREM. Bvuery class in Hb, (T(W, 2%(—k—2))) has a unique representative
of the form
H(w)W
CAN
where H(w) is a homogeneous polynomial of degree k.
Discussion. Given en as above, we have to show that the equation
G(z,w)Y q F(z,w) H(w)
(zAw)kt2 ~~ * @nw)kt2 CARD ee
has a unique solution where F’ has degree zero in z and degree k+2 in w and H(w)
is as above. Using Euler’s relation woF
wy,+ Wifw, = (k+2)F gives
q F(z,w) N 20 Fog Zw) P21 Foy (2; w)e
* Gin) ee Aare
Then the equation to be solved is, after a calculation,
Zo Fw (Z, w) + 21 Fw, (Z,
w) = (Zowi — 21W0)G(z, w) + (2ow1 — 21W0) H(w).
We shall first show that a solution is unique; i.e.,
20 fy + 21 lw, = owi — 21W0)H(w) > Aw) =0.
Taking the forms that are homogeneous of degree one in 20, 21 gives
eae = w,H
Tue = —woff.
Applying 0,,, to the first and 0,,, to the second leads to
H ar wiy, = —H — Wow, -
Euler’s relation then gives that H(w) is homogeneous of degree —2, which is a
contradiction.°
It is an interesting exercise to directly show by a calculation the existence of a
solution to be above equation. On general grounds we know that this must be so
because the map
H(w)W
CaN
has been shown to be injective and dim H! (Op:(—k — 2)) =k+1=dimS*.
The map (**) has the following interpretation: Let PZ and P!, be P! with
coordinates z and w respectively. Then we have a correspondence diagram
Ww
PL pl,
(**) H(w) —
°One may wonder why the degree —2 appears, when all that is needed is degree —1. The
philosophical reason is that H} (Opi (-1)) = (0).

Setting O.,(a,b) = m* Op: (a) X77, Op: (b) and using the theorem of EGW we obtain
a diagram
H° (Op: (k)) Z > H" (Op: (k — 2))
ai al
g eat
Hon (TW, Q2(Ork)); dy,)—— Hal
OV 2 t= 2c |
where the isomorphism f
H® (Ops,(k)) > H1 (Op: (—k — 2))
is termed a Penrose transform. Letting SLo(C) act on W Cc P!, x P! diagonally
in the above correspondence diagram we see that P is an isomorphism of SL2(C)-
modules.
In fact, it is a geometric way of realizing in this special case the isomorphism
in the Borel- Wezl-Bott (BWB) theorem. The line bundle L_;_2 has weight —k — 2,
euavel ione fs (0)
—k= 2p =—k
—1
———
is regular in the sense that its value on every root vector is non-zero. Moreover
# {positive root vectors X with (-k —1,X) <0} =1.
For w € W as above
w(—-k—-—1)—-p=k+1-1=k.
The BWB states that for k 2 0, H4(Op:(—k — 2)) 4 0 only for q = 1, and that
this group is the irreducible SL2(C) module with highest weight w(—k—2+p)—p =
k. The Penrose transform P realizes this identification.
The general discussion of the BWB will be given in the appendices to Lectures
5 and 7, where the special role of the weight p and transformation w(j + p) — p,
where pu is a weight, will be explained.
Penrose transform for elliptic curves. The mechanism of the EGW theo-
rem and resulting Penrose transform will be a basic tool in these lectures. We now
illustrate it for compact Riemann surfaces of genus g = | and then shall do the
same for genus g > l.
For reasons deriving from the work of Carayol that will be discussed in the last
lecture, it is convenient to take our complex torus
FY = C/Oy
where F is a quadratic imaginary number field and Of is the ring of integers in F;
e.g., F = Q(/—d). We set
W=CxC with coordinates (z’, 2’)
and consider the diagram
OpW
si Se

where a € Op acts by @ in the first factor and by —a in the second. It may be
easily checked that OrW is Stein and the fibres of x’, 7” are contractible (they are
just C’s). Thus the EGW theorem applies to the above diagram.
We will describe line bundles Li + E’ and Li! + E”, where r is a positive
integer, and then shall define the Penrose transform to give an isomorphism
A(R) He Ly.
For this we let 6 be a complex number with
B+B=|al?
HGS yi Us
Sections of L’ — E’ are given by entire holomorphic functions 6} (z’) where
Qni ‘
6!(z' + @) = 6.(z') exp (a (a?+ iY)
0
These are theta functions viewed as sections of Li > E’ where
Li = x Op C
with the equivalence relation
Qri :
(2',€)~ (<+ @, exp (= (a:4 2 é))
Bo 2
om
DO
G 2 uO (z exp Guza dz’
0
gives a relative differential for 7” : OrW — E”, and the functional equation
p(6')(2! +a, 2!" — a) = p(6")(2!, 2") exp (Aree" i 8)
Then
shows that p(6’) has values in 7’”"(L".). Thus
p(6") € Hp(T(OrW, 2%.,,(L%,)); dx”) & H1(E”, L",)
and defines the Penrose transform alluded to above.
Q2nIr ft Al
The relative 1-form exp qe 2 dz’ plays the role of the form w in the P!-
case. As suggested above the notation has been chosen to align with Carayol’s work
which will be discussed in the last lecture and in the appendix to that lecture.
Penrose transforms for curves of higher genus. We let [ Cc SL2(R) be a
co-compact, discrete group and set
Aelia Koel Ot
Here we take 7’ as coordinate in H and 7 as coordinate in H. The perhaps myste-
rious appearance of H and the complex conjugate H will be “explained” when in
Lecture 6 we discuss cycle spaces associated to flag domains Gp/T where G is of
Hermitian type. We set W = 1 x H and consider the diagram

It is again the case that ['W is Stein and the fibres of 1,7’ are contractible. The
Penrose transform will be an isomorphism
yes Fe OX Tees).
In order to have Li, + X’ be a positive line bundle we must have k = —1,—2,... .
Then
Ly-2 = Le @wx
where L;, — X is negative since X = TH.
We let f(r’) € H°(X’, Li.) be a modular form of weight —k, given by a holo-
morphic function on 1 satisfying the usual functional equation under the action of
I’. We then set
plA(r'.7) = 400) (a dr,
This is a relative differential for TW — X, and the transformation formula under
Y= Sean €T given by
i ae or
7* ( ) drs tepaed)-a (rid) 3 ( % ) dr’
i
shows that we obtain a class
(apologies for the double appearance of I). It is a nice exercise to show that
p(f) #0, and since
Gime Le initia
Cs ee)
we see that the resulting map H°(X’, Li) + H!(X, L,—2) is an isomorphism.
Orbit structure for P!. The main groups we shall consider acting on P! are
@ Gc = SL2(C);
e K =SO(2) and its complexification Kc;
e Gr = SL2(R) = real form of Gc.
The compact real form G, = SU(2) also acts on P', but in these lectures we shall
only make occasional use of it. The complex group Gc acts transitively on P!, but
Kc and Gp do not act transitively and their orbit structure will be of interest. The
central point is Matsuki duality, which is
the orbits of Kc and Gp are in a 1-1 correspondence.
We have already mentioned this in Lecture 1; here we formulate it in a manner
that suggests the general statement. The correspondence is defined as follows: Let
z €P! and Gp: z, Kc- z the corresponding orbits. Then
Gr:-z and Ke:z are dual exactly when their intersection consists
of one closed K orbit.

The following table illustrates this duality.
Gr-orbits Kc-orbits
open a. 1 closed
Gp orbits H —1 Kc orbits
closed jee 2 open
Gp orbit {Reo) ae i} Ke orbit
Description of the material in the later lectures and the appendices.
We will now informally describe the content of the remaining lectures in this series.
The overall objective is to present aspects of the relationship between Hodge theory
and representation theory, especially those that may be described using complex
geometry. One specific objective is to discuss and prove special cases of recent
results of Carayol, and some extensions of his work, that open up new perspectives
on this relationship and may have the possiblity to introduce new aspects into
arithmetic automorphic representation theory, aspects that are thus far inaccessible
by the traditional approaches through Shimura varieties. Whether or not this turns
out to be successful, Carayol’s work is a beautiful story in complex geometry.
Lecture 3 will introduce and illustrate the basic terms and concepts in Hodge
theory. We emphasize that we will not take up the extensive and central topic
of the Hodge theory of algebraic varieties.© Rather our emphasis is on the Hodge
structures as objects of interest in their own right, especially as they relate to
representation theory and complex geometry.
The basic symmetry groups of Hodge theory are the Mumford-Tate groups,
and associated to them. are basic objects of the related complex geometry, the
Mumford-Tate domains, consisting of the set of polarized Hodge structures whose
generic member has a given Mumford-Tate group G. In Lecture 4 we will describe
which G’s can occur as a Mumford-Tate group, and in how many ways this can hap-
pen. The fundamental concept here is a Hodge representation, consisting roughly
of a character and a co-character. As homogeneous complex manifolds the corre-
sponding Mumford-Tate domains depend only on the co-character. This lecture
will explain and illustrate this.’
Lecture 5 is concerned with discrete series (DS) and n-cohomology. The cen-
tral point is the realization of the DS’s via complex geometry, specifically the L?-
cohomology of holomorphic line bundles over flag domains.® The latter may be
realized, in multiple ways, as Mumford-Tate domains and this will be seen to be
an important aspect in Carayol’s work. The realization described above is largely
the work of Schmid.? An important ingredient in this analysis is the description of
the L?-cohomology groups via Lie algebra cohomology, in this case what is termed
n-cohomology. We will discuss these latter groups in some detail as they will play
an important role in the material of the later lectures and the work of Carayol.
®There are many excellent references for this subject. Three such ones are [Vol], [Vo2] and
[PS]. More recent sources are [Cat1], [Cat2], [Ke2], [Ca], and [ET].
"(GGK1] and [Rol1] are references for this.
® Another realization due to Atiyah and Schmid [AS], is via L? solutions to the Dirac equation
on the associated Riemannian symmetric spaces. This realization has advantageous aspects, but
since in these lectures our primary interest is in the complex geometric aspects of Hodge theory
and representation theory we will not discuss it here.
°cf. [Sch1] and [Sch2] and the references cited therein.

The flag domains fall into two classes, the classical ones that fibre holomor-
phically or anti-holomorphically over an Hermitian symmetric domain, and the
non-classical ones for which this not the case. For this work it is the non-classical
ones that are of the primary interest.
Lectures 6 and 7 will take up the basic constructions and results in the geometry
of homogeneous complex manifolds that will play a central role in the remaining
lectures, as well as being a very interesting topic in their own right. The main
point is that associated to a flag domain there are complex manifolds — including
the dual flag variety D — in which the group Gp acts, albeit non-transitively,
and these capture aspects of the complex geometry that provide the basic tools
for understanding the cohomology of homogeneous line bundles over flag domains.
One of these, the cycle spaces, are classical and originated from Hodge theory and
which have been the subject of extensive study over the years, culminating in the
recent monograph |[FHW]. The other tool, the correspondence spaces, are of more
recent vintage [GG1] and in several ways may be the object that for this work best
interpolates between flag domains and the various associated spaces. Their basic
property of universality will be introduced and illustrated in these two lectures. We
emphasize that even though the cycle and correspondence spaces may be defined
for any flag domain D, it is the case when D is non-classical that the geometry is
particularly rich.
Lectures 8 and 9 will introduce and study the Penrose transforms, which among
other things allow one to relate cohomologies on different flag domains and on their
quotients by arithmetic groups. The main specific results here are the analysis of
Penrose transforms in the case when G = U(2,1) studied by Carayol in [C1], [C2],
[C3] and when G = Sp(4), which is a new case that is discussed in [@GK2] and in
[Ke3]. Using the Penrose transform to relate classical automorphic forms to non-
classical automorphic cohomology, we discuss how the cup-products of the images of
Penrose transform reach the automorphic cohomology groups associated to totally
degenerate limits of discrete series (TDLDS), which are the central representation-
theoretic objects of interest in these lectures. This result for U(2,1) is due to
Carayol and for Sp(4) will appear in [Ke3].
In the last Lecture 10 we discuss some topics that were not covered earlier and
some open issues that arise from the material in the lectures, together with some
new results that have appeared since the lectures were given, and which are related
to questions posed in the lectures. Among the topics covered is the study by Carayol
of cuspidal automorphic cohomology expanded about boundary components in the
Kato-Usui completion, or partial compactifications, of quotients of of Mumford-
Tate domains by arithmetic groups in the case of SU(2,1). This seems to be a very
interesting area for further work (cf. [KP1)).
Turning to the appendices, the appendix to Lecture 5 discusses the Borel-Weil-
Bott (BWB) theorem, which is the basic result relating complex geometry and the
finite dimensional representation theory of complex semi-simple Lie groups. We
recall Kostant’s n-cohomology interpretation of the BWB theorem, which through
the use of the Hochschild-Serre spectral sequence and the decomposition of a general
Harish-Chandra module into its K-types plays a central role in the analysis of the
n-cohomology of those modules.
The rather lengthy appendix to Lecture 6 contains descriptions, with illustra-
tive examples, of the Gp-orbit structures of D and U, and of the Kce-orbit structure

of D that is dual to the Gp one (cf. [FHW]). Included are computations of in-
trinsic Levi forms for both D and U. These are interesting in the case of Gr-orbits
in OD where D is a Mumford-Tate domain, since as discussed in the appendix to
Lecture 10 these will relate to boundary components given by limiting mixed Hodge
structures. This provides a further connection between Hodge theory and repre-
sentation theory “at the boundary,” a topic that we suspect may have significant
further developments.
The main objective of the discussion of the Gg-orbit structure of U is to give
a proof of the fundamental result in [BHH] that there exist strongly plurisubhar-
monic exhaustion functions modulo Gp on U. This result implies that for C Gr
discrete and co-compact the quotient [U is Stein, a result that is basic to the use
of Penrose transforms to study automorphic cohomology relating those groups be-
tween the classical and non-classical cases. Along the way we identify the tangent,
normal and CR-tangent spaces to Gp-orbits in U. This is done in [FHW)], but for
the computational purposes in the present work we have proceeded in a somewhat
different way.
In the appendix to Lecture 7 we revisit the Borel-Weil-Bott theorem in the
context of Penrose transforms. Specifically, the BWB theorem gives the various
geometric realizations, indexed by the Weyl group W, of the same irreducible Gc-
module as cohomology groups over the flag variety D = Gc/B. In this appendix we
show how these abstract isomorphisms between cohomology groups may be realized
geometrically by Penrose transforms. The analogue of this for flag domains, where
now the Penrose transform is between Harish-Chandra modules with the same
infinitesimal character realized as cohomology groups over flag domains given by
open G'p-orbits D and D’ in D and where the complex structures of D and D’ may
be inequivalent, is fundamental to Lectures 8 and 9. The point here is that the
infinitesimal character is an invariant of W whereas the inequivalent homogeneous
complex structures on Gp/T are indexed by W/W x, so that that Penrose transform
enables one to relate geometrically classical and non-classical objects.
There are three appendices to Lecture 9. In the first we give the K-types
for the totally degenerate limits of discrete series in our two running examples
SU(2,1) and Sp(4); this was used in the lecture where among other things the
mechanism underlying the degeneration of the Hochschild-Serre spectral sequence
was presented. It will also be used in the paper |Ke3] where the results of the
lecture and the appendix will be used in the proof of the analogue for Sp(4) of
Carayol’s cup-product theorem. For Sp(4) this result is particularly subtle because
it involves the interplay between the two inequivalent TDLDS’s.
In the second appendix to Lecture 9 we have given an exposition of Schmid’s
proof of the degeneration of the Hochschild-Serre spectral sequences for the TDLDS’s
in the SU(2,1) and Sp(4) cases. This is a particular illustration of the use of
Zuckerman tensoring and the Casselman-Osborne theorem in the computation of
n-cohomology. The third appendix applies Zuckerman tensoring and the work of
Schmid to obtain a general construction of TDLDS via Dolbeault cohomology of
line bundles on nonclassical Mumford-Tate domains.
The lengthy appendix to Lecture 10 has two purposes. One is to combine the
root-theoretic analysis of the Gp-orbits in OD with the theory of limiting mixed
Hodge structures (LMHS) to give an analogue of the realization of certain open Gp-
orbits in flag varieties as Mumford-Tate domains for polarized Hodge structures on

(g,B). The main point here is the analysis of the period-type map
aaa boundary
components

ae {Gp-orbits in OD}.
In doing this we introduce and discuss mixed Hodge structures and limiting mixed
Hodge structures, Kato-Usui boundary components (nilpotent orbits), and Kato-
Usui extensions, all of which will be used in the proof of Carayol’s result mentioned
above. The underlying point is that much of the theory discussed in these lec-
tures relating Hodge theory and representation theory should be extended “to the
boundary.” This would be an analogue of the well-known principle in algebraic
geometry that Hodge theory is frequently simpler and more tractable when an
algebraic variety degenerates to a singular one.
Of particular note here is the quite different behavior of the differential ®..,,
between the classical and non-classical cases. In the former it only detects the
associated graded to the limiting mixed Hodge structures, analogous to the Satake-
Borel-Baily compactifications, whereas in the non-classical case some — but not all
— of the extension data is captured by ®o,x.
We would like to call attention to the papers [KP 1] and [KP2], where among
other things the Mumford-Tate groups associated to nilpotent orbits are defined
and criteria given for when a Kato-Usui boundary component is “classical,” thereby
preparing the way to extend Carayol’s result to other situations, including a case
where the arithmetically interesting automorphic cohomology group is associated
to an automorphic representation of G2. Some of the discussion in the first part of
this appendix (and part of the appendix to Lecture 6) overlaps with part of what
is discussed in the above works.
A second purpose of the appendix is to discuss the proof of the result of Carayol
[C3] where for SU(2,1) automorphic cohomology is expanded about a Kato-Usui
boundary component. This gives an analogue of the expansion of classical auto-
morphic forms about a cusp and suggests a possible definition of arithmeticity for
automorphic cohomology in this case.

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LECTURE 3
Polarized Hodge Structures and Mumford-Tate
Groups and Domains
In general we will follow the terminology and notation from [GGK1]. An
exception is that there Mumford-Tate groups were denoted by M,, whereas here
they will be denoted by Gy.
In this lecture we will introduce and explain the following terms:
e polarized Hodge structures (PHS);
e period domains and their compact duals;
e Hodge bundles;
e Mumford-Tate groups;
e Mumford-Tate domains and their compact duals;
e CM polarized Hodge structures.
We will also introduce three of the basic examples for this lecture series.
We begin with a general linear algebra fact. We define the real Lie group
S = Resc/p
Gm = C* =R*°x g!
where C* = {z = re*®} is considered as a real Lie group. If V is a rational vector
space with Vg = V @g R and we have a representation (a homomorphism of real
Lie groups)
@:S— Aut(Ve)
satisfying @ : Q* > Aut(V), then we have
Gy y= eV or) on Vv (weight decomposition);
(ii) V@= @ VP4, Ver = VP,
prq=n
(izle 2°22 on V's (eigenspace decomposition).
The V” C V are subspaces defined over Q, and the V4 C V@ are the eigenspaces
for the action of G(S) on V¢’. In (i) n is the weight, and in (ii) (p,q) is the type.
There are three equivalent definitions of a Hodge structure of weight n.
DEFINITIONS. (I) Ve ® VP4I,VIP=VP4 (Hodge decomposition);
p+q=n
(II) (0) Cc
F®? c++} C F™ 1 CF" =Ve_ (Hodge filtration) satisfying for
each p
ian eG.
(III) @:S > Aut Ve of weight n.
31

32 3. PHS AND MUMFORD-TATE GROUPS
The equivalence of the first two definitions is
FP = @ yrr id
p'2p
VP4d = FP OF NS.
We have seen above that the V?*% are the eigenspaces of Y(S) acting on Vc, which
gives
r= Ae
We shall primarily use the third definition and shall denote a Hodge structure by
(V, g).
In general, without specifying the weight a Hodge structure is given by V and
@ :S — Aut(Vp) as above. The weight summands are then Hodge structures of
pure weight n. Unless otherwise stated we shall assume that our Hodge structures
are of pure weight.!
We define the Weil operator C on Ve by C(v) = y(t)v. Then C = 7?~4 on V4.
Hodge structures admit the usual operations
®, ®, Hom
of linear algebra. A sub-Hodge structure is given by a linear subspace V’ C V with
p(S)(Vg) C Vg. An important property of Hodge structures is that morphisms are
strict: Given
Oe Ve =v"
where V,V’ have weights r,r’ = n + r (r may be negative) and
(FP) CF Pt,
which is equivalent to
RV ee) Vara Cas
there is the strictness property
W(Ve) VBP = (FP).
That is, anything in the image of w that lies in F?*+” already comes from something
in F?, The property of strictness implies that Hodge structures form an abelian
category.
HODGE’S THEOREM. For X a compact Kahler manifold the cohomology group
H"(X,Q) has a Hodge structure of weight n.?
As remarked in the first lecture, the decomposition into (p,q) type of the C'®
differential forms
A™(X)= @® AP4(X)
pt+q=n
Av? (X) = APq(X)
1In the appendix to Lecture 10 we will discuss mixed Hodge structures. A direct sum of
polarized Hodge structures is a mixed Hodge structure, but not conversely.
Cf. [Cat1] for a recent treatment of this result.

3. PHS AND MUMFORD-TATE GROUPS 33
where
AD tix) — { > fra(z, Z)dz" naz’,
{1|=p
|J|=¢
and for I = (i1,...,%)) we have dz’ = dz" A--- A dz’», induces via de Rham’s
theorem the Hodge decomposition on cohomology.
An example of a different sort is given by
Tate Hodge structure Q(1). Here the Q-vector space is 277Q, the weight
n = —2 and the Hodge type is (—1, —1).
One sets Q(n) = Q(1)®” and V(n) = V @g Q(n) (Tate twist). Then
H*(C*,Q) = Q(-1) with generator SS
where for y = {|z| = 1} € H,(C*,
Q)
dz
Acco pas
y =
gives an isomorphism H;(C*,Q) ~ Q(1). In general, for Y C X a smooth hyper-
surface and
H"(Y,Q) > H"4?(X,Q)
the Gysin map, defined to be the Poincaré dual of the map on homology induced
by the inclusion and which is dual to the residue map (where the 277 comes in),
one has a morphism of Hodge structures of the same weight n + 2
H"(¥,Q(-1)) > H™+?(X,Q).
This is useful for keeping track of weights in formal Hodge theory.
For these lectures a main definition is the following
DEFINITION. A polarized Hodge structure (V,Q,y) (PHS) is given by a Hodge
structure py: S > Aut(Vp) of weight n together with a non-degnerate form
Q:VEV>Q, Qv,0') = (-1I)"Q(v',
v)
satisfying the Hodge-Riemann bilinear relations
Re Es
Cseee
Gem ass Fe
(II) Q(v, Cv) > 0, OAVE Ve.
These are equivalent to the more classical versions
HAVES VR = One PEP TP
PHO WVEt V 4) > 0:
A sub-Hodge structure V’ Cc V of a polarized Hodge structure is polarized by the
restriction
Olas
yk a
or Odo V7 fand setting V7 = V5 OQ” =O),
(ViQAWV OVO QO)
is a direct sum of PHS’s. As a consequence, PHS’s form a semi-simple abelian
category.

For polarized Hodge structures we set y = 9 |
gi and have the
PROPOSITION. y: S' + Aut(Vp, Q).
Proor. Q € V* @V* and by Hodge-Riemann (1) it has Hodge type (—1, —1).
O
In general for a Hodge structure of even weight n = 2m we define the Hodge
classes Hg,(V) to be those rational vectors of Hodge type (m,m). We will return
later to the resulting algebra of Hodge tensors
He"(V)= @ He(V® @v™®).
k=1(2)
An important observation is
Given a polarized Hodge structure (V,Q, y), Hom(V, V) = V*®
V has a polarized Hodge structure. Moreover, the Lie algebra
g = Homg(V,V) Cc Hom(V, V)
is a sub-Hodge structure.
For the Hodge decomposition we have
gc = Og"
where
got = {X € gc: X(V"4) C eA
We note that
[aet,gh 2] = gith— G42),
The case of Shimura varieties [Ke2], which includes PHS’s of weight n = 1, is
when
g”* =0 unless i = 0, +1.
Period domains and their compact duals ({CM-SP] and [Ca]). For a
Hodge structure (V, ~) of weight n we set
hP-4 = dim V?4 (= Hodge numbers)
fh ae eee.
DEFINITION. (i) A period domain D is the set of PHS’s (V,Q,y) with given
Hodge numbers h?4. (ii) The compact dual D is the set of filtrations F’ of Ve with
dim F? = f? and satisfying
OE aires
te 10)
The group Gr := Aut(Vp, Q) is a real, simple Lie group that acts transitively
on D. The isotropy group H of a reference PHS (V,Q, yo) preserves a direct sum
of definite Hermitian forms, and therefore it is a compact subgroup of Gp that
contains a compact maximal torus T. The following exercises give details.
EXERCISE. D = {p: S' > Gr: p= g~'og for some g € Gr}. That is, D is
the set of Gp-conjugacy classes of the circle yp : S' > Gp.

It follows that H = Z.,(Gr) is the centralizer in Gg of the circle yo(S!). The
centralizer of a circle in a real Lie group always contains a Cartan subgroup, which
is isomorphic to the identity component of a product of R*’s and S!’s. Since in our
case Z,(Gr) C H is compact only $!’s occur.
EXERCISE. For n = 2m+ 1 odd
He Chae Sedu Se ekg eae
is a product of unitary groups, and for n = 2m even
be eNO linc BSSUNG use
aay Sere un)
is a product of unitary groups and an orthogonal group.®
The group Gc = Aut(Vc,
Q) is a complex, simple Lie group that acts transi-
tively on D. The subgroup P in Ge that stabilizes a Fp is a parabolic subgroup
with
m= Ge Ne.
Usually we choose Fo to be Fp, where yo € D is a reference point.
Since the second Hodge-Riemann bilinear relations are strict inequalities, the
period domain is an open orbit of Gp acting on D. The orbit structure of Gp’s
acting on D’s will be one theme in Lectures 6 and 7.
EXERCISE. For n = 1 show that
DAH,
where dimV = 2g and Hy, Siegel’s generalized upper half space, is = {Z € Mgxq :
Z ='Z, ImZ > 0}. For the PHS associated to H1(X,Q) where X is a compact
Riemann surface of genus, the associated Z is the classical period matrix of X.
(Here we use Z instead of Q.)
EXERCISE. For n = 2 and h®:® = h, h?>? = 1 show that
Dee Gre POUR ry Ou Gr, (nc 1),
and that Gg acting on D has two open orbits, one of which is the period domain.
This is the case that arises in the period matrices of the 2"? primitive cohomology
of smooth algebraic surfaces.*
Hodge bundles. Over D these are the Gc-homogenous vector bundles
Rel)
whose fibre at a given point F is F?. Restricting to D C D we have the Hodge
bundles
Pod ;— RP /RPT)
These are homogeneous vector bundles for the action of Gg. Importantly, they are
Hermitian vector bundles with Gp-invariant Hermitian metrics given in each fibre
by the second of the Hodge-Riemann bilinear relations. Their general differential
geometric properties are discussed in [GS1] and in [CM-SP]. In Lecture 5 we will
discuss the special case of homogeneous line bundles.
3It is frequently convenient in the even weight case to take V to be oriented, so that Gp is
connected and the last factor is then SO(h™"™).
4We shall use the notation Gry (H@®) for the Lagangian Grassmannian of Q-isotropic h-
planes in a C* on which we have a non-degenerate symmetric form Q.

At a reference point y € D with the PHS on g described above, we have for
the Lie algebras hc of Hc and P
and the holomorphic tangent space
i) = ~ @ gi”.
v gc/P ers
We shall sometimes write g, and Gon when we wish to emphasize the circle
YQ: sis GR.
The real tangent space is the G-homogeneous vector bundle whose fibre of
T,y,rD at the reference point ¢ is
@ grt) 5
@ ae Ne
Setting T5'°D = T,D, we have
1,0 0,1
TryD ®C=T, DOT, D
where TD = POE This gives a Gp-invariant almost complex structure on
D, which is integrable by the bracket relations given above. The Hodge-Riemann
bilinear relations for gg induce a Gg-invariant Hermitian metric on D.
Mumford-Tate groups. These are the basic symmetry groups of Hodge the-
ory, encoding both the Q-structure on V and the complex structure (Hodge decom-
position) on Vc.
DEFINITIONS. (i) Given a Hodge structure (V, ~) the Mumford-Tate group Gg
is the smallest Q-algebraic subgroup Gg C GL(V) such that
P(S) C Gor.
(ii) Given a PHS (V,Q,y) the Mumford-Tate group Gy is the smallest Q alge-
braic subgroup G, C Aut(V,Q) such that
y(S*) e. Gor:
It may be shown, and we will explain why this should be so, that
Go = Get AutlV, @):
It is also the case that
Gg and Gy are reductive, Q-algebraic groups.
For Gy, we may see this as follows: If we have a G,-invariant subspace V’ Cc V,
then since (St) C Gy.p there is an induced action y’ of ($1) on Vg and therefore
(V’,y’) is a sub-Hodge structure. We have observed earlier that it is polarized by
(6 ee Ot and that setting (V",Q”,»") = (V’,Q’,y’)+,
(V,Q, 9) = (V',Q',¢') 8(V",Q",~")
is a direct sum of PHS’s. Then by minimality of the Mumford-Tate group, since
y(S") C Gy p x Gy p we have that G, C Gy x Gov; this inclusion is in general
strict. In particular, G, preserves the direct sum decomposition V = V’@V”.
°Here, and throughout, we shall use the notation (W)g = set of real points in a complex
vector space W on which there is a conjugation; that (W)p = {w € W,w = vw}.

We note that
Gy 1s a sub-Hodge structure of Home(V,V).
In case Gy is semi-simple, the polarizing form will, up to scalings, be induced by
the Cartan-Killing form of gy.
The extreme cases are
e y € Dis a generic point > G, = Aut(V,Q);
e G, C Hy = stability group of (V,Q, ~) > Gg is a Q-algebraic torus.
The second statement is a result whose proof will he given just before the next sec-
tion. When G¢ is an algebraic toruss (V, ¢) is by definition a complex multiplication
(CM) Hodge structure. If (V, ) is simple, i.e., it contains no non-trivial proper sub-
Hodge structures, then Homg(V, V) is a division algebra acting on (V,@). We shall
discuss more about CM PHS’s below.
EXAMPLE. Let X, = C/Z+7Z be as in the first lecture. Then
H'(X,,Q) is CM & 7 is a quadratic imaginary number.
Then L = Q(r) is a number field and Gg = L* is the group of units with G, being
those of norm one.
Since Gy is a Q-algebraic group it is natural to ask:
What are the Q-algebraic equations that define Gy C Aut(V,Q)?
This question has a very nice answer as follows. Recall the algebra of Hodge tensors
He. © Vv @vr®,
k=l(2)
It is an exercise to show that G, fixes Hgy*.
THEOREM. Gy is equal to the subgroup Fix(Hg%°) that fixes the algebra of
Hodge tensors.
The reverse inclusion
Fix(Het*) C Gy
is based on a theorem of Chevally:
A linear reductive Q-algebraic group is defined by stabilizing a
line LC ove @V*®'),
Using this the basic idea behind the reverse inclusion is that if it happens that
LCV® @V*®’, then since y(S") C Gyr we have that y(S') acts trivially on Le.
Thus the weight 1 — k = 2m and Lc = Le’, which says that LC Hee We refer
to [GGK1] for the general argument.
The above characterization of G, holds in a suitably modified form for G¢.
The modification is that on Hodge classes of weight n, Y(re) acts by r”. Thus
the condition of fixing tensors must be replaced by scaling them, and when this
is done the above result extends to general Hodge structures. In particular, given
(V,%) and a polarization Q, g(re’’)-Q =r-?Q. Thus for Hodge structures that
are polarizable the difference between Gg and Gy is just in the scaling action.
The theorem “explains” why for a direct sum (V,y) = (V’,y’) + (V",p”) of
Hodge structures, the inclusion
Gy C Gy x Gyn

is in general strict. The inclusion holds because the direct sum has at least as many
Hodge tensors as those that come from the two factors. It will be strict if there are
additional Hodge tensors that relate (V’, yp’) and (V",y”).
EXAMPLE. For the PHS (gy, B,y) where B is the Cartan-Killing form, both
B and the bracket [ , | are Hodge tensors. They essentially generate the algebra
of Hodge tensors in a manner to be explained below.
PROOF OF G,CH,=>Gg IS AN ALGEBRAIC TORUS. Note that End(V, y), the
endomorphisms of V that commute with the action of y(S') on Vp, is just the space
Hg!" of Hodge tensors in V @ V*. Next, the assumption Gy, C Hy, i.e. that Gy
preserves the Hodge structure (V, y), implies that
Gy C End(V, y).
Then G, = Fix(HgZ*) says that G, is commutative, which is what was to be
shown. oO
Mumford-Tate domains and their compact duals.
DEFINITION. Given a PHS (V,Q,y) the associated Mumford-Tate domain is
Dg, the Gy,R-orbit of the corresponding point in the period domain.
Thus for H, C Gyr the stability group of (V,Q, y) the quotient space
Dy = Gor/ Ay
is a homogeneous complex manifold. As a set
Dy = {97 99:9 € Gor}
is the set of Gyp-conjugacy classes of y : S' + G,x. From this we may infer that
Hy = Ze,,.2((S")) is the centralizer of p(S') in Gyr.
Since H, is compact we have that
Hg, contains a compact maximal torus T.
From general properties of Q-algebraic groups we obtain the result
A Mumford-Tate group contains an anisotropic, Q-mazximal
torus.
One may think of a split Q-maximal torus in a reductive Q-algebraic group as a
product (Q*)™ x (S(Q))” where
s@)={( 5 fy:a,b€Q and +e ai},
Anisotropic means that m = 0.
Among reductive Q-algebraic groups this is a very special property. For exam-
ple, GL, (Q), SL,(Q) for n 2 3 are not Mumford-Tate groups. It is a more subtle
matter to rule out other simple groups as being Mumford-Tate groups.
EXAMPLE (continued). Given a PHS (V,Q,~) there is an associated PHS
(gy, B, y). It defines a point Ad y in the corresponding period domain Dag. In case
Gy is simple it may be shown that the Mumford-Tate domain Diag © Dia is the
connected component containing (g,, B,y) of the variety defined by imposing the
condition that B and [ , |are Hodge tensors. The essential point is the equality
Gc = Aut?(gc,[ , ])

between the adjoint group and the identity component of the subgroup of Aut(gc)
that preserves [ , ] (cf. [Kn1)).
In general, it does not seem to be known in what degrees the algebra of Hodge
tensors are effectively generated.
EXAMPLE. We shall show how to realize the unitary group U(2, 1)p as the real
Lie group associated to a Q-algebraic group U(2,1), and we will see that U(2, 1)
is the Mumford-Tate group of three PHS’s, including one of weight n = 3 with
h3° = 1, h®-1 = 2. For this we proceed in three steps:
(i) determine Hodge structures of a certain type;
(ii) put a real polarization on’ them;
(iii) ensure that the polarization is rational.
Let F = Q(V—d) where d > 0 is a squarefree positive rational number (d = 1
will do), and let V be a 6-dimensional Q-vector space with an F-action; i.e., an
embedding
F > Endg(V).
Setting Ve = V @g F, we have over F the decomposition into conjugate eigenspaces
Ve = Vi @V_
where Vi = V_. We will show how to construct polarized Hodge structures of
weights n = 4, n = 3, and n = 2 with respective Mumford-Tate groups U(2, 1),
U(2,1), and SU(2,1). For this we write Ve = Vic @ V_.c. We shall do the n = 4
case first, and for this we consider the following picture:
* * * Vic
| | * * | * | Vc
(4,0) (3,1) (2,2) (1,3) (0,4)
The notation means this: Choose a decomposition V4¢ = Vee ® Vex @ Ve into
1-dimensional subspaces for the action of F. Then define V_.¢ = Vere ee
where V2? = ve Setting VP" = VP? @ V2" gives a Hodge structure. The
number of *’s in a box denotes the dimension of the complex vector space.
Next we define a real polarization by requiring Q(V;,V;)=0=Q(V_,V_), then
choosing a non-zero vector w4” € V{? and setting
OG oa. atl, wo eV Oe)
LE” = Pes
Queah)=-1, ot ev”
DO) OX} =) (2,2)
Vey @, )=1 Oi GV os.
All other Q(x, *) = 0.
Finally, we may choose the V?"? to be defined over F and wi? € Vi". Then
Lah? +007) =e5s» p=4,3,2
aa (we" — i") = e7-p p =3,2,1
gives a basis €),...,€¢ for Ve Ve = V. In terms of this basis, the matrix entries
ofQ arein RN F=Q.
We observe that, by construction, the action of F on V preserves the form Q.
We set
U = Autr(V,Q).

This is an F-algebraic group, and we then set
UG. 1) = Resp/q Ul.
PROPOSITION. (i) U(2,1) is a Q-algebraic group whose associated real Lie group
is U(2,1)g. (ii) If we operate on the reference polarized Hodge structure conjugated
by a generic g € Autp(Vpr,Q) = U(R), the resulting polarized Hodge structure has
Mumford-Tate group U(2, 1).
PROOF. Setting J = ('=1 4 ),the matrix of Q in.the Q-basis e1,...,eg for V
a= (5 Q)4)
In terms of this basis, Vi 7 is spanned by the columns in the matrix
is
I
(yaar):
If g € Autg(V), then the extension of g to Vr commutes with the projections onto
V,¢ and V_y. A calculation shows that these are equations defined over Q. The
conditions that g preserve @ are further equations defined over Q. Thus, U is a
Q-algebraic group. Moreover, g is uniquely determined by its restriction to the
induced mapping
Ga Vash Va he
eee jen? = =a
In terms of the basis ae ws? BOO: &OLAV aig C3, g, preserves the Hermitian form
J; 1.e.,
"G4.Jg4 = J.
This shows that the real points U(IR) have an associated Lie group isomorphic to
U(2, 1), and therefore proves (i). The proof of (ii) will be omitted (cf. [GGK1)).
The reason that the Mumford-Tate is U(2, 1) and not SU(2, 1) is that the circle
(ze Cun|z)= I }eacts'on wit by 2872 and 2*= 27> 2° =e 1. 0
To obtain a polarized Hodge structure of weight n = 2 with Mumford-Tate
group SU(2,1) we do the construction as shown in this figure:
“| + * Vic
* * * Vic
We are in SU(2,1) because z? - 29 - z-? =1.
To obtain a polarized Hodge structure of weight n = 3 with Mumford-Tate
group U(2,1) we do a similar construction
* * * Vy Cc
* * * V_c
h39 h24 h}2 h2.3
A difference is that, in order to have Q alternating, we set
3,0 —3,0
fete yO) ee
iQ (oie) = lk

All of the above give Mumford-Tate domains that are of the form G/T where
T is a compact maximal torus. The picture when n = 1
me : Vic
* ee Ve
fio hp?
gives a Mumford-Tate domain U(2,1)/U(2) x U(1), which as a complex manifold
is SU(2, 1)/S(U(2) x U(1)). It is an Hermitian symmetric domain B parametrizing
polarized abelian varieties of dimension 3 with an F-action. The corresponding
quotient Gr/T, where T C K is a maximal torus, may be thought of as the set of
Hodge flags lying over the Mumford-Tate domain B. Here, for F! € B a Hodge flag
is given by 0 C LC F! where L is a line in F'.
Returning to the general discussion, we note that Mumford-Tate domains D =
Gy.r/H, have compact duals
Di Cae) P.
where Gy,c is the complex Lie group associated to G, and P, is the parabolic
subgroup of G, that stabilizes the Hodge filtration F',. The Mumford-Tate domain
is an open orbit of Gy.z acting on D.
We will next obtain “pictures” of the D above corresponding to the weight
n = 3 PHS’s and of its compact dual. For this we identify Vic with C? using the
basis wey oe red above. We take as Hermitian form H(u,v) = —iQ(u,
0) with
the matrix
1
=]
1
Writing vectors in C? as z = (2) with [z] = |=|€ P?, the condition
Hige2) = 0
defines the unit ball B c C? c P?, where C? is given by 2 = 1.°
The compact dual D = GL3(C)/B where the Borel subgroup B stabilizes the
flag
* * *
| € les) € We
0 0 *
6 This will be one of the “running” examples in the lectures. For computational purposes it
=4 il 1 Ann
will be more convenient to use each of ( 1 ; ): ( -1 ; )
, and ( 1 EA )for our Hermitian forms
in the different lectures where this example appears. We will specify which one is used each time
the example is discussed.

in P?. We may picture D as the incidence variety in Wel aa
where p € P?, | € P** is a line and p € |. The Mumford-Tate domain is the open
set of all configurations
B
where, setting B° = P?(closure of B), we have
p € BS
INBF90.
EXAMPLE. We will describe the period domain D for PHS’s of weight n = 3
and with all Hodge numbers h?4 = 1. This example is of considerable importance
in mirror symmetry, as it parametrizes possible PHS’s for mirror quintic varieties
(cf. [GGKO] and the references cited therein).
The construction we now give is an extension of the SU(1,1), or unit disc,
construction of PHS’s of weight n = 1 with h!° = 1.
We consider a complex vector space Vc with an alternating form Q where
=
@ there is a basis 722, (Ues,
Uc, Ve, tar Ve such that'Q:= ( , a1 ;
e there is a complex conjugation 0 : Ve > Vc where :
G(VEei ise 102)
O (Uae = We,
and then. oa.) tse 5 (Ue) = Vers
e There is a Q-form V C Ve given by V = spang{w1, w2, wW3, wa} where
Wis Fa (Ver Th WVe,)
Uy = 5 (0-e1 Ay 1Ve, )
ee Fj (Ves S We, )
wa = 5 (U-en + ie,);

The matrix Qw of Q in this basis is
e H : Ve. ®@ Ve — C is the Hermitian form H(u,v) = iQ(u,ov). It has
signature (2, 2);
e H(v,ov) = 0 defines a real quadratic hypersurface Qy in PVc
which we picture as
© Gc = Aut(V, Q);
e Gp = Aut,(V,Q). Then Gp is a real form of Gc containing a compact
maximal torus T;”
e Gp is also the subgroup of GL(Vc) that preserves both Q and H.
Proof. For g € Gc = Aut(Vc, Q) we have
H(9(v), 9(w)) = iQ(g(v), a (g(w))
= 1Q(9(v), ((79)(ow)))
where g € GL(Vc) and og is the induced conjugation;
e the complexification of the maximal torus T C Gp is given by the set of
Ile PS,
Ne
© V_—¢,;U—e; Vey; Ve, are the eigenvectors for the action of T on Ve.
The compact dual D may be identified with the set of Lagrange flags
(OWE ELE AAD EATEN Ge
where dim F* = i and Q(F?, F?) = 0. In P* = PVc such a Lagrange flag is given
by a picture
Pee
Pp
where E (= PF?) is a Lagrange line in P? and p (= PF") is a point on E.
TIn fact, Gp = Aut(Ve, Qw) = Sp(4)r-

The period domain D may then be pictured as the set of Lagrange lines
l
(1,1)
P
<0
where the notation means H(p) < 0 and the restriction Hj := fen has signature
(1,1). This translates into the condition that the corresponding flag F satisfy the
second Hodge-Riemann bilinear relation.
EXAMPLE. The “first” non-classical PHS occurs with weight n = 2 and Hodge
numbers h?? = 2, ht! = 1. Then dim V = 5 and the symmetric bilinear form
Q:VeEeV—Q
has signature (4,1). For example, we might take V = Q° and Q to have matrix
a=-(4 ah
For convenience we choose an orientation on V.
The period domain may be described as
D={We Gr2Ve) PO, F)=0, ORF) =] 0
Here, Gr(2, Vc) is the Grassmannian of 2-planes in Ve & C°, or equivalently the set
G(1, 4) of lines in PVc = P*. The compact dual is
D=4IF €GrOwe)- Of. F)=0),
It is sometimes convenient to denote it by Gr; (1,4), thought of as Lagrangian lines
in P*. As a homogeneous complex manifold
1D Gril
where H ~ U(2)p with A € U(2)g mapping to (49) € SO(4, 1)g using the standard
inclusion U(2)r — SO(4)g where U(2)p is given by the orthogonal transformation
on R* preserving J = ce - ).
Variation of hodge structure and Mumford-Tate groups. We will only
briefly touch on this, referring to [@GK1] and [GS2] for details.
Let D bea period domain for PHS’s (V, Q, ~) of weight n and where V = Vz@Q.
We set [Tz = Aut(Vz,Q). In the tangent bundle TD there is a homogeneous sub-
bundle W whose fibre at y € D is
Wy = he.
In terms of Hodge filtration we may think of the fibre
Wee Sale De A) Cobety.
The condition in the brackets will be called the infinitesimal period relation (IPR).
Next, let S be a connected complex manifold. Usually S will be a quasi-
projective algebraic variety. A variation of Hodge structure (VHS) is given by a
locally liftable, holomorphic mapping
6:5 31zD

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As he spoke he fancied he saw in her eyes the glimmer of a
haunting fear. But it vanished so swiftly he doubted he had ever
glimpsed it. The big eyes reading his were heavy with grief. With
sudden impulse he crushed her in the shelter of his great arms.
"I should not have breathed the thought," said he penitently.
"Nothing conceivable can ever strike our love, Mary. You are not
afraid?"
"Not of that," was the reply as she nestled contentedly within the
strength of him. "Many things may happen, but not that. Just now
Father is obsessed with his new friendship. It is a thousand pities that
the friend should be Chesley Sykes. His presence in Pellawa is an
ominous mystery to me. So far he has deported himself with desirable
aloofness. May he continue to do so. He is completely outside of this
beautiful moment. Let us forget him."
"And ride away together," suggested Ned.
"I have an hour yet," calculated Mary.
"We'll spend it riding No-trail Gulch," tempted Ned.
"Let us away," laughed the girl gaily. "For the trail——"
"Is luring," completed Ned, leading her to the horses.
A moment later they clattered over the gravel bed of the brook
and into the trees.
III
BOUQUETS
The month of October sped swiftly away in one long attack on oceans
of stooks amid the blue blaze of cloudless skies. The threshers were

having a run of "great weather" as the blank fields and the piles of
straw averred. The matter of the McClure-Pullar wager had of course
leaked out and become the one thrilling feature of the annual wind-
up. Aside from the two gangs there was a keenly interested and, alas,
gaming public. The sympathy of the plains went to Ned Pullar; the
odds to Rob McClure. Jack Butte had become an inhuman sphinx. Into
Jack's elevator had come the steady stream of grain from the
contending mills but to no one had he divulged the respective
records. No system of tapping his books had yet succeeded. This was
due to the fact that Jack Butte was an irreproachable and resourceful
stakeholder. As rare evidence of his unique qualifications he had
sworn the secrecy of every farmer threshed by the rivals. It was a
tribute to the sporting public that with but three days to run only one
man knew of the interesting situation.
The Valley Outfit was resting. Ned Pullar was oiling-up and
cleaning his engine during the dinner interim. Every bit of brass about
her was gleaming gold while the friction surfaces shone clean like new
silver. The "Old Lady" had established a personal reputation in the
Valley as a "mighty good engine," and her engineer was justly proud
of her. To Ned she had become a living thing. Mounting on the
footboard he grasped the throttle. During the pounding grind of the
past month he had formed the habit of communing with this thing of
power that he controlled with so masterful a hand. As his eyes read
gauge and water-glass with satisfaction he spoke to the engine,
addressing her not by word of mouth but with the voice of his
reflection.
"Just a couple of days more and we'll ease up on you, old girl.
You've been a game old Pal and you'll not throw me down now."

The Old Lady made violent protest at even the hint of such
infidelity by throwing a hissing cloud of steam from her exhaust. Ned
smiled, gripping the throttle with a fond clutch.
"Same old ready bird!" said he. "Eager to get at it, are you? Just
five minutes, Old Lady, and we'll set you purring again."
With the flames roaring through her flues the thing of steel
waited restively for the thing of will that held her levers in sinewy
grasp.
At the separator the men resting for a few minutes upon the
straw were looking up into the face of Andy Bissett, the separator
man, listening to him as he worked away with wire prod and oil can.
"I tell you, lads, we are up against a stiffer proposition than any
of you fellows think. Ned's out for blood. He doesn't care a whiff for
that wager Butte holds. But he's got to win it."
"Hold on, Andy!" cried Lawrie, the big feeder. "You've got me up
in the air. I thought the Valley Outfit was after McClure's long green."
"So they be," agreed Dad Blackford belligerently. "And Ned, 'e's
a-goin' to get hit."
But Andy shook his head.
"You don't get me," said he, pausing in his work. "And I can't
explain for I'm as much at sea as the rest of you. But we've got to win
this little bet. If we put it over McClure it will only be by a thousand or
two. Ned says he won't push the Outfit any harder, but I've taken the
liberty to put on the squeeze play for a couple of days. Grant's putting
on two extra stook wagons and a couple of men. Here they come
now. We're going to slam through a couple of thousand above the
regular. If Grant can bung this old fanning mill I don't know it."
The men leaped to their feet, for the extra wagons had rattled
up. There was a fresh determination in every face. They had been

working at high pressure for the long run, but they were right on their
toes in the face of the challenge. Each man went to his place
addressing himself to the struggle in the workmanlike fashion of the
Valley Outfit. Jean Benoit, the little French bagger, plucked the
tankman's sleeve as the group broke up.
"What Ned hole on hees cheek?" questioned the Frenchman
excitedly.
Easy Murphy looked at him a moment deeply puzzled. Suddenly
light broke.
"Begobs, 'tis the tongue in his chake yer dappy about. Why, sez
you, does not the sly divil be afthur-r showin' the hand uv him? Shure
Ned's not wearin' his heart on his lapel, me frind from Montmorenci."
Jean searched the Irishman's face as it went through the
contortion of an excessively wise and secretive wink.
"Mon Gar!" exclaimed the confused fellow. "De boss wan
woodhead! Why he de debble not squeal? Eef we know, den lak wan
blankety busy bee we work de whole gang. Eef we not know, Ned he
ged him on de neck."
"You're right, Jean!" was the emphatic pronouncement. "And yit
Ned wull not be afthurr tellin' his saycrits till the gintle lugs uv the
Valley Gang. Can't ye see whut's eggin' him on? 'Tis not the wee
wager. 'Tis a man." Tapping the Frenchman wisely on the breast he
whispered tragically, "The boss is thrailin' a varmit be the cognomin
uv Robbie McClure and he'll be afthurr gittin' his man dead or aloive.
Put that intill the poipe uv ye and smoke ut, not forgettin' till wur-rk
like —— in the manetoime. Farewell!"
Jean did not understand quite all but he turned to the bagger
with fierce resolution. As he knocked the filling bag with his knee he

caught sight of McClure's smoke through the cloud of dust enveloping
him. His dark eyes shone.
"We lick heem! We lick heem!" was his low soliloquy. Then he
added joyously as he gave the bag a vicious jab, "Ha! Eet will be
good!"
The thought energized him mightily. Deftly settling the bag and
closing it he seized it adroitly and by united force of arms, knees and
back hurled it up into the wagon, remarking ferociously:
"So we give McClure the beeg fall. We give him beeg scare too,
eh? And mebbe leetle licking also."
Smiling gleefully he settled to the grind.
Easy Murphy was absorbed in a brown study as he climbed up on
his water tank and started his horses over the stubble. Suddenly he
came out of the maze of his cogitations and called fiercely at his
horses.
"Arrah, me beauties, shake the legs uv ye or I'll be afthurr pokin'
yer rumps wid me number tins."
The horses took the hint and broke into a lumbering trot. They
were making a trip to the water-hole and at the moment were passing
through a field of oats into which they would soon be hauling the
Outfit. As he drove through the wire gate out into the road-allowance
he saw a buckboard pull up at the fence some distance away. The
sole occupant dropped out of the vehicle and passing through the
strands of wire walked for a considerable distance into the stocks.
Pausing for a moment the stranger knelt down beside a stock, then
rising walked on to another, where he knelt again. His actions excited
a keen curiosity in his observer.
"Begobs, me hearty!" exclaimed Easy. "Ye're not pickin' pansies in
an oat-field. Nathur are ye adorin' the Almighty, for ye're almighty

loike Snoopy Bill Baird, head foozler of McClure's bums. I'll hail yuh,
Bill, till I find out yer tack."
He was about to yell when he checked himself, muttering:
"Howld yer jaw, ye owld fool."
The other had noticed his approach and loitered a few minutes
shelling the grain, interested evidently in the yield. This matter duly
settled, he climbed back through the fence and reëntering the
buckboard drove slowly along toward the tank. It was Snoopy Bill all
right. As they drew abreast Easy pulled up his horses. A roguish
twinkle played in his eyes as he said:
"'Tis a foine day wur-r havin', Bill. A pleasant day indade for
pluckin' swate bokays."
"Great day! Great day! Murphy!" was the jocular reply,
"Bin pickin' pansies the day," continued Easy naïvely, curious to
discover what he could.
Snoopy Bill looked at him sharply. But no guile could he discover
in the face grinning down at him.
"No such luck, Murphy," said he casually. "I was taking a squint at
the yield. Pretty durn good, eh?"
"And it's the yield ye're afthurr meddlin' with and not the swate
and gowlden daisies. I saw yuh pokin' around among the stooks as I
pulled through the gate."
The smile on Snoopy Bill's face ceased to deepen while the whole
man became suddenly alert. Easy Murphy caught the change.
"Ye're Snoopy Bill, shure enough," blurted he. "And I'll lay ye a
tin-spot ye were up to no godly devowshuns kneeling in the muck by
the stooks. Ye're not prominint for religion, are ye, Snoopy?"
Snoopy Bill's tone was galling to Easy's inflammable spirit as he
replied imperturbably:

"Leaving the matter of the 'swate daisies' aside, Murphy. I was
praying for you, honest. I was putting in a lick for the Valley Gang
asking the good Lord to have a look to Pullar's Outfit when we clean
them up."
Easy's jaw set, a sign that an ultimatum was imminent.
"Ye blatherin' spalpeen!" he cried, his hands opening and shutting
convulsively. "I'll be afthurr spilin' yer sassy mug if ye open it agin."
Snoopy Bill opened his "mug" with commendable lack of
hesitation. An impudent drawl pointedly accentuated did not tend to
reduce Easy's evident irritation.
"Talking about mugs, Murphy," said he confidentially, "it seems to
me we have some curious and fine large samples hereabouts gopping
wide open for free inspection."
The sardonic grin that accompanied the casual observation
touched off a whole magazine of high explosive. Easy's mouth was a
generously ample specimen and his posture of attention was to sit
with it ajar. The amplitude of that particular area of his facial map was
a source of constant regret. Hence the remark rankled.
"Ye've said it!" was his angry utterance as he threw down the
lines. With a leap he was off the tank. They dropped to the road
together, but Snoopy Bill having a shorter descent recovered first and
rushing at his antagonist swung swiftly and struck, planting a
powerful blow on the chest, hurling the other against the tank. He
followed quickly for the head with his other hand but Easy's native wit
acted with surprising speed and he ducked. Snoopy Bill's closed fist
rapped on the hard surface of the tank, skinning the knuckles.
"Thry agin!" yelled the Irishman mockingly, with a vicious thrust
into his enemy's ribs. The blow staggered his opponent. Swiftly he

followed it with a jolting up-cut, yelling again, "Take wan yersilf and
be hanged!"
The blow made Snoopy Bill's head bob back and he dropped to
his knees. Easy stood over him furiously triumphant. Stooping he
called into the other's ear:
"Git busy at yer devowshuns, me hearty. Put in a wur-rd for
McClure and his divils."
With a weak smile Snoopy Bill staggered to his feet.
"You are a hard hitter, Murphy," said he dazedly.
Picking his late antagonist up bodily Easy bundled him into his
buckboard and slapping the horse smartly on the hip sent him off at a
trot. Placing his hands to his mouth the tankman shouted:
"If ye want anny more forgitmenots come back the morrow, the
garden's full."
With this parting shot he climbed up on his tank and resumed his
trip to the water-hole.
IV
THE MAN, ROB McCLURE
Rob McClure sat before his roll-top desk, his head resting upon his
hands. He was perturbed. Occasionally his head would sink into a
posture of dejection. In a moment he would straighten, shrug his
shoulders and look out of the window, his face swept by the irony of
an uncouth smile.
He was a man of powerful physique, large of frame, possessor of
a presence singularly impressive. He was conscious of his power. An

habitual, impatient shrug revealed a restive spirit deeply antagonistic
to baffling elements. A relentless, implacable expression inwrought
the face that exhibited even in the act of smiling the dominance of an
over-riding will. There was something cruel in the hard lines about the
mouth, while the deep little wrinkles about the eyes more than hinted
brutal cunning. One felt that given sufficient pressure Rob McClure
was capable of the unspeakable. There were, however, relieving
features to the hard visage, most prominent of all a high, expansive
brow and great, volcanic eyes.
Looking out of the window his eyes fell on the yellow stretches of
stubble, empty now save for the huge piles of straw thrown up by the
blower. In the west the plain was gulfed by the blue depths of The
Qu'Appelle Valley. His glance swept over the autumn landscape all
unseeing, for his gaze was fixed on two streams of distant smoke that
rose for a little in straight columns, then floated off in long parallel
lines to the west. Clenching his fist he brought it down on the desk.
"I've got him nailed!" he breathed fiercely, smiling his strange
smile.
Then his confidence seemed to shake. The two lines of smoke
were streaming over the fields evenly abreast.
"Pullar's a silent devil," he whispered darkly. "He is deep—deep
as ——, and he cleans up a pile of stuff."
He meditated for a little then added decisively:
"But I've got him nailed tight."
The irresolution disappeared and the cruel smile stole out again.
"If he should win," was the jocular reflection. "We'll take a look at
the little game proposed by Reddy Sykes. Reddy has a way—a
fetching way." The name brought a certain merriness to his face. The
humour was not attractive.

With a satisfied shrug he rocked back in his chair. As he did so his
eyes rested on a photograph above his desk. Down upon him gazed
two beautiful faces. Instantly a tender light softened the hard
features. His lips moved, shaping involuntarily the names:
"Helen! Mary!"
The picture held his searching gaze until the sound of
approaching footsteps broke the spell. At the sound the tender light
vanished and a conflict surged over his face. Gradually his jaw set and
the steel of the unyielding will revealed itself. The door opened quietly
and in a moment a hand rested gently on his head. The voice that fell
on his ear was sympathetic and affectionate. Mary had broken into his
sanctum.
"Why, Daddy," she cried, "you are looking very serious. Are you
troubled about something?"
The very solicitude of the voice seemed to chafe him.
"No," he exclaimed abruptly.
Nothing daunted she fondled his hair.
"Is the mill not running well, Daddy?"
The appeal in the voice caused a relenting of his face but his tone
was forbidding as he replied:
"Yes. She's running along fine. I must go out to her right away."
Submitting brusquely to her kiss he rose and snapping the roll-
top shut took his departure.
Mary McClure sat down in the vacated chair, resting her head on
her hands as her father had done.
"Poor Daddy!" she murmured. "You are so busy, so preoccupied."
There was a trace of pain in the voice, a great wistfulness in the
eyes. Once again she was confronted with the tragedy of affection
unrequited.

Looking at the father one would expect in his daughter the
robust, ample type. But she was small and fragile, a delicate bloom of
young womanhood. Out of the bright face looked lustrous brown
eyes, a seriousness lying in their playful depths. In appearance only
was she fragile, for the small form was well compacted, lithe and wiry,
capable of really great endurance. She was more than equal to
exhausting rides along the ravine and the trails of the upper country.
Sitting by the desk she was a diminutive, disconsolate figure. She had
drooped into a pensiveness that of late visited her all too frequently.
Nose and chin had the dainty grace of the spirituelle and such was
Mary McClure. Yet was she human, fired with an intense passion for
people. A quick, light glance of her eyes or the flash of her smile
threw the spell that was irresistible. Life opened to her on all sides.
The girl was fortunate in her mother. The glory of a great affection
enveloped her. In the mother appeared the culture of Old Varsity,
giving to the McClure home a distinguishing atmosphere not often
found on a Western farm. Helen McClure was a fine companion for
the vivacious girl, and the two enjoyed a delightful camaraderie.
In her father Mary was presented with the most cruel enigma.
Here lay the secret of the solemnness that so often filled her eyes. By
him all affectionate approach was resented. He seemed deliberately
striving to quench her natural attachment. But Mary's affection knew
no repulse. Patiently she pressed the attack, intent on destroying the
barrier he would insist on building between them. At times she
fancied a relenting had rewarded her efforts.
Rising, she walked to the window and looked out pensively upon
the autumn fields. Her heart was conscious of a dearth as great as
that of the barren stubble. Her lips trembled as she whispered
musingly:

"Daddy doesn't seem to want my love. Why is he so busy—so—so
unfriendly? So buried from us in a hundred cares?"
As she pondered she shuddered, for she remembered times when
he was well-nigh brutal. Then the fetid odour flowed from his breath.
Rapt in the poignant moment her face drew into sad lines and a mist
stole over her eyes, blurring the autumn vision.
McClure had made all haste and drew near his machine. As he
approached the engine slowed up and stopped and the pitchers,
jabbing their forks into the sheaves, lay down on the loads. Urging his
horse to great speed he rode up to the machine. A lively altercation
was in progress. A knot of excited men were gathered about Snoopy
Bill Baird and Sid Smithers, the farmer. Smithers' voice rose high in
angry tones.
"She stops right now," he cried vehemently. "And you pull your
Outfit off my farm."
Throwing down the lines McClure strode in among the men. His
heavy voice rose above the hubbub.
"What's the kick?" was his demand.
"Smithers is trying to put a crimp in this job," replied Snoopy Bill.
"He's ordered the mill off the farm. He contends we're throwing over
his grain."
Smithers interposed warmly.
"And you are doing it," said he wrath fully. "It's a cussed shame. I
can prove it. Come back to the straw pile."
He promptly led the way and the crowd moved back quickly to
the blower. Reaching into the straw pile Smithers drew out a coal
shovel. His voice was indignant as he said:
"Here's what I caught in five minutes at the mouth of that
blower."

The men crowded round. Cleaning the straws away he disclosed
a layer of plump yellow grains covering the bottom of the shovel. As
the sight met his eye McClure gave an involuntary start and his face
grew dark. His voice was mollifying, however, when he spoke.
"That looks pretty bad, Smithers," said he quietly. "But you just
happened to catch a shoal of grain thrown over on a bunch of straw.
I'll bet you ten to one we haven't thrown over five bushels in the last
three days."
But Smithers stood firm.
"You can't pull the wool here, McClure," was the menacing retort.
"There is a heap of my stuff going over and you quit. Easy Murphy
gave me a line on Grant's yield and he's beating me bad. My crop's as
good as Grant's and you know it. Haul your Outfit off my farm."
Smithers was determined. For a moment McClure was silent.
Then he spoke in an appeasing tone.
"I don't want to quit this job right now," said he. "I'll tell you
what I'll do. Let me finish this run in my own way and if your yield
doesn't equal Grant's I'll make up the shortage and not charge you a
sou for your threshing. Is that square?"
Smithers turned the matter over deliberately.
"Make it law," said he shrewdly, "and I'll hook up with you."
"Agreed!" was the quick response. "I'll sign the papers to-night.
Meet me at Reddy Sykes' at ten and we'll put it through."
"Go ahead on that condition," said Smithers, climbing into his
wagon.
Quickly the men were in their places and the machine went
roaring into the twilight. As McClure stood by the separator he
signalled to Snoopy Bill.

"Let her rip, Bill," was his shout. "Crowd through a couple of
thousand extra before to-morrow night."
Snoopy Bill passed the word and the engineer opened the
throttle. The gang responded with a will and soon a great stream of
straw was gushing from the blower.
At that moment Mary McClure was standing up in her stirrups with
eyes fixed intently on a spur of the north bank of the Valley. As she
watched, a yodling scream came over the rounded hilltops. She
smiled delightedly. On the tip of the lofty spur she caught sight of a
red flash that she knew instantly as the shining coat of a certain bay
broncho.
"It is Flash with Margaret up!" was the pleased exclamation. "I
believe she wants me."
Forming a horn with her hands she called back in the cry of the
hills. The rider on the spur waved her gauntlet in reply, beckoning to
the rider in the Valley. Instantly Mary turned Bobs into the trees,
sending him up a steep bridle path to the left. In a few minutes the
girls were together and they set out through the stubble to where the
Valley Gang was finishing the wheat.
"We are just in time to see the move," said Margaret. "For you, of
course, the engineer is the whole gang. You will be able to see Ned in
action."
"And you will be absorbed in the rest of the gang, that is in the
antics of the separator man," countered Mary.
"At present," laughed Margaret, "I am going to make a raid on
your preserves and talk to Ned."
She rode up to the engine.

At that moment there was a boisterously gallant salute from the
gang, accompanied by a vigorous waving of caps and the shrill
scream of the engine. The girls acknowledged the reception by a gay
flourish of gauntlets.
"We are going to time the move, Ned," shouted Margaret above
the roar of the engine, showing him her watch. "Let us see what the
Valley Outfit can do."
Drawing his watch from his pocket Ned blew the whistle,
promptly gaining the attention of the whole gang. Waving his hand
toward the site of the new setting, he lifted high his watch and
pointed to Margaret. With a ringing cheer they accepted the challenge
and addressed themselves to the race against time. One of the feats
of a crack outfit is the swift move to a new setting without mishap or
confusion.
Already the last stock teams have pulled away from the separator
and are careering in wild race to the adjacent field. With the tossing
in of the final shovelful of chaff the separator stands clean and naked
above the stubble. As the last bit of wheat dribbles into the bag Ned
signals the stop and Margaret lifts her watch aloft.
"It is up to the Valley crew now," comes the silvery challenge,
and the boys respond with a merry shout and the address that marks
the discipline of the gang.
As the fly-wheel slows up the pitchers deftly throw the belt, roll it
up and hang it in place. At the same time the carriers are lowered and
secured and the two waiting grain-teams hooked to the separator.
Leaning well on the lines the drivers give the word. With a sharp gee
and a steady pull they haul the mill up on the stubble and head in a
curved line for the site of the new setting a quarter of a mile away.

There a space has been already cleared and a circle of loaded stook-
wagons is beginning to form, awaiting the arrival of the machine.
The feat par excellence of all the teaming about a threshing mill
is that of pulling the engine out of the holes into which she has
settled and over the intervening stubble. Usually two teams are
detailed to this duty, but here the big tank team is sufficient. At the
drop of the belt Easy Murphy hitched the grays. The two big beasts
stand expectant. Seizing the lines Easy gives the inspiration of his
invigorating brogue. Thrusting their great shoulders at the collars the
team leans steadily forward. Straining with their mighty muscles they
sink their toes deep into the turf. The traces stretch into tense,
vibrating thongs. Hawing sharply the real pull commences. The mass
begins to move. Swaying slightly as his horses' heads go down, Easy
heartens them.
"Stiddy now, me beauties, and aisy ut is or the stubble wull be
afthurr ticklin' the bellies uv ye."
Suddenly the wheels rise out of the holes and the heavy mass
rolls along.
"Aye, 'tis an aisy waltz fer yez, me bantams!" crows the tankman
as the big team swings through the soft muck with the weighty Old
Lady in tow. At precisely the same instant the separator has made its
start. Glancing at her watch Margaret is surprised to observe that
barely a minute has elapsed.
Arriving at the cleared area the separator, under the guidance of
Andy Bissett, circles to the east, coming up to position in the teeth of
the wind. The engine takes a curve to the west, swinging east to
meet it. With the separator in place and blocked, every man springs
to his task. Carriers are swung into proper elevation, feeder and band-
cutter's stands dropped and the belt run out to the engine.

Ned stands on the rear of his engine with eye sighting along the
fly-wheel. Now is the critical moment. An inch too much to right or
left means the loss of minutes.
"Gee a little!" comes the crisp command. "Steady ahead! Let her
swing to gee! Easy now! Hold!"
At the final order Easy Murphy brings his horses to a dead stop.
Quickly the belt is slipped on and tautened. Every man stands in his
place poised for work. Two short shrieks of the siren and the whole
scene leaps into animation. Volumes of smoke belch from the funnel,
the big belt speeds flapping along to the separator, starting the
whirring of a maze of lesser belts and the spinning of countless
pulleys. In a moment the cylinder is devouring an endless flood of
sheaves. From the side of the mill the oats gush out while the straw
rolls up over the carriers in a golden stream.
The girls ride up to the engine, admiration in their eyes.
"What time did we kill?" inquired Ned, smiling through his layers
of grease.
"You made time," corrected Mary, flashing a bright smile down
upon him. "That was wonderful work, quite worthy of the Valley
Outfit."
"Time," said Margaret with official dignity, "is the surprising
record of eight minutes and twenty seconds."
"I must let the gang know," said Ned in high elation. "That is a
pretty decent record." Reaching out he blew eight screeching calls.
The threshers paused long enough to respond with a trio of husky
cheers. Then back they went with a will to the grind.
"What a furiously busy gang you have, Ned," was Mary's
ingenuous observation, her eyes on the lively sight. "You all work as if

we are to have a two-foot fall of snow, during the night. Why this
haste?"
Ned smiled peculiarly and was silent. Margaret came quickly to
his relief. She was aware of the exact situation and entirely
disapproved, but she knew Ned wished to hold the truth from Mary.
"The Valley Outfit have been rushing along at this breakneck
speed for the whole of October," said Margaret. "They are gambling,
Mary. The boys have a wager that they can pile up a record output for
the month. The trial winds up to-morrow night. Ned Pullar and his
vaunted Valley Gang are a company of very foolish gentlemen."
"There are exceptions in the case, I suspect," insinuated Mary.
"Our little Miss Grant exempts all tall, good-looking separator men.
Hum!"
Ned laughed.
"Were it not for the dust," said he, "I would take you girls over
for a chat with our rather handsome fellow. I have a hunch, however,
that Margaret would scarcely enjoy it."
"What? The handsome fellow?" posed Mary mischievously.
"No. The dust," replied Ned.
"It is a little matter," agreed Margaret.
"The handsome fellow?" teased Ned.
"No. The dust," prompted Mary archly.
All three laughed.
"Here, White!" called Ned to his fireman. "You handle the throttle
while I take the girls to the mill."
In spite of the dust the four-cornered interview though
necessarily brief resolved itself into a charming "little matter." Andy
was back in his place on top of the mill oiling near the carriers. Ned
stood beside the girls, who were sitting their horses just beyond the

cloud of dust. They were enjoying a few moments' contemplation of
the lively scene before departure for the Grant homestead when
suddenly a vivid light flashed red in the twilight, flaring on the
sweating face of Lawrie, the big feeder. Instantly followed a loud
metallic crashing. With a strange, muffled shout Lawrie threw up his
hands and fell on the feed table, pitching forward into the jaws of the
machine. An instant more and he must be seized by the deadly teeth
of the whizzing cylinder.
At the blare of fire Ned uttered a cry of alarm and rushed toward
the separator. Realizing Lawrie's horrible plight he shouted to White at
the throttle and taking a lightning leap drew himself up on the
separator above the whirring teeth. Already they were fanning the
hair of the insensible feeder as his head settled nearer to the blurred
shine of the hideous jaws. Reaching over, Ned seized the helpless
man and lifted him by the sheer strength of his powerful arms out of
the fangs of the machine. But the weight of his inert burden swinging
suddenly overbalanced him. Poised over that maw of whirling death
the two men hung for an awful instant as Ned fought to recover. But
the weight was too much; Lawrie began to sink. It was evident the
two men were falling back into the cylinder. A scream of terror leaped
from the lips of the horror-stricken band-cutters. Then it was Ned felt
his shoulder clutched in a mighty grip and he with his precious burden
was dragged back to the roof of the mill.
"Thank God you were there, Andy!" exclaimed the big fellow
breathlessly as they composed the huddled form of the unconscious
Lawrie.
"A touch and go, Ned!" was the solemn rejoinder. "I did not know
anything was amiss—until I heard your shout. It took me an instant to
spot you in the dust. Lawrie's badly smashed."

And so it seemed, for the man's face was washed with blood.
Meanwhile White had shut down and willing hands helped them
move the wounded man to the ground. Water was speedily applied
and the blood sopped up, revealing a deep gash along the forehead
gouged by some missile thrown out by the rotating cylinder. Under
the steady bathing there were soon signs of returning consciousness.
Slowly opening his eyes Lawrie was surprised to find Ned bending
over him, looking at him with anxious, sober gaze. A gleam of
intelligence crept into the man's face and he smiled faintly.
"Oh, yes!" he said reminiscently. "I remember. I felt it slip in and
tried to draw it back but it got away." After a moment's pause he
added: "I am afraid it has played hob with the cylinder and concave.
Have you taken a look, Ned?"
"You Lawrie!" cried Ned, smiling at the game fellow. "It's the man
first here, you know. How are you feeling?"
"O.K., Ned, though by gum I seem to have taken the count."
Recovering he rose on his elbow and looked around curiously.
The gang were gathered about him, a circle of solemn faces. Giving a
little laugh he said naïvely:
"What's got your goat, pals?"
"Shure 'tis the lucky, quare divil ye are," said Murphy, "till be dead
wan minute and assistin' at your own post mortin the nixt."
A hearty laugh passed round the circle relieving the tension. No
more was said, but Lawrie understood the grip of Ned's strong hand.
"We must fix that cut, Lawrie," said he, looking helplessly about.
"This dirt will never do."
The moment the girls realized the accident they had dismounted
and assumed the official duties of Red Cross first aid. Mary McClure

smiled at Ned's words. She had already arrived at a solution. Rising
from her place beside Lawrie she spoke.
"Ned," said she curiously, "have you a knife?"
"Here," was the prompt response as he produced a jack-knife.
"Margaret, you take it," said the girl, "and if the Valley Gang will
close their eyes for a minute I'll direct you what to do."
At the words she lifted her skirt daintily, revealing the snowy
white edge of the petticoat beneath. With dancing eyes the gang
made the right about turn and Lawrie decided on an immediate
snooze. A few minutes later his brow was bound with a clean bandage
and he was making his way shakily to the feed-board. Calling a
farewell the fair riders rode away over the stubble, followed by the
applause of the grateful fellows.
Meanwhile at the machine there were interesting developments.
Jean Benoit, who was working in on the shakers, gave a sudden shout
and popped up out of the separator holding something in his hand. It
was a heavy wrench. He examined it in a puzzled manner for a
moment then handed it to Easy Murphy. The tool was minus one of its
jaws. On the remaining jaw some initials had been punched, but they
had been almost obliterated through the recent offices of a file.
"Dat no Valley wrench!" exclaimed Jean.
"Probably one of Grant's left on the stock during the binding,"
said Ned.
Easy Murphy shook his head sceptically.
"Ah!" was his fierce cry as he tipped the tool at a new angle to
the light. "So I think. By the Howly St. Paddy! Take a look, Ned. Can
you see?"
Ned took a look and there in the bright shine of the filed surface
were good traces of the punch marks forming plainly the letters, R-M.

Over him swept an ominous conviction. Without a word he placed the
wrench carefully in the tool-box.
"'Tis the hand uv Snoopy Bill," said Easy Murphy darkly. "And 'tis
his foul plot near did fer Lawrie and Ned." Clenching his hands he
dropped suddenly into a vengeful silence.
A desire for revenge swept through the gang like an electric
shock. Even Ned's cool eyes emitted a dangerous glare. Andy Bissett
saw the dire change in his companion. Laying his hand on Ned's
shoulder he said quietly:
"Ned, it's a dastardly trick but Lawrie will be well in half an hour.
It's up to the Valley Outfit to call the bluff and play the winning card.
Half a dozen teeth are gone in the concave and several others
twisted. The cylinder is about as bad. With fast work it will mean only
a two-hour stop. Let us finish strong."
"Very well!" agreed Ned. But his face did not resume its usual
imperturbable demeanour.
There was no more threshing that night. Morning found them out
an hour earlier, however, pounding grimly ahead, bent on recovering
the lost time. As Ned stood at the throttle, a masterful shadow in the
gray dawn, he thought over the adventure of the night before. It
seemed to hold some sinister portent. Easy Murphy had in the
meantime recounted to him the episode with Snoopy Bill Baird. Two
more heavy tools had been discovered in one of the loads. Suddenly
he became conscious of the malignant nature of the foe with whom
he was striving. His jaw set tightly and a mighty resolution shot from
his eyes. Unconsciously he opened the throttle and the power
throbbed with a fresh leap along the great belt. As he did so a vision
flitted unexpectedly before him. He saw Mary McClure standing amid
the gang, her eyes alight with laughter while she held her skirt

daintily lifted to disclose the snowy fabric for Lawrie's wound.
Suddenly his face lost its seriousness and he laughed delightedly.
"Mary!" he cried softly.
Shutting off the throttle he curbed the engine in her impulse to
race.
"I guess we have a bunch of pressure left, Old Lady," said he
confidently, as he guided her into steadiness. The thing of power
steamed on into the strenuous day while the thing of will threw down
the challenge of youth.
V
AT THE WATER-HOLE
Easy Murphy shaded his eyes from the sun as he gazed eagerly over
the prairie. After a prolonged look he remarked:
"Begobs, I belave he's coming!"
A further scanning of the landscape elicited a cry of satisfaction.
"Nick's headin' fer the howl all right," said he elatedly.
The Irishman was standing on the tank, his hand on the pump-
handle. He had backed the grays into a pool fed by a small creek that
here expanded into a miniature pond some dozen yards across. In
Western threshing the tankman draws his water from the nearest hole
or stream. For some days both Easy and Nick Ford, the McClure
tankman, had been filling their tanks at the same pool.
Nick Ford was known familiarly as Boozey Ford, a self-explanatory
sobriquet. Whiskey aside, he was one of the most reliable tankers

along the Valley. With whiskey by his side his water-wagon was apt to
receive a diluted attention.
As the days sped by the struggle between the two outfits became
intense. The two tankmen were nearing the point of interpersonal
complications in their heated conversations on the issue. Easy Murphy
was feeling irrepressibly loquacious on this occasion, for he had not
met Boozey since the affair of the R-M wrench. However, as Nick
drove up he began a foxy approach, greeting him in a friendly
manner.
"Nick! How is the wur-r-rld using you?" was his opening.
"So, so!" was Nick's no less friendly response.
"Ye'll be afthurr faylin' a demi-semi-quaver in yer boots, Nick,
since till-night's the night the Valley Outfit take the candy from the
kid."
"There's sure going to be a lark to-night," agreed Nick. "We'll
have a howling time putting the kibosh on your little, old Outfit. You
mark my words, Murphy, when Jack Butte hands out his estimates
you'll freeze stiff. I'll bet you even money we lick you by a thousand."
"Just cover that wee trifle," said Easy, revealing a ten-dollar bill.
"Sorry to rob you, Murphy," said Nick, "but it's awfully decent of
you to accommodate me. We'll hand it to Butte just before the curtain
goes up."
"'Tis a great pleasure till contribute," agreed Easy light-heartedly.
Then he added slyly, "By the way, Nick, did ye miss anny tools from
yer tool-chist lately?"
"Not that I know of," was the frank reply.
"Shure we found wan uv Rob McClure's wrenches in our
separator yisturr-day."
Nick's interest perceptibly increased.

"'Tis not the act uv a gintleman, but a dirty trick uv Snoopy Bill
Baird, and 'tis achin' I am till spile the impudint jaw of the Snoopy
wan fer the same foul act."
Nick's blood began to sweep into his animated face. But the other
continued:
"Howld yer timper, lad. I'm not afthurr blamin' you, Nick. Yer as
innocent as the lambs in the spring."
His voice grew sweet as honey and he made a suspicious motion
to his breast pocket.
"We'll just have a wee dthrop as gintlemen together on the head
uv the divilmint, and part—frinds."
He drew an amber-coloured flask from his pocket.
"'Tis the rale Irish, Nick. Be afthurr washin' down a swate
swallow."
He extended the bottle convivially.
Nick took in the sight with fascinated and thirsty eyes. All hostility
magically vanished and a supreme joy capered shamelessly into his
face.
"Don't care if I do," said he, with a too casual unconcern. "Dad,
that's prime stuff!" was his genuine approval as he handed back the
flask.
"Shure I'm afthurr sayin' the same mesilf. Yer over modest, lad.
Take a sip that wull tingle the toes uv ye."
So gracious a pressure was not to be resisted, and Nick
responded with a ready acquiescence that left nothing to be desired.
Easy emulated in pantomime, tipping the flask adroitly but permitting
no drop to pass his lips. Taking another "sensation," Nick scurried off
to his own tank and began pumping vigorously. Soon, however, he felt
the desire for still another touch and was back at the flask. Easy

Murphy kept the bottle supplied from some mysterious source about
his person. So the best part of an hour passed and signs began to
appear that Nick was rivalling the tanks in the quantity of liquid he
was carrying. In the meantime Easy had leisurely filled his own tank.
Suddenly The Mogul, McClure's giant engine, sounded the water call.
Nick recognized the signal and, dropping the pump-handle, seized the
lines and started off, urging his amazed horses in a line of patter that
was new to them. As he drove away Easy slipped down off his own
wagon and, stealing craftily after, tapped the bung of Nick's tank with
a stone. One or two skillful knocks and the peg fell out, letting the
water away in a heavy gush. Throwing the bung into the grass, Easy
climbed up on his tank and followed.
Ahead drove Nick, supremely unconscious of the fact that his
tank was fast emptying. When they reached the road-allowance he
became suddenly confused. His trail lay directly across the road and
into a field. His horses would have taken the right way, but Nick
pulled them up sharply. His eyesight was temporarily impaired. He
could see only the good road running east and west. Pulling on the
left line, he turned into the east. Yet he was not sure, and drew up his
horses once more. His tongue was thick as he called back:
"Hello, Eashy! (hic) Ish the trail (hic) all right?"
"Shure and indade it is that," came the wily response. "Go right
ahead to yer outfit, Nick, man. It's a foine road, the smoothest in the
howl counthry."
With a flourish of his whip Nick sent the unwilling team on down
the road. Crossing the road-allowance, Easy entered the oat-field
through the wire fence and made straight for his own machine. As he
hit the stubble trail he heard the Mogul whistle impatiently for water.
A moment later she called again. Turning around, he looked at Nick.

He, too, had heard the urgent calls and was standing up driving like
Jehu. The tank was now empty and the horses responded by breaking
into a smart trot. The sight was hugely entertaining to the watcher.
He slapped his thigh, shouting in unholy glee.
"Be the wake uv me grandmother!" he cried exultingly, "it's now
we get back the swate and precious minutes they filched by their
rascalities uv yisterday."
Away in the distance Nick was driving like mad while the Mogul
tattooed her calls for water with an angry insistence that drove him
from her at accelerated speed. The circumstance was too much for
the delighted Irishman. Laughing till the tears rolled down his cheeks
he called after the disappearing Nick:
"Go it, me hearty! Kape it up, bye, and ye'll soon reach the broad
Atlantic. Begobs! Call in at Winnipeg. They're shy on water-wagons in
the Gateway uv the Gowlden Wist."
Never a word of the matter did he give to his young boss as he
emptied his tank in preparation for the next trip. His wickedly radiant
face attracted Ned, however, stirring his curiosity.
"What's tickling you, Easy? Been filling your boiler at Louie's
tank?"
"Niver the dthrop, Ned. Not wanct since the twilfth uv July have I
shined up till the dementin' crathur. 'Tis the whistle uv the Mogul
that's drivin' me tipsy. Somehow the thirsty screamin' uv it tickles me
since uv the rediculous."
"Rob's engine is out of water. She's been callin' for over half an
hour," observed Ned, looking over the stubble at the rival outfit.
"Indeed, Easy, she's hung up. Their blower is stopped."
At an unusual hearty chuckle from the tankman, Ned eyed him
sharply, a suspicion leaping into his mind.

"Shtopped's the wurrd!" exclaimed Easy in feigned surprise,
shading his eyes the better to study the Mogul. "Rob wull be afthurr
havin' a brathin' spell. May it last a wake."
Ned's eyes detected an unusual excitement on his companion's
averted face. His suspicion took a sudden definite form.
"Easy," said he seriously, "you are mighty pleased about
something and yet not at all surprised. Let me into the secret."
"Shure 'tis plazed I am this minute, Ned, and the most astonished
critter on the Valley Gang."
"Steady, lad," cautioned Ned. "You can't fool me. You know more
about the water shortage at Rob's outfit than Rob himself. What's
keeping Nick?"
Easy found a matter for precipitate occupation in the barrel he
was filling and did not reply at once. He was seized with sudden
panic, for he had caught sight of Ned's face. The unsmiling eyes filled
him with trepidation. When he at length looked up Ned's clear eyes
looked through him. For once the garrulous Irishman was speechless
while a blush flamed slowly over his brown face.
"Tell me," said Ned simply.
Hitching his overalls nervously and somewhat forcefully, Easy let
a broad, sheepish grin play on his ample face. He attempted
jocularity.
"'Tis a lugoobrius confession ye'll be draggin' out uv me wid the
third degree uv yer blazin' eye."
"Tell me," repeated Ned.
"Wull," said Easy, scratching his head with obvious regret, "since
'tis implacabul ye are, I'll make it short and swate. Nick and yer
humble sarvint meets at the mud puddle. We pass the complimints uv
the sayson, git intill a small fracas uv the tongue and out uv it by the

bottle. We had a wee dthrop. That is, Nick had. Thin he took another
and another, et cetra and so on. Nick was oncommon thirsty. In a
wurrd, I filled Nick till the neck and pulled the bung uv his tank. The
one is impty and the other full. 'Tis the Mogul and mesilf knows which
and,—yersilf, begobs, since ye tapped me wires. To sum up fer ye, me
inquisitive frind, Rob's tank is impty and his tankman full, and the pair
uv thim is headin' fer salt water at a spankin' trot. 'Tis comin' till the
blackgards if ye ask Easy Murphy."
Easy stood before his boss with hanging head. His confession had
not stimulated any risible emotions in Ned. Ned, on his part, said
nothing, but stood looking for a little at the culprit, a kindly light
mingling with the flash of his eyes. Then he stepped over to his
engine and, seizing the whistle-cord, gave it a jerk, blowing the one
sharp shriek that signals stop. Instantly the work ceased and the
outfit slowed to rest. Amid the shouts of the men demanding the
cause of the stop, Easy Murphy ran swiftly to Ned.
"Ye're not afthurr killin' the outfit," cried he, a peculiar pleading in
his voice.
"Easy," said Ned quietly, "the Valley Outfit is running this little jig
on the square. Not a wheel turns on this mill until McClure makes up
every minute we've killed for him."
The Irishman looked into Ned's face. There had been the glimmer
of an accusing look but it was gone. In its place was something big
and honest that hushed the angry protest about to leap forth. Their
eyes held for a moment, then the tankman's fell while the flush swept
his face once again.
"I'll explain to the boys," said Ned, moving away toward the
separator.

"No, lad," cried Easy, impulsively seizing his arm. "'Tis the hot
curse I was nearly givin' ye. Ye're too white, Ned, fer a divil the loikes
uv wan Easy Murphy. Shure 'tis right ye are, though I'm hatin' the
idea. I'll hike till the mill and make me diplomatical defince before the
gang. Sind me carcas till Belfast whin the boys git through wid ut."
Making a comical grimace, he set off to the separator to do the
hardest thing he had ever attempted.
The men listened silently while Easy made his brief and self-
accusative explanation. At the abrupt conclusion there resulted a most
awkward pause. The gang were dumb at the unexpectedness of it.
Each man was torn by several desires. He wanted to laugh, to howl,
in fact. But something fine in him rendered him mute. There was a
great admiration for their game boss and an even greater admiration
for their game and artful culprit. The embarrassment had about
reached the explosive point when Jean Benoit let out a scream.
"Ze res' do moche good, I tink," said he, shaking with laughter.
"Wan, two, tree cheer on de boss an' dees ver bad Irish fellow."
At his words there broke out a jolly shout while the gang lay back
on the straw and laughed to their heart's content.
Through the long wait there was not a murmur.
Meanwhile in McClure's gang consternation reigned. The last drop of
water had been sucked up by the inspirator and the water was sinking
in the glass. The men were perched on all vantage points on the
lookout for the delinquent. No sign of him could they discover.
"Get Smithers to haul these barrels filled at the slough," directed
McClure to Snoopy Bill, pointing to the barrels about the engine.
"They'll keep her going until I can find that blankety Nick."

McClure had barely set off on his quest when one of the
teamsters called the attention of the gang to the sudden "hang-up" of
the Valley machine. As an hour passed and there was no sign of the
Valley men resuming work, Snoopy Bill and his companions grew
jubilant to a degree.
Nearly two hours later McClure appeared riding the tank and
towing his buggy, in which lay the inebriate tanker.
A few minutes after, the Mogul was driving ahead under full
pressure, joined shortly by the distant hum of the Valley Gang. Into
the dark they raged, fighting ahead until eight, when the defiant
whistles of the rival engines told that the great run was over.
VI
THE THRESHING CHAMPIONS
Louie Swale's restaurant was full, choked with threshers agog for the
result of the great struggle. Almost every individual present had a
stake involved. The building was a uniquely composite plant,
comprising department store, café, bar, club, all under the solitary
genius of the rotund and active Swale. He combined the offices of
proprietor, manager, floor-walker, bartender, chef, cashier, possessing
an innocent smile of friendliest amenity and the obsequious
deportment of a suave head-waiter. He had certain periodic fines to
meet for the vending of ancient beverages that fell without the code.
These he paid promptly with sanguine light-heartedness. Louie Swale
was universally liked, as are all good fellows whom careless Nature

throws into life incomplete in the entire central osseous system of the
vertebrate. He was a fat, juicy, even companionable earthworm.
The store carried a thorough line from roots to ribbons,
occupying the front section of the building. Out of the store one
wandered into a long room, low and rectangular, where Louie
dispensed the quaffable and edible mysteries of his bar-café. The rear
apartment was a blind room some twenty feet square, containing a
few rough chairs and a round table covered with a green baize cloth.
A well-thumbed pack on the centre of the table was the only
purposeful article visible. There were two doors, both provided with
heavy bars on the inside. One opened into the outshed; the other into
the bar. This door was locally renowned as The Green Baize Door, and
was believed to secrete behind its baize-covered panels a barrel of
mysteries unco', cabalistic and otherwise. Since it was windowless,
two dirty lamps did duty night and day. Obviously, when the "Square
Room" was occupied seriously the Green Baize Door was to be found
shut. At such times a peculiar knock was the sesame.
Store and café were crowded with men anxious to hear the
momentous decision of Jack Butte. Suddenly there arose a stamping
and shouting. The stakeholder had climbed up on a table and was
calling order. Glasses were set down and cards stacked.
"Gentlemen!" he cried. "There is a little preliminary or two I must
pull off before I can announce the winner of the threshing bout
between Rob McClure and Ned Pullar. Whatever the result, I appeal to
the winners and losers to take their medicine. I want the word of both
bosses that they will not stand for any sorehead business or rough
house. I'll not hand out the totals until I get that word."
Butte paused significantly.
"Go ahead," said Ned, with a grin. "We'll be good."

"Agreed!" exclaimed McClure. "My gang is no bunch of squealers.
Spit it out."
"Thank you, gentlemen," said Butte. "That is satisfactory. But
there is another matter. Before I hand out the stakes I want you to
choose two rank outsiders from this crowd who shall go into the
Square Room with me and verify my figures. When they have made
an audit I will come out and give you the facts."
Speedily the arrangement was effected and the three men went
in behind the Green Baize Door.
During the interim Easy Murphy shuffled close to Snoopy Bill
Baird. Grinning insolently into his face he addressed him in a
cavernous stage whisper.
"How's the buttercups, Snoopy?" said he. "Ye did not consarn
yersilf wid a second bokay."
Andy Bissett, standing near, placed his hand deterringly on Easy's
shoulder.
"Steady, lad!" he whispered. "Ned's given his word. Keep in line."
Snoopy Bill ruffled instantly at the thrust. With a quick snatch at
his breast pocket he drew out a bunch of bills and fluttered them
flauntingly in Easy's face.
"How about a bokaa-y of these nice green shamrocks?" said he,
with an exasperating laugh. "Have you the eye for a fresh fifty?"
"Indade, and they are the purty flowers," was the quick response.
"They're to be had fer the pickin'. I'm wid ye, Snoopy."
Quickly he covered the bet, placing the stake with a bystander.
The incident stimulated an emulation in the crowd, and by the time
Butte appeared again the excitement had risen to the point of
explosion.

"Hold your horses for a little!" he cried, smiling into the glaring
eyes of the gamesters. "I'll go right to the point. For a month past
these two gangs have been hammering away to roll up a big total,
and I want to tell you they have done it. The gangs have worked
twenty-seven full days and have made the record runs of the Pellawa
country."
Butte's deliberate manner was too slow for his strained audience.
"Cut the talk, Jack! Cough up the totals!" yelled a voice.
"Hear, hear!" came an applauding roar.
"To resume," said Butte, bowing pleasantly, "in estimating the
oats I reduced them to a total weight and then dividing by sixty,
found the equivalent in weight of wheat. The total is therefore stated
in terms of wheat. This was agreed upon by the two bosses. Rob
McClure's machine has turned out a total of seventy thousand, eight
hundred and twenty-one bushels."
At the announcement the McClure gang and their partisans lifted
a shout of elation. Above the ensuing hubbub rose the brogue of Easy
Murphy:
"Shure, Johnny Butte, 'tis a swell towtal. But ye'll hev till open yer
mug wider, begobs, whin ye give the Valley count."
In spite of the extreme tension a boisterous roar greeted the
defy.
"Against this," said the stakeholder amid a breathless silence,
"the Valley Outfit have rolled up the huge total of seventy-one
thousand, nine hundred and fifty-five bushels——"
His words were drowned in a wild ringing cheer. Led by Murphy's
deep bass roar, the Valley Outfit let go. As the rumpus died down
Andy Bissett lifted his cap and shouted:
"Three cheers for Rob McClure's gang. They made a great run."

Ere they could raise the shout McClure yelled:
"No! Saw off your blankety howl. We want none of it. You doped
one of my men or you would never have turned the trick."
Easy Murphy's lips were framing a reply when Ned spoke up.
"I want to state," said he with quiet deliberateness, "that as far
as my knowledge goes, the Valley Gang has run this thing as straight
as a whip. I appeal to Jack Butte. Do we win on our merits?"
A chorus of applause greeted Ned's words.
"Gentlemen!" replied the stakeholder. "This game has been run
on the square. My figures have been verified and are open to the
public. The Valley Outfit are the undisputed champions of The
Qu'Appelle. Come up to the counter and I'll pay over the cash."
The convivial spirit ran high as the wagers were collected. In the
rear of the room McClure and his men held angry concourse.
Suddenly they pushed their way to the counter. McClure spoke loudly,
his face and eyes aflame.
"Come, Swale," commanded he. "We set up the drinks for the
house. Make it hard stuff all round."
His manner was offensive. Ostensibly the host, he was really the
bully. The Valley Outfit made no move to accept the proffered treat.
Ned Pullar stepped up to his sullen opponent.
"No, Rob McClure!" was his crisp exclamation, accompanied by a
flash of indignant eyes. "We don't drink with gentlemen who insult us
in the same breath. The Valley Outfit, with their little thirty-six inch
mill, beat you to a frazzle. You'll never have a chance like this again,
for next fall will find The Qu'Appelle Champions capering about the
finest mill on the Pellawa plains. You look, Rob, almost mad enough to
fight. Very well. I have given Jack Butte my word to keep quiet. The
Valley Outfit is going to get out and leave you the whole house. If you

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