Arithmetic Theory of Symmetrizable Split Maximal Kac-Moody ...cass/research/hesam-thesis.pdf ·...

Arithmetic Theory of Symmetrizable SplitMaximal Kac-Moody Groups

by

Hesameddin Abbaspour Tazehkand

B.Sc., Sharif University of Technology, 2004M.Sc., Simon Fraser University, 2006

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

The Faculty of Graduate Studies

(Mathematics)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

November 2012

© Hesameddin Abbaspour Tazehkand 2012

Abstract

In this thesis we present a reduction theory for the symmetrizable split maximal

Kac-Moody groups. However there are many technical difficulties before one can

even formulate a reduction theorem. Combining the two main approaches com-

monly seen in the literature we define a group, first over any field of characteristic

zero and then on any commutative ring of characteristic zero. Then we prove a

number of structural properties of the group such as representation in the highest

weight modules, existence of a Tits system and an Iwasawa decomposition over R

and C. Finally we arrive at reduction theory which can only hold for part of the

group.

ii

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 The Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Kac-Moody Algebras: A Primer . . . . . . . . . . . . . . . . . . . . 4

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 A Classification of GCMs . . . . . . . . . . . . . . . . . . . . . 5

2.3 Root System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 An Analogue of the Killing Form . . . . . . . . . . . . . . . . . 9

2.5 Integrable Modules . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6.2 Action on the Weight Space . . . . . . . . . . . . . . . . 14

2.6.3 W as a Coxeter Group . . . . . . . . . . . . . . . . . . . 15

2.7 Geometry of the Weyl Group . . . . . . . . . . . . . . . . . . . . 15

2.7.1 Real and Imaginary Roots . . . . . . . . . . . . . . . . . 15

2.7.2 The Tits Cone . . . . . . . . . . . . . . . . . . . . . . . 15

3 Kac-Moody Algebras: Highest Weight Modules . . . . . . . . . . . 18

3.1 Verma Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

iii

Table of Contents

3.2 The Irreducible Quotient . . . . . . . . . . . . . . . . . . . . . . 19

3.3 The Shapovalov Bilinear Form . . . . . . . . . . . . . . . . . . . 20

3.4 A Positive Definite Inner Product . . . . . . . . . . . . . . . . . 22

4 Kac-Moody Algebras: Arithmetic Theory . . . . . . . . . . . . . . 24

4.1 An Integral Form for UC.g/ . . . . . . . . . . . . . . . . . . . . 24

4.1.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.2 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . 25

4.2 The Chevalley Lattice . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.2 Proof of Theorem 4.18 . . . . . . . . . . . . . . . . . . . 30

5 Groups over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 HQ.m/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.2 Hopf Algebra Structure . . . . . . . . . . . . . . . . . . 35

5.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Peter-Weyl Type Theorems . . . . . . . . . . . . . . . . . . . . . 40

5.3 The Unipotent Subgroup . . . . . . . . . . . . . . . . . . . . . . 42

5.4 The Split Maximal Kac-Moody Group . . . . . . . . . . . . . . . 45

6 Groups over Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.1 HZ.m/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Integrality Conditions . . . . . . . . . . . . . . . . . . . . . . . 47

6.2.1 Hopf Algebra . . . . . . . . . . . . . . . . . . . . . . . 47

6.2.2 Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3 Integral Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . 50

6.4 The Arithmetic Group . . . . . . . . . . . . . . . . . . . . . . . 52

7 Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.1 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . 53

7.1.1 Constructing the Map . . . . . . . . . . . . . . . . . . . 53

7.1.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1.3 The Arithmetic Subgroup and the Chevalley Lattice . . . 56

iv

Table of Contents

7.2 Existence of a Tits System . . . . . . . . . . . . . . . . . . . . . 57

7.3 Iwasawa Decomposition . . . . . . . . . . . . . . . . . . . . . . 59

7.4 The Orbit of the Highest Weight Vector . . . . . . . . . . . . . . 62

8 Reduction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.1 The Four Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.2 The Arithmetic Set . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.3 Points with Minima . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.3.1 Proof of Reduction Theorem . . . . . . . . . . . . . . . 70

8.4 A Subset of �GMO . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.4.1 A Spectral Characterization of INT.T / . . . . . . . . . . 73

8.4.2 Decay and Minima . . . . . . . . . . . . . . . . . . . . . 75

8.5 �G[ is not � -invariant . . . . . . . . . . . . . . . . . . . . . . . . 77

8.5.1 Iwasawa Decomposition in SL2 . . . . . . . . . . . . . . 77

8.5.2 The Counterexample . . . . . . . . . . . . . . . . . . . . 78

8.6 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Appendices

A Formulas in Associative Algebras . . . . . . . . . . . . . . . . . . . 83

B Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

C Hopf Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

C.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

C.2 Enveloping Algebra . . . . . . . . . . . . . . . . . . . . . . . . 89

C.3 The Dual of the Enveloping Algebra . . . . . . . . . . . . . . . . 89

D Amalgams and Tits Systems . . . . . . . . . . . . . . . . . . . . . . 93

D.1 Direct Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

D.2 Tits Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

D.3 Tits’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

v

Acknowledgements

This thesis would not exist without the advice and encouragement of my advisor,

Bill Casselman, who suggested the topic and insisted I finish. I should also mention

the generous financial support he has provided me for more than six years, which

made this thesis possible in the first place.

I am very much indebted to Julia Gordon. Her patience, concern and care

went beyond anything required or expected of her. The very least of which was to

carefully read an early draft of this thesis and point out many small errors.

I would like to thank H. Garland for providing me with original copies of [10]

and [13], J. Humphreys for mailing me hard copies of [26] and [27] (which are very

hard to find) and S. Kumar for scanning and emailing [23] (another text which is

not widely available). While these are not cited in the body of the thesis, reading

them was very helpful to me as it enabled me to find the right frame to deal with

the problems that are the subject of the thesis.

vi

Chapter 1

Introduction

1.1 The Aim

The goal of this Thesis is to state and prove a reduction theorem for Kac-Moody

groups which arise from generalized Cartan matrices (GCMs) which are sym-

metrizable and invertible. In fact the thesis generalizes the earlier work by H.

Garland in [11] and [12] which only deals with untwisted affine GCMs. The only

exceptions are §18 and §21 in [12]. The former constructs a fundamental domain

for the unipotent subgroup while the latter deals with intersection of the � -orbits

with the Siegel set. The question of constructing a fundamental domain does not

generalize since his construction relies on the realization of untwisted affine Kac-

Moody algebras as central extensions of loop algebras. As for the intersection of

the � -orbits with the Siegel set there is a analogous theory in the general sym-

metrizable case however this is not covered in the thesis.

1.2 Structure of the Thesis

The thesis has seven chapters beside the current one and each chapter in the thesis

begin with short overview of what will follow. The seven chapters can be divided

in four parts:

� Chapters 2 and 3 chapters provide an introduction to theory of Kac-Moody

algebras and their representation theory.

� The material in chapter 4 corresponds to [11], it provides an arithmetic the-

ory for Kac-Moody algebras.

� In chapters 5, 6 and 7 we define the maximal Kac-Moody group. Most of

1

1.3. Notation

the material is from chapter I of [17]. After defining a number of subgroups

(including the minimal parabolics and the Borel) O. Mathieu employs tech-

niques from algebraic geometry to define a group by constructing every sin-

gle Bruhat cell and constructing global product and inverse maps. Instead

of doing this we simply use Tits’s work and take the maximal Kac-Moody

group to be the product of parabolic subgroups amalgamated along their in-

tersections.

� Chapter 8 deals with reduction theory of the group constructed earlier and

contains all the new results. First we encounter the problem that unlike the

finite dimensional case we can not have a reduction theory for the entire

group. So one has to find a � -invariant proper subset of the group which is

well behaved. There are various candidates, we examine two in detail, prove

reduction theory for one of them and show that it contains a large subset.

Finally we conclude by listing some open problems.

1.3 Notation

Here is some of the notation used in the thesis:

� C;N;Q;R;Z: the usual suspects.

� ˝ always indicates˝Z.

� F is a field of characteristic zero.

� R is a commutative ring with a unit.

� For any F-vector space VF we let V �F denote its linear dual: HOMF.VF; F/.

� h�; �i W V �F � VF! F is the natural pairing between VF, and its dual, V �F.

� HOMR.�; �/: the set of R-module homomorphisms.

� HOMR�alg.�; �/: the set of R-algebra homomorphisms.

� HOMR�lie.�; �/: the set of R-Lie algebra homomorphisms.

2

1.3. Notation

� UR.l/ the universal enveloping algebra of the R-Lie algebra lR.

� zUR.l/ is the augmentation ideal in UR.l/.

There is also an index of notation which gives the page on which the symbol

was first introduced or defined.

3

Chapter 2

Kac-Moody Algebras: A Primer

Kac-Moody algebras are a class of Lie algebras generalizing the notion of finite

dimensional semi-simple Lie algebras (see Theorem 2.11 below). In this chapter

we define the Kac-Moody algebras, and introduce some elementary concepts.

Our references for basic theory of Kac-Moody algebras are [7] (chapters 14 -

16 and 19) and [15].

2.1 Definitions

Notation 2.1. Let n 2 N and set I D f1; � � � ; ng.

Definition 2.2. A generalized Cartan matrix (or GCM for short) is a square matrix

A D�Aij�i;j2I

satisfying the following:

(1) Aij 2 Z for all i; j 2 I .

(2) Ai i D 2 for all i 2 I .

(3) Aij � 0 if i ¤ j .

(4) Aij ¤ 0 if and only if Aj i ¤ 0 for all i; j 2 I .

Definition 2.3. Let A be a GCM of size n and corank r . A realization of A is a

triplet: .aC; ˘;˘_/, where:

(1) aC is a C-vector space of dimension nC r .

(2) ˘ D f˛1; � � � ; ˛ng is a linearly independent subset of a�C .

(3) ˘_ D˚˛_1 ; � � � ; ˛

_n

is a linearly independent subset of aC .

(4)D˛i ; ˛

_j

ED Aj i for all i; j 2 I .

4

2.2. A Classification of GCMs

Remark 2.4. If dim.aC/ < nCr then one can not find linearly independent subsets

˘ � a�C and ˘_ � aC which satisfy (4) in Definition 2.3.

Remark 2.5. For a given GCM, a realization always exists and is unique up to

isomorphism of vector spaces, see Proposition 14.2 and 14.3 in [7].

Definition 2.6. Let A be a GCM with realization .aC; ˘;˘_/. The associated

Kac-Moody algebra gC is the C-Lie algebra generated by aC and 2n generators

fe˙i W i 2 I g, subject to the following relations:

ŒaC; aC� D 0�ei ; e�j

�D ıij˛

_i 8i; j 2 I

Œ´; e˙i � D ˙h˛i ; í e˙i 8i 2 I;8´ 2 aC

And the Serre relations:

ad.e˙i /1�Aij .e˙j / D 0; 8i; j 2 I W i ¤ j:

Remark 2.7. Note that this is not the definition given in [7] and [15], however

for the particular class of GCMs we are interested in (symmetrizable Kac-Moody

algebras) the two definitions coincide (Theorem 19.30 in [7] or Theorem 9.11 in

[15]).

Proposition 2.8 ([7] Proposition 14.17). There is an automorphism ! of gC satis-

fying !2 D 1 determined by:

!.e˙i / D �e�i ; !jaC D �1aC :

2.2 A Classification of GCMs

Definition 2.9. Two GCMs, A and B are called equivalent if they have the same

size n and there is a permutation � of I such that:

Bij D A�.i/�.j /; 8i; j 2 I:

5

2.2. A Classification of GCMs

Definition 2.10. A GCM, A, is called decomposable if it is equivalent to a diagonal

sum A1 0

0 A2

!of smaller GCMs A1;A2. A GCM that is not decomposable is called indecompos-

able.

If A is a GCM so is its transpose. Moreover A is indecomposable if and only if

its transpose is indecomposable.

Let v D .v1; � � � ; vn/ be a vector in Rn. We write v � 0 (v > 0) if vi � 0

(vi > 0) for each i . Then one has the following classification of indecomposable

GCMs (see [7] Corollary 15.11) into three classes each of which is closed under

taking transpose:

(1) A has finite type if and only if there exists u > 0 with Au > 0.

(2) A has affine type if and only if there exists u > 0 with Au D 0.

(3) A has indefinite type if and only if there exists u > 0 with Au < 0.

The following justifies our claim that Kac-Moody algebras generalize the no-

tion of finite dimensional simple Lie algebras:

Theorem 2.11 ([7] Theorem 15.19). Let A be an indecomposable GCM. Then A

has finite type if and only if it is the Cartan matrix of a finite dimensional simple

Lie algebra.

At any rate from now on we will assume that the GCM is invertible, in partic-

ular r D 0. This is not an essential assumption for what we want to do but it will

simplify our task.

6

2.3. Root System

2.3 Root System

Definition 2.12. We define two discrete additive subgroup of a�C and aC generated

by ˘ and ˘_:

Q D Z˛1 ˚ � � � ˚ Z˛n;

Q_ D Z˛_1 ˚ � � � ˚ Z˛_n :

Q is called the root lattice, and Q_ the coroot lattice. Finally set:

Q˙ D Z˙˛1 ˚ � � � ˚ Z˙˛n:

Definition 2.13. For ˛ DPki˛i 2 Q the height of ˛ is the number ht.˛/ DP

ki .

Definition 2.14. Introduce a partial ordering � on a�C by setting � � � if �� 2

QC.

Definition 2.15. For every aC-module VC and every � 2 a�C we define the weight

space associated to � as :

VC;� D fv 2 VC W ´ � v D h�; í v; for all ´ 2 aCg :

The elements of VC;� are called the weight vectors corresponding to � and the

dimension of VC;� is the multiplicity of the weight �. The set:˚� 2 a�Cnf0g W VC;� ¤ f0g

;

is called the weights of VC .

Definition 2.16. With the adjoint action, ´�x D Œ´; x�, gC becomes an aC-module.

In this particular case the weight vectors and weight spaces are referred to as roots

and root spaces respectively, while the weights of gC will be denoted by � and

called the root system of gC .

Proposition 2.17 ([7] Proposition 14.18).

7

2.3. Root System

(1) gC DL˛2Q gC;˛.

(2) dim�gC;˛

�<1 for all ˛ 2Q.

(3) gC;0 D aC .

(4) If ˛ ¤ 0 then gC;˛ D 0 unless ˛ 2Q.

(5)�gC;˛; gC;ˇ

�� gC;˛Cˇ for all ˛; ˇ 2Q.

Definition 2.18. Proposition 2.17 shows that � � Q. The roots in QC are called

positive roots and denoted by�C and the roots in Q� are called negative roots and

denoted by ��. If we set:

n˙C DM˛2�˙

gC;˛

then we have a direct sum of C-vector spaces:

gC D n�C ˚ aC ˚ nCC ;

This direct sum decomposition is referred to as the triangular decomposition of

gC . The subspaces n�C , aC and nCC are in fact Lie subalgebras of gC . In accordance

with the finite dimensional case aC is called the Cartan subalgebra of gC .

Proposition 2.19 ([7] Proposition 14.19).

(1) dim�gC;˛i

�D dim

�gC;�˛i

�D 1.

(2) If k > 1 then dim�gC;k˛i

�D dim

�gC;�k˛i

�D 0.

Definition 2.20. For each i 2 I we define the following subalgebras:

n˙˛i ;C D Ce˙i D gC;˙˛i :

The subalgebra bC D aC ˚ nCC is called the Borel algebra of gC . For each i 2 I

the corresponding minimal parabolic algebra is defined as: pi;C D n�˛i ;C ˚ bC .

8

2.4. An Analogue of the Killing Form

2.4 An Analogue of the Killing Form

For the finite dimensional semi-simple Lie algebras the Killing form on a finite

dimensional semi-simple Lie algebra is defined as:

.xjy/ D trace.ad.x/ ı ad.y//:

The Killing form has very desirable properties: it is a non-degenerate symmetric

bilinear form that is invariant, i.e. one has: .Œx; y�j´/ D .xjŒy; ´�/. We would

like to define an analogue of the Killing form for Kac-Moody algebras that are

not of finite type. However since these are infinite dimensional, the expression

trace.ad.x/ ı ad.y// is not always defined. Therefore we will impose a further

restriction on our GCM so that gC has an analogue of the Killing form.

Definition 2.21. A GCM, A, is called symmetrizable if there exists a nonsingular

diagonal matrix D D diag.d1; � � � ; dn/ and a symmetric matrix B, such that A D

DB.

Remark 2.22. Indecomposable GCMs of finite or affine type are symmetrizable,

see Theorem 15.17 in [7].

Definition 2.23. On aC define:�˛_i

ˇ̌̌˛_j

�D

�˛_j

ˇ̌̌˛_i

�D didjBij :

Since ˘_ is a basis for aC by extending using linearity we obtain a symmetric

bilinear form on the Cartan subalgebra.

Proposition 2.24 ([7] Proposition 16.1). The symmetric bilinear form on aC de-

fined above is non-degenerate.

Based on Proposition 2.24 we define a bijection a�C ! aC given by: ˛ 7! ´˛

where ´˛ is defined by:

.´˛j´/ D h˛; í ; 8´ 2 aC;

in particular we have ˛_i D di´˛i . Using this bijection we can define the induced

9

2.5. Integrable Modules

bilinear form on a�C:.�j�/´

�´�ˇ̌´��:

In particular we have:�˛iˇ̌j̨

�D Bij .

Theorem 2.25 ([15] Theorem 2.2, Exercise 2.2). If A is symmetrizable then gC has

a non-degenerate symmetric bilinear C-valued form such that:

(1) .�j�/ is invariant.

(2) When restricted on aC , .�j�/ is given by Definition 2.23.

(3)�gC;˛

ˇ̌gC;ˇ

�D 0 unless ˛ C ˇ D 0.

(4) Suppose x 2 gC;˛; y 2 gC;�˛ then Œx; y� D .xjy/ ´˛.

(5) The pairing gC;˛ � gC;�˛ given by .x; y/ 7! .xjy/ is non-degenerate.

(6) For each 0 ¤ x 2 gC;˛ there exists y 2 gC;�˛ with Œx; y� ¤ 0.

(7) .�j�/ is uniquely determined by (1) and (2).

Definition 2.26. The form of Theorem 2.25 is called the standard invariant form

on gC .

Remark 2.27. When A is of finite type the standard invariant form is a multiple of

the Killing form.

2.5 Integrable Modules

Definition 2.28. A linear endomorphism F of vector space VC is called locally

nilpotent if for every vector v 2 VC there exists N 2 N such that FN .v/ D 0.

Definition 2.29. A representation, � W gC ! gl.VC/, is called integrable if:

VC DM�2a�C

VC;�;

and if �.e˙i / are locally nilpotent endomorphisms of VC for all i 2 I .

10

2.5. Integrable Modules

Lemma 2.30 ([7] Proposition 7.17). For all i 2 I , ad.e˙i / are locally nilpotent

endomorphisms of gC , in other words the adjoint module is integrable.

Proof. Since we already know that gC decomposes into aC weight spaces, we only

need to show that ad.e˙i / are locally nilpotent linear endomorphisms of gC for all

i 2 I . We will show ad.ei / is locally nilpotent, the proof for ad.e�i / is similar.

First we claim that if ad.ei / acts locally nilpotently on x and y then it also locally

nilpotently on Œx; y� to see this consider:

ad.ei /N .Œx; y�/ DNXkD0

N

k

! �ad.ei /k.x/; ad.ei /N�k.y/

�;

ad.ei /k.x/ will be 0 if k is sufficiently large and ad.ei /N�k.y/ will be 0 if N �

k is sufficiently large. Thus ad.ei /N .Œx; y�/ will be 0 if N is sufficiently large.

Therefore the set of elements of gC on which ad.ei / acts locally nilpotently is a

subalgebra. However

ad.ei /.ei / D 0

ad.ei /1�Aij .ej / D 0 i ¤ j

ad.ei /2.˛_j / D 0

ad.ei /3.e�i / D 0

ad.ei /.e�j / D 0 i ¤ j

and therefore the subalgebra contains all the generators, so it is the whole of gC .

11

2.6. Weyl Group

2.6 Weyl Group

2.6.1 Definition

Definition 2.31. For a locally nilpotent endomorphism F of a vector space VC we

define its exponential, EXP.F /, as the formal sum:

EXP.F / D

1XkD0

F k

kŠ:

Definition 2.32. If � W gC ! gl.VC/ is an integrable module for gC then for each

i 2 I we can define:

r�i D EXP.�.ei // EXP.�.�e�i // EXP.�.ei // 2 GL.VC/:

In particular since the adjoint module is integrable, for each i 2 I we have an

automorphism radi 2 GL.gC/.

Lemma 2.33 ([7] Propositions 16.11). radi .aC/ D aC . For ´ 2 aC we have:

radi .´/ D ´ � h˛i ; í˛

_i :

Proof. Let ´ 2 aC , we will compute the action of radi term by term. First we have:

EXP.ad.ei //.´/ D .1C ad.ei //.´/ D Ć Œei ; ´� D ´ � h˛i ; í ei :

Based on this we add the second term:

EXP.ad.�e�i // EXP.ad.ei //.´/ D EXP.ad.�e�i //�´ � h˛i ; í ei

�D

�1 � ad.e�i /C

ad.e�i /2

2

��´ � h˛i ; í ei

�D ´ � h˛i ; í ei � Œe�i ; ´�C h˛i ; í Œe�i ; ei �

C12

ad.e�i /�Œe�i ; ´� � h˛i ; í Œe�i ; ei �

�D ´ � h˛i ; í ei � h˛i ; í e�i � h˛i ; í˛

_i

C12

ad.e�i /�h˛i ; í e�i C h˛i ; í˛

_i

�12

2.6. Weyl Group

D ´ � h˛i ; í ei � h˛i ; í e�i � h˛i ; í˛_i

C12

�0C h˛i ; í .2e�i /

�D ´ � h˛i ; í ei � h˛i ; í˛

_i

Finally:

radi .´/ D EXP.ad.ei // EXP.ad.�e�i // EXP.ad.ei //.´/

D EXP.ad.ei //�´ � h˛i ; í ei � h˛i ; í˛

_i

�D .1C ad.ei //

�´ � h˛i ; í ei � h˛i ; í˛

_i

�D�´ � h˛i ; í ei � h˛i ; í˛

_i

�C�Œei ; ´� � 0 � h˛i ; í

�ei ; ˛

_i

��D ´ � h˛i ; í˛

_i � 2 h˛i ; í ei C h˛i ; í .2ei /

D ´ � h˛i ; í˛_i

Definition 2.34. Let ri denote the restriction of radi to aC , then one gets: r2i D 1aC

and ri .˛_i / D �˛_i . In fact we have:

ri .´/ D ´ � h˛i ; í˛_i :

ri are called the fundamental reflections, the group they generate (as a subgroup of

GL.aC/) is called the Weyl group and is denoted by W .

Proposition 2.35 ([7] Proposition 16.13). The bilinear form .�j�/ on aC is W -

invariant.

Proof. Let ´; ´0 2 aC . Then:�ri .´/

ˇ̌ri .´

0/�D�´ � h˛i ; í˛

_i

ˇ̌´0 �

˝˛i ; ´

0˛˛_i�

D�´ˇ̌´0�� h˛i ; í

�˛_iˇ̌´0��˝˛i ; ´

0˛ �´ˇ̌˛_i�C˝˛i ; ´

0˛h˛i ; í

�˛_iˇ̌˛_i�

D�´ˇ̌´0�� h˛i ; í

�di´˛i

ˇ̌´0��˝˛i ; ´

0˛ �´ˇ̌di´˛i

�C˝˛i ; ´

0˛h˛i ; í .2di /

D�´ˇ̌´0�� h˛i ; í di

˝˛i ; ´

0˛�˝˛i ; ´

0˛di h˛i ; í C

˝˛i ; ´

0˛h˛i ; í .2di /

D�´ˇ̌´0�

13

2.6. Weyl Group

In fact a converse is true that is if there exists a non-degenerate symmetric

W -invariant bilinear form on aC then A is symmetrizable ([15] Exercise 3.3).

2.6.2 Action on the Weight Space

Definition 2.36. We define a W -action on a�C , let w 2 W and � 2 a�C then the

weight w.�/ is defined as follows:

8´ 2 aC W hw.�/; í D˝�; w�1.´/

˛:

Lemma 2.37. The W -action on a�C is compatible with the bijection ˛ 7! ´˛ and

hence the induced bilinear form .�j�/ on a�C is W -invariant as well.

Proof. Let w.�/ D � for �;� 2 a�C and take ´ 2 aC to be arbitrary:

.w.´�/j´/ D�´�ˇ̌w�1.´/

�D˝�; w�1.´/

˛D hw.�/; í D h�; í D

�´�ˇ̌´�D�´w.�/

ˇ̌´�

Since .�j�/ is non-degenerate and ´ 2 aC is arbitrary we have: w.´�/ D ´w.�/.

For the W -invariance, let �;� 2 a�C and observe:�rj .�/

ˇ̌rj .�/

�D�´rj .�/

ˇ̌´rj .�/

�D�rj .´�/

ˇ̌rj .´�/

�D�´�ˇ̌´��D .�j�/ :

Lemma 2.38 ([7] Proposition 16.14). The action of ri on a�C is given by: ri .�/ D

� �˝�; ˛_i

˛˛i .

Proof. Let � 2 a�C; ´ 2 aC:

hri .�/; í D h�; ri .´/i

D˝�; ´ � h˛i ; í˛

_i

˛D h�; í �

˝�; ˛_i

˛h˛i ; í

D˝� �

˝�; ˛_i

˛˛i ; ´

˛Proposition 2.39 ([7] Proposition 16.15). If ˛ 2 �;w 2 W then w.˛/ 2 �.

Moreover dim�gC;˛

�D dim

�gC;w.˛/

�.

14

2.7. Geometry of the Weyl Group

2.6.3 W as a Coxeter Group

Theorem 2.40 ([7] Theorem 16.17). The Weyl group W is a Coxeter group gen-

erated by r1; � � � ; rn with relations:

r2i D 1�rirj

�2D 1 if AijAj i D 0�

rirj�3D 1 if AijAj i D 1�

rirj�4D 1 if AijAj i D 2�

rirj�6D 1 if AijAj i D 3

2.7 Geometry of the Weyl Group

2.7.1 Real and Imaginary Roots

Definition 2.41. ˛ 2 � is called a real root if there exist ˛i 2 ˘ and w 2W such

that ˛ D w.˛i /, the set of all real roots is denoted by �re. A root that is not real

is called imaginary and the collection of all imaginary roots is indicated by �im.

Finally �re˙; �im˙

are defined as the 4 possible intersections of real and imaginary

roots with positive and negative ones.

Remark 2.42. Real roots behave very much like the roots of finite dimensional

semi-simple Lie algebras: they have multiplicity 1 and the only multiples of a real

root ˛ that themselves are roots are ˙˛ ([7] Proposition 16.18). Imaginary roots

on the other hand have no counterpart in the finite dimensional Lie algebras (see

Proposition 16.27 in [7]). Moreover if ˛ 2 �imC

then k˛ 2 �imC

for all k 2 ZC

([7] Corollary 16.25) which in turn implies that gC is infinite dimensional exactly

when A is not of finite type.

2.7.2 The Tits Cone

Definition 2.43. Given a GCM A of size n let aR be an n-dimensional R-vector

space such that�C ˝R aR; ˘;˘

_�

is a realization for A. In our case, since we

15


assume A to be invertible, we may take aR to be the R-subspace in aC generated

by simple coroots.

Definition 2.44. We define the Tits cone and the open Tits cone as the following

subsets in aR:

T D˚´ 2 aR W h�; í < 0; for finitely many � 2 �re

C

INT.T / D

˚´ 2 aR W h�; í � 0; for finitely many � 2 �re

C

Here INT.T / is the interior of T in the metric topology of aR.

Remark 2.45. A GCM A has finite type if and only if T D INT.T / D aR.

The subsets T ; INT.T / are closely related to the action of the Weyl group:

Definition 2.46. For each subset J � I the corresponding face in aR is defined as

follows:

FJ D f´ 2 aR W h˛i ; í D 0;8i 2 J and h˛i ; í > 0;8i … J g :

Given any subset J � I we say J has finite type if the principal submatrix corre-

sponding to J has finite type. Now define:

D D[J�I

FJ

Dfin D[J�I

J has finite type

FJ

Proposition 2.47 ([2] Proposition 4.4.9).

(1) T DW �D.

(2) INT.T / DW �Dfin.

The following characterization of the closure of the Tits cone in the metric

topology of aR will be useful later:

16


Proposition 2.48 ([29] Proposition 5.6). If A is of indefinite type then:

CL.T / D˚´ 2 aR W h�; í � 0; 8� 2 �

imC

:

17

Chapter 3

Kac-Moody Algebras: HighestWeight Modules

In this chapter we introduce the concept of a highest weight module. Almost all of

the theory of this class of representations is similar to that of the finite dimensional

case; one major difference is that when our GCM is not of finite type then the

irreducible quotient of the Verma module is not finite dimensional.

One of the most important aspects of the theory is the Shapovalov bilinear

form (see Definition 3.11), originally introduced in [22]. This provides us with

a non-degenerate symmetric contravariant bilinear form on any irreducible high-

est weight module (see Proposition 3.13). Furthermore, if the irreducible highest

weight module is integrable as well, we get a positive definite inner product.

3.1 Verma Modules

Definition 3.1. Let � 2 a�C and define K�C to be the left ideal of UC.g/ generated

by nCC and all elements of the form ´ � h�; í where ´ 2 aC . The Verma module

with highest weight � is defined as:

M.�/C D UC.g/=K�C :

Proposition 3.2 ([7] §19.1). Let 1� 2M.�/C be the image of 1 2 UC.g/. Then:

(1) Every element of M.�/C is uniquely expressible in the form of u � 1� for

some u 2 UC.n�/.

(2) M.�/C DL�2a�C

M.�/C;�.

18

3.2. The Irreducible Quotient

(3) M.�/C;� ¤ 0 if and only if � � �.

Another way of defining the Verma module is as follows: let C� be a 1-

dimensional vector space on which aC acts via h�; �i. We extend this to a rep-

resentation of bC D aC ˚ nCC by requiring that nCC act trivially. Now we have:

M.�/C ´ INDgCbC

�C�

�D UC.g/˝UC.b/ C�:

Lemma 3.3. The product map gives us an isomorphism of C-vector spaces:

UC.n�/˝C UC.a/˝C UC.n

C/ Š UC.g/:

Proof. This follows from the triangular decomposition of gC and the PBW Theo-

rem.

In particular we have: UC.g/ Š UC.n�/ ˝C UC.b/. Therefore as UC.n

�/-

modules, M.�/C Š UC.n�/, where UC.n

�/ acts on itself via left multiplication.

In other words as UC.n�/-modules, all Verma module look the same.

3.2 The Irreducible Quotient

Definition 3.4. The Verma module has a unique maximal proper submodule ([7]

Theorem 10.9) which we shall denote by M 0.�/C . We define:

L.�/C DM.�/C=M0.�/C:

Then L.�/C is an irreducible module and therefore it is called the irreducible

highest module with highest weight �.

Remark 3.5. From Proposition 3.2 we see that

L.�/C DM�2a�C

L.�/C;�

and that every weight � that appears in L.�/C has to be of the form �� ˛, where

˛ 2QC.

19

3.3. The Shapovalov Bilinear Form

Definition 3.6. The set of weights of L.�/C will be denoted by P�. The depth of

the weight � D � � ˛ 2P�, denoted by dp.�/, is taken to be the ht.˛/.

Definition 3.7. � 2 a�C is called integral if˝�; ˛_i

˛2 Z for all i 2 I . It is called

dominant if˝�; ˛_i

˛> 0 for all i 2 I .

Proposition 3.8 ([7] Proposition 19.14; [16] Corollary 2.1.8). L.�/C is integrable

if and only if � is dominant and integral.

3.3 The Shapovalov Bilinear Form

Definition 3.9. Consider the involution ! W gC ! gC we have from Proposi-

tion 2.8 and let U.!/ W UC.g/ ! UC.g/ be its lift to the universal enveloping

algebra. Now define: � D U.!/ ı , where is the principal anti-automorphism

of UC.g/ (see Appendix C).

Definition 3.10. Based on Lemma 3.3 we can write UC.g/ as a direct sum of two

vector spaces:

UC.g/ D UC.a/˚�n�C � UC.g/C UC.g/ � n

C

C

�:

Let � denote the projection on the first factor, this map is commonly referred to as

the Harish-Chandra map.

Definition 3.11. The Shapovalov bilinear form is defined as follows:

˚S W UC.g/ � UC.g/! UC.a/

S.x; y/´ �.� .x/y/

Proposition 3.12 ([18] §2.8 Proposition 1).

(1) S is symmetric.

(2) For all x; y; u 2 UC.g/;S.ux; y/ D S.x; � .u/y/.

(3) For ˛ ¤ ˇ 2Q;UC.g/˛?UC.g/ˇ .

20

3.3. The Shapovalov Bilinear Form

(4) S.1; 1/ D 1.

Since ˘_ is a basis for the abelian Lie algebra aC (recall our assumption that

the GCM A is of full rank), any weight � 2 a�C can be extended to the polynomial

ring: UC.a/ Š C�˛_1 ; � � � ; ˛

_n

�in a natural way:

��˛_i ˛

_j

�D˝�; ˛_i

˛ D�; ˛_j

E:

Now we define a bilinear form:

S� WM.�/C �M.�/C ! C;

as follows. Let v;w 2M.�/C then by Proposition 3.2 there exist x; y 2 UC.n�/

such that v D x � 1�; w D y � 1� and set:

S�.v; w/ D �.S.x; y//:

Proposition 3.13 ([16] Proposition 2.3.2).

(1) S� is symmetric.

(2) S� is contravariant, that is:

S�.x � v; w/ D S�.v; � .x/ � w/;

for all v;w 2M.�/C and all x 2 UC.g/.

(3) S�.M.�/C;�; M.�/C;�/ D 0 if � ¤ �.

(4) S�.M 0.�/C; M.�/C/ D 0.

(5) S� induces a non-degenerate symmetric contravariant bilinear form onL.�/Calso denoted by S�.

(6) Any contravariant bilinear form on L.�/C is a scalar multiple of S� and

hence it is automatically symmetric.

21

3.4. A Positive Definite Inner Product

Remark 3.14 ([15] §9.4). For any highest weight vector v� 2 L.�/C;�, we define

a corresponding functional: Ev� Œ�� W L.�/C ! C as follows:

v D Ev� Œv�v� C v0; v0 2

M�¤�

L.�/C;�:

Then one may write:

S�.x � v�; y � v�/ D Ev� Œ� .x/y � v��:

A normalization such as S�.1�; 1�/ D 1will determine the bilinear form uniquely,

we will use this normalization from now and we will abbreviate E1� Œ�� to EŒ��.

3.4 A Positive Definite Inner Product

Definition 3.15. Let gR be the real subalgebra of gC generated by fe˙i W i 2 I gand aR. This gives us a conjugate linear involution of gC denoted by u 7! u which

we lift to UC.g/. Since the involution ! satisfies: !.gR/ � gR and we have:

!.u/ D !.u/ we can define a conjugate linear anti-automorphism of UC.g/ of

order two by setting �0 D � (� was introduced in Definition 3.9).

For any � 2 a�R we get a real form: L.�/R ´ gR � 1� � L.�/C and hence a

conjugate linear involution of L.�/C , denoted by v 7! v. We define a Hermitian

form f�; �g on L.�/C by:

fv; wg D S�.v; w/:

Since S� was a contravariant bilinear form, f�; �g becomes a contravariant Hermi-

tian form, this means that we have:

fx � v; wg D fv; �0.x/ � wg :

In order to get an inner product on L.�/C we need f�; �g to be positive definite.

Using the contravariance of f�; �g we can calculate some inner products. For exam-

22

3.4. A Positive Definite Inner Product

ple:

fe�i � 1�; e�i � 1�g D f1�; �0.e�i /e�i � 1�g

D f1�; eie�i � 1�g

D f1�; .Œei ; e�i �C e�iei / � 1�g

D˚1�; ˛

_i � 1�

D˚1�;

˝�; ˛_i

˛1�

D˝�; ˛_i

˛f1�; 1�g

If we use a normalization such as f1�; 1�g D 1 we see that˝�; ˛_i

˛� 0 is a nec-

essary condition for f�; �g being positive definite. In fact with similar calculations

one can show that � being dominant and integral is necessary. However it turns

out that this condition is sufficient as well:

Theorem 3.16 ([16] Theorem 2.3.13). f�; �g is positive definite on L.�/C if and

only if � is dominant and integral.

Notation 3.17. For v 2 L.�/C we set: kvk Dpfv; vg.

23

Chapter 4

Kac-Moody Algebras: ArithmeticTheory

In any integrable highest weight module, L.�/C , we would like to define a lattice

which is compatible with the inner product defined on L.�/C . That is, the inner

product of any two elements in the lattice is an integer, in particular the length of

any element is a positive integer. How should we define such a lattice? Recall that

the highest weight vector generates the module: L.�/C D UC.g/ � 1�. Based on

this we may define L.�/Z D UZ.g/ � 1� where UZ.g/ itself is a lattice in UC.g/

and we show that this is the lattice in L.�/C with the desired properties. This

chapter is divided in two sections: in §1 we first define what we mean by a lattice

in UC.g/ (see Definition 4.1) and then construct one by giving generators. In §2

we show that the subset defined in the highest weight module is a lattice with all

the desired properties.

The material of §1 is based on [28] §4.4 while §2 follows [11], we note that

while the [11] only deals with the specific case of affine GCMs the same proof can

be used for the general case as we show here.

4.1 An Integral Form for UC.g/

4.1.1 Construction

Definition 4.1. An integral form of a C-algebra AC is a subring A � AC such that

the canonical map: C ˝ A! AC is bijective.

Notation 4.2. In order to simplify our notation in this section we will use UC;U0C

and U˙C as a shorthand for UC.g/;UC.a/ and UC.n˙/ respectively.

24

4.1. An Integral Form for UC.g/

Definition 4.3. We define the following subrings of UC (for notation see Appendix

A):

U˙i DMp2N

ZeŒp�˙i

U0 D

* ´

p

!W ´ 2Q_; p 2 N

+U˙ D

˝U˙1; � � � ;U˙n

˛U D

˝U�;U0;UC

˛Theorem 4.4 ([28] §4.4). U is an integral form for UC .

The proof of Theorem 4.4 can be divided in two steps:

(1) U�;U0 and UC are integral forms for U�C;U0C and UCC respectively.

(2) The product map: U� ˝ U0 ˝ UC ! U is bijective.

(1) and (2) combined with Lemma 3.3 would imply that U is an integral form for

UC .

4.1.2 Proof of Theorem 4.4

Proof of (1)

Proposition 4.5. U0 is an integral form for U0C .

Proof. U0C Š C�˛_1 ; � � � ; ˛

_n

�and ˛_1 ; � � � ; ˛

_n 2 U0.

Proposition 4.6. U˙ is an integral form for U˙C .

Proof. Since UC is a subring of UCC , we only need to show that the canonical map

C˝UC ! UCC is bijective. It is surjective because UC contains all the generators

of UCC . Assume it is not injective, that is:

0 ¤

kXiD1

ci ˝ xi 7!

kXiD1

cixi D 0;

25

4.1. An Integral Form for UC.g/

where xi are monomials in UC and ci 2 C are all nonzero. Using the grading

of UCC we see that all xi have the same degree m. Therefore fx1; � � � ; xkg � UCC

is linearly independent over Z but not over C. This contradiction proves that the

canonical map is injective as well and hence UC is an integral form for UCC . The

proof for U� is identical.

Proof of (2)

Lemma 4.7. For ´ 2 aC and p; q 2 N we have: ´

p

!eŒq�˙i D e

Œq�˙i

´˙ q h˛i ; í

p

!:

Proof. Note that Œ´; e˙i � D ˙h˛i ; í e˙i and then use Lemma A.3 with P.X/ D�Xp

�.

Lemma 4.8. U0U˙i D U˙iU0.

Proof. This follows from Lemma 4.7.

Lemma 4.9. UiU0U�i D U�iU0Ui .

Proof. Use Lemma 4.8 to arrive at:

UiU0U�i D UiU�iU0

Using Lemma A.6 with x D ei ; y D e�i and ´ D ˛_i we see that: eŒp�i eŒq��i can be

written as a sum, where each summand belongs to U�iU0Ui . Hence:

UiU�iU0 D U�iU0UiU0

Using Lemma 4.8 one more time gives us the result.

Lemma 4.10. If j; i1; � � � ; im is a sequence of mC 1 elements in I then:

Ui1Ui2 � � �UimU�j � U�jU0UC:

Proof. We prove this by induction on m:

26

4.2. The Chevalley Lattice

� m D 1: If j D i then UiU�i � U�iU0UC follows from Lemma 4.9. If

m D 1 and j ¤ i then UiU�j � U�jU0UC is a consequence of the fact

that ei and e�j commute.

� m > 1:

Ui1Ui2 � � �UimU�j D Ui1�Ui2 � � �UimU�j

�� Ui1

�U�jU0UC

�induction hypothesis

D

�Ui1U�j

�U0UC

�

�U�jU0UC

�U0UC base of induction

D U�jU0�UCU0

�UC

D U�jU0U0UCUC Lemma 4.8

� U�jU0UC

Proposition 4.11. The product map U� ˝ U0 ˝ UC ! U is bijective.

Proof. Lemma 3.3 implies the injectivity of the product map. In order to prove

surjectivity let U0 denote the the image of the product map. Then:

U0U�i � U0 Lemma 4.10

U0Ui � U0 Ui � UC

U0U0 � U0 Lemma 4.8

So U0U � U0 which implies that U0 contains U.

4.2 The Chevalley Lattice

4.2.1 Construction

Definition 4.12. Set:

gZ ´ UZ.g/ \ gC

27


n˙Z ´ UZ.n˙/ \ n˙C

aZ ´ UZ.a/ \ aC DQ_

In particular this allows us to define all these Lie algebras over any commutative

ring of characteristic zero with a unit.

Definition 4.13. Let � be an integral and dominant weight and define the Cheval-

ley lattice, as L.�/Z ´ UZ.g/ � 1� � L.�/C . Then Chevalley lattice is a UZ.g/-

invariant Z-module in L.�/C , below we will show that it is indeed a lattice in

L.�/C (see Theorem 4.18).

Lemma 4.14 ([11] Lemma 11.4). EŒ�� takes integer values on L.�/Z.

Proof. Let v 2 L.�/Z, by definition there exists a 2 UZ.g/ such that v D a � 1�.

Define:zUZ.n

˙/´ zUQ.n˙/ \ UZ.n

˙/;

which is the integral span of all monomials in UZ.n˙/ of strictly positive degree.

Then we have:

UZ.g/ D zUZ.n�/UZ.g/CUZ.g/zUZ.n

C/C UZ.a/

zUZ.nC/ � 1� D 0

zUZ.n�/UZ.g/ � 1� �

M�¤�

L.�/C;�

Hence there exists a0 2 UZ.a/ such that:

EŒv� D EŒa � 1�� D EŒa0 � 1�� :

But by definition a0 2 UZ.a/ is an integral linear combination of products of

elements of the form:�˛_im

�which act on v as follows: ˛_i

m

!� 1� D

˝�; ˛_i

˛m

!1�:

Now since � is dominant and integral˝�; ˛_i

˛2 ZC for all i 2 I .

28


Theorem 4.15. If v;w 2 L.�/Z then fv; wg 2 Z.

Proof. By definition there exist a; b 2 UZ.g/ such that v D a � 1�; w D b � 1�.

Hence:

fv; wg D S�.v; w/

D S�.v; w/

D �.S.a; b// D ��.� .a/b/

�where � is the Harish Chandra map. Since UZ.g/ is � -invariant, � .a/b 2 UZ.g/.

Moreover we may define a map �Z W UZ.g/ ! UZ.a/ such that the following

diagram commutes:

UZ.g/� //

�Z

��

UC.g/

�

��UZ.a/ �

// UC.a/

Therefore we have:

fv; wg D ��.� .a/b/

�D �

��Z.� .a/b/

�:

But � is dominant and integral so when extended to UZ.a/ it will only produce

integer values.

Corollary 4.16. If v 2 L.�/Z then kvk > 1.

Definition 4.17. An admissible basis for L.�/C is an ordered basis consisting of

weight vectors ordered such that the depth of basis elements is non-decreasing in

this basis. That is if fv1; v2; � � � g is a an admissible basis with vk 2 L.�/C;�k then

i < j implies dp.�i / � dp.�j /. Note that the first basis element of any admissible

basis has to be a highest weight vector, that is it belongs to L.�/C;�.

Theorem 4.18 ([11] Theorem 11.3). L.�/C has an admissible basis such that its

Z-span is UZ.g/-invariant.

29


4.2.2 Proof of Theorem 4.18

Set: L.�/Z;� D L.�/Z\L.�/C;�. ThenL.�/Z has a direct sum decomposition:

L.�/Z DM�2P�

L.�/Z;�:

Let B� be a Z-basis for L.�/Z;� and set:

B D[�2P�

B�;

we claim B is the basis we are looking for. There are 3 points to prove:

(1) The Z-span of B is UZ.g/-invariant.

(2) B spans L.�/C .

(3) B is linearly independent over C.

SinceL.�/Z is the Z-span of B, (1) is true. (2) follows fromL.�/C D UC.g/�1�

and the fact that UZ.g/ is a lattice in UC.g/. That leaves (3), however this is

equivalent to to proving that for any finite subset of L.�/Z linear independence

over Z implies linear independence over C. We prove the latter by contradiction,

so suppose there exist v1; � � � ; vr 2 L.�/Z which are linearly independent over Z,

but not over C, that there exist c1; � � � ; cr 2 C not all zero such that:

rXjD1

cj vj D 0: (4.19)

Moreover we assume that r is minimal, in other words any other subset of L.�/Zof size smaller than r is not a counterexample to our claim. In order to derive a

contradiction we first need the following Lemma:

Lemma 4.20. Let v1; � � � ; vr 2 L.�/Z be such that v1 ¤ 0 andPrjD1 cj vj D 0

with cj 2 C. Then there exist integers n1; � � � ; nr with n1 ¤ 0 satisfying:

rXjD1

cjnj D 0:

30


Proof. Choose a 2 UZ.g/ such that EŒa � v1� ¤ 0. Such an element exists,

otherwise v1 would generate a non-trivial submodule of L.�/C which did not

intersect L.�/C;�, and by definition of L.�/C , this is impossible. Applying

first a and then EŒ�� toPrjD1 cj vj D 0, we get:

PrjD1 cjE

�a � vj

�D 0. Since

vj 2 L.�/Z; a 2 UZ.g/ we have E�a � vj

�2 Z from Corollary 4.14. Now take

nj ´ E�a � vj

�, note that n1 ¤ 0 due to our choice of a.

Applying Lemma 4.20 to (4.19) we see that there exist integers n1; � � � ; nr 2 Z

with n1 ¤ 0 satisfying:rX

jD1

cjnj D 0: (4.21)

Now from (4.19) and (4.21) we have:

0 D

rXjD1

cj vj

!n1 D c1n1v1 C

rXjD2

cjn1vj

0 D

rXjD1

cjnj

!v1 D c1n1v1 C

rXjD2

cjnj v1

Eliminating c1n1v1 using the two equations we get:

rXjD2

cj�n1vj � nj v1

�D 0

Set wj WD n1vj �nj v1. Now w2; � � � ; wr 2 L.�/Z are linearly dependent over Z

since v1; � � � ; vr were linearly independent over Z however we have:

rXjD2

cjwj D 0:

which implies w2; � � � ; wr is linearly dependent over C which contradicts the min-

imality of r .

31

Chapter 5

Groups over Q

In this chapter we give a definition of the split maximal Kac-Moody group over Q

(and any field of characteristic zero) associated to a given GCM. One should note

that given a GCM that is not of finite type one can associate at least two different

groups with it (maximal vs minimal). Moreover in either case there are several

definitions in the literature (see [20] for various definitions of both the maximal

and minimal groups and their comparison). The definition given below is new

but it is a synthesis of two main ways of approaching the subject. Our starting

point is the result that any complex semi-simple Lie group can be expressed as

the amalgamated product of the minimal parabolic subgroups and the normalizer

of a maximal torus (see Theorem D.2). We use this theorem as our definition:

first define the minimal parabolic subgroups and the normalizer of the maximal

torus and then we define the split maximal Kac-Moody group as their amalgamated

product.

The first step (defining the subgroups) is the subject of §1, for any subalgebra

mQ � gQ which is invariant under the adjoint action of aQ we construct a Q-

algebra: HQ.m/. Then we show that HQ.m/ is in fact a Q-Hopf algebra and so we

have an affine group scheme over Q. In §2 we compute the Hopf algebra HQ.m/ in

terms of the representations of mQ where mQ D aQ; nC

Q; bQ; li;Q; pi;Q (for these

two sections we follow Mathieu, pages 19-20 and 24-25 in [17]). In §3 we give an

explicit description of the unipotent group, NCQ , as a subset of a completion of the

universal enveloping algebra of nCQ. Finally in §4 we note that while gQ itself is

invariant under the adjoint action of the torus, when the GCM is not of finite type

gQ becomes infinite dimensional and while this process yields us a group, it is too

small to be of any use, see Remark 5.32. The second step (amalgamation) was the

approach championed by Jacques Tits, an exposition can be found in [16].

Finally another way of constructing Kac-Moody groups is to use integrable

32

5.1. HQ.m/

representations of the Kac-Moody algebra, see [6].

5.1 HQ.m/

5.1.1 Definition

Notation 5.1. For a subalgebra mQ � gQ, set:

m˙Q ´ mQ \ n˙Q

m0Q ´ mQ \ aQ:

Definition 5.2. The character lattice is defined as follows:

P´ HOMZ.aZ; Z/:

P is a lattice in a�Q which contains the root lattice, Q. For each i 2 I we also set

the following subset of the character lattice:

Pi ´˚� 2P W

˝�; ˛_i

˛� 0

:

Let mQ � gQ be a subalgebra such that:�aQ;mQ

�� mQ: (?)

Let L.u/;R.u/ W UQ.m/ ! UQ.m/ be the left and right multiplications by u.

Corresponding to these maps we have the left regular representation of UQ.m/ W

u 7! L.u/ and the right regular representation: u 7! R. .u//, where is the

principal anti-automorphism of UQ.m/. The transpose of these maps give us an

action of UQ.m/ on its dual:�L�.u/.�/

�.x/ D �.ux/�

R�.u/.�/�.x/ D �.x .u//

33

5.1. HQ.m/

For ´ 2 aQ let ad.´/ W UQ.m/ ! UQ.m/ denote the lift of ad.´/ W mQ ! mQ.

Again the transpose gives us an action of aQ on UQ.m/:�ad�.´/.�/

�.x/ D �.ad.´/.x//:

Definition 5.3. � 2 UQ.m/� is called L�-finite (resp. R�-finite) if the span of the

maps L�.u/.�/ (resp. R�.u/.�/), as u varies over UQ.m/, is finite dimensional in

UQ.m/�.

Definition 5.4. Let HLQ.m/ (resp. HRQ.m/) be the set of all linear combinations of

elements � 2 UQ.m/� that satisfy:

(1) � is L�-finite (resp. R�-finite).

(2) There exists � 2Q such that ad�.´/.�/ D h�; í� for all ´ 2 aQ.

(3) There exists � 2 P such that L�.´/.�/ D h�; í� (resp. R�.´/.�/ D

h�; í�) for all ´ 2 m0Q.

Lemma 5.5. HLQ.m/ D HRQ.m/.

Proof. Let .�; �; �/ be a triplet satisfying the conditions of Definition 5.4 with �

non-zero. By definition showing � 2 HRQ.m/ will imply HLQ.m/ � HRQ.m/.

Set:

IQ ´˚u 2 UQ.m/ W L

�.u/.�/ D �.u�/ D 0: (5.6)

Since � isL�-finite, IQ is a right ideal of finite co-dimension in UQ.m/. Therefore

IQ contains, JQ, the annihilator of UQ.m/=IQ, which is a two-sided ideal of

of finite co-dimension (it is the largest two sided ideal contained in IQ). Now

consider the following:�R�� .JQ/

�.�/�.x/ D �.xJQ/ D �.JQ/ D 0:

Since JQ is invariant under and has finite co-dimension � is R�-finite.

Next let ´ 2 m0Q, by definition:

ad�.´/.�/ D h�; í�;

L�.´/.�/ D h�; í�:

34

5.1. HQ.m/

Therefore:

R�.´/.�/ D L�.´/.�/ � ad�.´/.�/ D h� � �; í�;

since � 2P; � 2 Q �P we have � � � 2P. The proof of HLQ.m/ � HRQ.m/ is

similar.

Notation 5.7. Based on Lemma 5.5 we set: HQ.m/´ HLQ.m/ D HRQ.m/.

Example 5.8. For mQ D aQ the three conditions enumerated in the definition

of HQ.a/ collapse to one. So HQ.a/ is the linear combination of elements � 2

UQ.a/� that satisfy:

9� 2P W L�.´/.�/ D h�; í�:

Since UQ.a/ Š Q�˛_1 ; � � � ; ˛

_n

�we see that � is determined by � up to a scalar

constant, that is �.´/ D h�; í�.1/. Now the map:

� 7! �.1/ı�;

where ı� WP! Q is the map that sends� to 1 and is zero on the rest of the lattice,

shows that HQ.a/ D QŒP�, where QŒP� is the group algebra of the discrete group

P.

5.1.2 Hopf Algebra Structure

Definition 5.9. Let XQ.m/ be the set of all left ideals, IQ, in UQ.m/ that satisfy:

(1) IQ is of finite co-dimension in UQ.m/.

(2) IQ is stable under the adjoint action of aQ.

(3) There exists a finite subset� �P such that their restriction to m0Q satisfies:

8´ 2 m0Q WY�2�

´ � h�; í 2 IQ:

Lemma 5.10. For � 2 UQ.m/� the following are equivalent:

35

5.1. HQ.m/

(1) � 2 HQ.m/.

(2) There exists a two-sided ideal JQ 2XQ.m/ such that �.JQ/ D 0.

Proof. First note that if IQ 2XQ.m/ then UQ.m/=IQ is a finite dimensional Q-

vector space and any linear map W UQ.m/ ! Q such that .IQ/ D 0 belongs

to HQ.m/.

For the converse, let � 2 HQ.m/ be arbitrary, then we may write it as a linear

combination of non-zero elements:

� D c1�1 C � � � C cm�m;

where for each k; 1 � k � m, the triplet .�k; �k; �k/ satisfies the conditions of

Definition 5.4. In the proof of Lemma 5.5 for each �k we have defined a right ideal

Ik;Q and a two sided ideal Jk;Q contained in it. Now set:

JQ D J1;Q \ � � � \ Jm;Q:

Clearly JQ is a two sided ideal of finite co-dimension and �.JQ/ D 0.

Since JQ is a two sided ideal it is stable under the adjoint action in particular

by elements of aQ.

Finally for condition (3) of Definition 5.9 we take:

� D f�1; � � � ; �mg :

Lemma 5.11. HQ.m/ is a subalgebra of the commutative algebra UQ.m/�.

Proof. Recall that UQ.m/� is a commutative algebra with the unit map �� and

product map �� (see Appendix C). Since HQ.m/ is a Q-subspace of UQ.m/� we

only need to show that it is closed under ��. In other words:

��HQ.m/˝Q HQ.m/

�� HQ.m/:

Suppose �; 2 HQ.m/ and let � denote ��.� ˝ /. By Lemma 5.10 � 2

HQ.m/ is equivalent to finding a two-sided ideal PQ 2XQ.m/ such that .� /.PQ/ D

0. Since �; 2 HQ.m/ there exist two-sided ideals IQ;JQ 2 XQ.m/ such

36

5.1. HQ.m/

that �.IQ/ D .JQ/ D 0. Let PQ D IQJQ, then one immediately has:

.� /.PQ/ D 0, all that is left to show is PQ 2XQ.m/.

PQ has finite co-dimension: take fx1; � � � ; xpg 2 UQ.m/ to be a basis for the

Q-vector space UQ.m/=JQ and let fy1; � � � ; yqg be a set that generates IQ as an

ideal. Then t 2 IQ can be written as:

t D

qXiD1

uiyi ; (5.12)

where ui 2 UQ.m/. For each ui we can write:

ui D

pXjD1

cj vj

!C JQ; (5.13)

where cj 2 Q. Combining (5.12) and (5.13) we arrive at the following:

t D

qXiD1

pXjD1

cj vj

!C JQ

!yi

D

qXiD1

pXjD1

cj vjyi

!C JQyi

D

qXiD1

pXjD1

cj vjyi

!C IQJQ

which shows that IQ=IQJQ is finite dimensional and since:

dim�UQ.m/=IQJQ

�D dim

�UQ.m/=IQ

�C dim

�IQ=IQJQ

�;

we see that PQ D IQJQ has finite co-dimension.

PQ is stable under the adjoint action of aQ: let a 2 IQ; b 2 JQ and ´ 2 aQ

then:

Œ´; ab� D áb � ab´

D áb � a´b C a´b � ab´

37

5.1. HQ.m/

D Œ´; a�b C aŒ´; b�:

Since IQ;JQ both are stable under the adjoint action of aQ we have: Œ´; a� 2

IQ; Œ´; b� 2 JQ.

Finally from IQ;JQ 2XQ.m/ we have two finite sets: �1; �2 �P. For PQ

the union �1 [�2 satisfies the desired conditions.

Theorem 5.14. If the subalgebra mQ � gQ satisfies .?/ then�HQ.m/; �

�; ��;��; ��; ��

is a commutative Hopf algebra over Q.

Proof. Based on Lemma 5.11 and the fact that UQ.m/ is itself a cocommutative

Hopf algebra we only need to prove that HQ.m/ is closed under the transpose

maps: �; �� and ��. The first two are easy to verify and so turn our attention to

the transpose of the product map in UQ.m/:

˚�� W UQ.m/

� !�UQ.m/˝Q UQ.m/

��.�/.u˝ u0/ D � .�.u˝ u0// D � .uu0/

On the other hand � 2 HQ.m/ and so by Lemma 5.10 there exists JQ 2XQ.m/

such that �.JQ/ D 0. Therefore we get:

��.�/�JQ ˝Q UQ.m/C UQ.m/˝Q JQ

�D 0:

Combining this with:

��.�/ 2 HOMQ�UQ.m/˝Q UQ.m/; Q

�;

we get:

��.�/ 2 HOMQ

�UQ.m/˝Q UQ.m/

JQ ˝Q UQ.m/C UQ.m/˝Q JQ; Q

�D HOMQ

�UQ.m/=JQ ˝Q UQ.m/=JQ; Q

�D HOMQ

�UQ.m/=JQ; Q

�˝Q HOMQ

�UQ.m/=JQ; Q

�38

5.1. HQ.m/

But since JQ 2XQ.m/ we have:

HOMQ�UQ.m/=JQ; Q

�� HQ.m/:

Therefore:

��.�/ 2 HQ.m/˝Q HQ.m/:

Definition 5.15. Based on Theorem 5.14 for any subalgebra mQ � gQ that satis-

fies .?/ we may define an affine group scheme over Q:

MQ D HOMQ�alg.HQ.m/; Q/:

5.1.3 Examples

Example 5.16. Let us return to the case when mQ D aQ, we have already shown

that HQ.a/ D QŒP�. Therefore:

AQ D HOMQ�alg.QŒP�; Q/ D HOMZ.P; Q�/:

So our definition agrees with that of a classical finite dimensional split torus.

Example 5.17. For each i 2 I define:

li;Q D n�˛i ;Q ˚ aQ ˚ n˛i ;Q:

Then li;Q satisfies .?/ and so we get a group Li;Q, this is a finite dimensional

reductive group of semi-simple rank 1 which contains AQ as a subgroup. Let W i;Q

denote the normalizer of AQ in Li;Q. Then the quotient W i;Q=AQ is a group with

two elements: f1; r ig and the conjugation action of r i on AQ is induced from the

action of the fundamental reflection ri on aQ. Let a D EXP.´/ 2 AQ then:

ri .a/ D r iar�1i D ri .EXP.´// D EXP.ri .´//:

39

5.2. Peter-Weyl Type Theorems

Moreover we have r2i 2 AQ, more precisely:

˚r2i WP! Q�

r2i .�/ D .�1/h�;˛_i i

Remark 5.18. Earlier we have defined several subalgebras of gQ W aQ; bQ; n˙Q; n˙˛i ;Q

and pi;Q. Now for each i 2 I we define:

ni;Q DM

˛i¤˛2�C

gQ;˛;

b˙i;Q D aQ ˚ n˙˛i ;Q;

ci;Q D n�˛i ;Q ˚ aQ ˚ ni;Q:

These subalgebras of gQ all satisfy .?/ and so we have the corresponding groups

over Q:

AQ;BQ;B˙i;Q;Ci;Q;Li;Q;N˙Q ;Ni;Q;N˙˛i ;Q;Pi;Q:

5.2 Peter-Weyl Type Theorems

Definition 5.19. Given � 2 P, let M.�/_Q denote the subspace generated by

weight vectors in the coinduced module:

COINDbQaQ

�Q�

�D HOMUQ.a/

�UQ.b/; Q�

�;

where Q� is the 1-dimensional UQ.a/-module with weight �. Then as UQ.b/-

modules we have:

M.�/_Q DM.0/_Q ˝Q Q�:

Therefore all these modules are isomorphic as UQ.nC/-modules, in fact as UQ.n

C/-

modules we have:

M.�/_Q Š UQ.nC/_;

40

5.2. Peter-Weyl Type Theorems

where the latter is the restricted dual with respect to the QC-grading, that is:

UQ.nC/_´

M˛2QC

UQ.nC/�˛ �

M˛2QC

UQ.nC/˛

!�D UQ.n

C/�:

Definition 5.20. Suppose � 2 Pi then˝�; ˛_i

˛� 0 and hence there exists a

unique irreducible UQ.li /-module of dimension˝�; ˛_i

˛C1, which we will denote

by ì .�/Q. We define Mi .�/_Q to be the subspace generated by weight vectors in

the coinduced module:

COINDpi;Qli;Q

�ì .�/Q

�D HOMUQ.li /

�UQ.pi /; ì .�/Q

�:

Lemma 5.21 ([17] page 25, Lemma 7).

(1) For each i 2 I we have natural isomorphisms:

Pi;Q D Ni;Q � Li;Q;

BQ D Ni;Q � Bi;Q;

Ci;Q D Ni;Q � B�i;Q;

BQ D NCQ �AQ:

(2) If mQ � m0Q are two subalgebras mentioned in Remark 5.18 then there

exists a natural morphism MQ ! M0Q which is a closed immersion, in

other words HQ.m0/! HQ.m/ is a surjection of Q-algebras.

(3) We have the following isomorphism:

HQ.a/ DM�2P

Q� ˝Q Q�� As UQ.a/ � UQ.a/-modules

(5.22)

HQ.li / DM�2Pi

ì .�/Q ˝Q ì .�/�Q As UQ.li / � UQ.li /-modules

(5.23)

HQ.nC/ DM.0/_Q As UQ.n

C/-modules (5.24)

41

5.3. The Unipotent Subgroup

HQ.b/ DM�2P

M.�/_Q As UQ.b/-modules (5.25)

HQ.pi / DM�2Pi

�Mi .�/

_Q

�h�;˛_i iC1 As right UQ.pi /-modules (5.26)

Proof. (1) is easy and implies (2). For (3) we note that (5.22) and (5.23) are known

from finite dimensional theory. (1) and (5.24) together imply (5.25) and (5.26).

Therefore we only need to show (5.24).

We claim HQ.nC/ D UQ.n

C/_. First note that HQ.nC/ can not be any bigger

than the restricted dual since any functional not in the restricted dual is not L�-

finite and can not be written as a finite linear combination of L�-finite functionals.

Pick a basis fu˛ W ˛ 2QCg consisting of the weight vectors of the adjoint action

of aQ, and let f�˛ W ˛ 2QCg be a corresponding dual basis which spans the re-

stricted dual. Evidently all the elements in the dual basis are L�-finite, now by

definition we have: �ad�.´/.�˛/

�.x/ D �˛.ad.´/.x//:

This expression is zero unless x 2 UQ.nC/˛ in which case we have:

�˛.ad.´/.x// D �˛.h˛; í x/ D h˛; í�˛.x/:

Since nCQ\ aQ D f0g we have: �˛ 2 HQ.nC/. Finally we note that aQ annihilates

the unit in HQ.nC/, which is the map: 1� W UQ.n

C/ ! Q, characterized by

1�.1/ D 1 and 1��zUQ.n

C/�D 0. The isomorphism between HQ.n

C/ andM.0/_Qis then given by sending 1� to 1�0 the dual weight vector to the highest weight

10 2M.0/Q.

5.3 The Unipotent Subgroup

While (5.24) gives us some information about the structure of NCQ in this section

we will give an explicit construction of this group.

Notation 5.27. In this section UQ.nC/ will be denoted by UQ.

42


Definition 5.28. Consider a completion of UQ based on the root lattice decompo-

sition:

UcQ DY˛2QC

UQ;˛ �M˛2QC

UQ;˛ D UQ:

Let fu1; u2; � � � g be a Q-basis for UQ with fu�1; u�2; � � � g as the corresponding dual

basis. Then from Appendices B, C and the proof of Lemma 5.21 we have:

UQ Š QŒu1; u2; � � � �; UcQ Š QŒŒu1; u2; � � � ��;

U_Q Š QŒu�1; u�2; � � � �; U�Q Š QŒŒu�1; u

�2; � � � ��;�

U_Q

��D UcQ:

Note that UQ;U_Q are dense subsets of UcQ and U�Q, respectively.

Definition 5.29. Let ncQ be the completion of nCQ in UcQ and define the exponential

map EXP W ncQ ! UcQ as the formal sum:

EXP.x/ D

1XnD0

xn

nŠ:

Lemma 5.30. NCQ can be identified with EXP�ncQ

�� UcQ.

Proof. 1 Since elements of NCQ are Q-algebra homomorphisms, U_Q ! Q, we can

consider NCQ as a subset of UcQ.

THE MAP: Since U_Q is dense in U�Q we can extend any � 2 U_Q to UcQ by

continuity. Now for y 2 ncQ define:

˚nEXP.y/ W U

_Q ! Q

nEXP.y/.�/ D �.EXP.y//

INJECTIVITY: This follows from U_Q being dense in U�Q.

HOMOMORPHISM: We may assume y to be primitive, non-primitive elements

of ncQ can be expressed as limits of primitive elements. Now the following calcu-

1This proof was communicated to me by D. H. Peterson.

43


lation proves the claim:

�.EXP.y// D EXP.y/˝ EXP.y/:

Finally we claim that nEXP.y/ is a Q-algebra homomorphism:

nEXP.y/.� / D .� /.EXP.y//

D .� ˝ /.�.EXP.y//

D .� ˝ /.EXP.y//˝ .EXP.y//

D .�.EXP.y//˝ . .EXP.y//

D nEXP.y/.�/nEXP.y/. /

SURJECTIVITY: Let n 2 NCQ be arbitrary, we will inductively construct y 2

ncQ such that n D nEXP.y/, that is:

n.�/ D �.EXP.y//; 8� 2 U_Q Š QŒu�1; u�2; � � � �:

Let P 2 QŒu�1; u�2; � � � � and assume Xk D x1 C � � � C xk is such that

n.P / D P.EXP.Xk//; deg.P / � k:

We would like to find xkC1 that satisfies:

n.P / D P.EXP.Xk C xkC1//; deg.P / � k C 1:

But based on the Campbell-Hausdorff formula we have:

P.EXP.Xk C xkC1// D P.EXP.Xk//P.EXP.xkC1//; deg.P / � k C 1;

which implies that we should have:

n.P / D P.EXP.xkC1//; deg.P / D k C 1:

But this determines xkC1 and then we can repeat the same with XkC1 D Xk C

xkC1. Finally we take y D limk Xk .

44

5.4. The Split Maximal Kac-Moody Group

5.4 The Split Maximal Kac-Moody Group

Identify the n copies of AQ in W 1;Q; � � � ;W n;Q and define W Q as the group

generated by all W i;Q subject to one additional relation.

For all i ¤ j 2 I where rirj 2W is of finite order mij : .r irj /mij D 1.

Then we have a short exact sequence of groups:

f1g ! AQ !W Q !W ! f1g with r i 7! ri :

Definition 5.31. Identify all the copies of BQ in P1;Q; � � � ;Pn;Q, then we define

the split maximal Kac-Moody group as the product of P1;Q; � � �Pn;Q and W Q

amalgamated along their intersections. The resulting group is not a group scheme,

it is the direct limit of groups (see Appendix D), where each group is the Q-points

of a group scheme.

Remark 5.32 ([17] page 27). gQ itself satisfies (?), and if dim.gQ/ < 1 then

one obtains Chevalley’s simply connected group. However when gQ is infinite

dimensional the Hopf algebra HQ.g/ is too small to give us a suitable group. To

see this let � 2 HQ.g/ � UQ.g/�, and consider the Q-grading:

UQ.g/ DM˛2Q

UQ.g/˛:

Suppose � is non-zero on the subspace UQ.g/˛0 , however this would violate the

L�-finiteness of � since there are infinitely many different ways of writing ˛0 D

ˇ0C 0 with ˇ0; 0 2Q. Therefore we denote the group defined in Definition 5.31

by �GQ to distinguish it from GQ discussed above.

Remark 5.33. Similarly we may define �GF where F is any field of characteristic

zero. Set:

HF.m/´ F˝Q HQ.m/;

where mF D p1;F; � � � ; pn;F; aF then we have groups over F: P1;F; � � � ;Pn;F

and AF. The definition of W F is identical to W Q since the quotient group W

does not depend on F.

45

Chapter 6

Groups over Z

Starting with the group MQ defined in the previous chapter, we aim to define a

group scheme over Z: MZ. In order to do so we first define a natural subring of

the Hopf algebra HZ.m/ � HQ.m/ (see Definition 6.1). If HZ.m/ becomes a Hopf

algebra over Z with the maps inherited from HQ.m/ (so that we have a compatible

subgroup) and is a lattice in HQ.m/ (to ensure the group is large enough) we say

that mQ is an integral subalgebra of gQ and define MZ to be the spectrum of

HZ.m/.

However not every subalgebra which satisfies .?/ (which is needed for MQ to

exist in the first place) is integral so we have to impose further conditions on mQ.

In §1 we introduce HZ.m/ and the concept of an integral subalagebra, in §2 we

investigate which conditions are needed for mQ to be an integral subalgebra, in §3

we try to establish which subalgebras of gQ are integral and finally in §4 we define�GZ in an analogous way to �GQ since the subalgebras used in the definition of �GQ

are integral as shown in §2.

The material of §1 and §2 are based on [17] pages 21 - 23. It should be noted

that the theory of Kac-Moody groups over integers has become a very active field

of research, two recent and noteworthy references are [1, 5].

6.1 HZ.m/

Definition 6.1. Throughout this chapter mQ � gQ is a subalgebra that satisfies .?/

and hence HQ.m/ is a commutative Hopf algebra. Set:

UZ.m/´ UZ.g/ \ UQ.m/;

HZ.m/´˚� 2 HQ.m/ W � .UZ.m// � Z

:

46

6.2. Integrality Conditions

Example 6.2. Let us try and compute HZ.a/. By definition HZ.a/ is the set of

linear maps in HQ.a/ that take integer values when restricted to UZ.a/. Since

elements of UZ.a/ are integral linear combination of products of elements of the

form�˛_im

�, we conclude that � 2 HQ.a/ belongs to HZ.a/ if and only if �.˘_/ �

Z. This combined with the map we used to show HQ.a/ D QŒP� shows that

HZ.a/ D ZŒP�.

Definition 6.3. A subalgebra mQ � gQ is called integral, if:

(1) HZ.m/ is a commutative Hopf algebra with the maps inherited from HQ.m/,

(2) HZ.m/ is a lattice in HQ.m/, i.e. HQ.m/ D Q˝ HZ.m/.

Definition 6.4. Given an integral subalgebra, mQ, we may define a group over Z:

MZ D HOMZ�alg.HZ.m/; Z/:

Moreover if R is any commutative ring of characteristic zero with a unit, by setting

HR.m/´ R˝ HZ.m/ we can define a group over R:

MR D HOMR�alg .HR.m/; R/ :

This is compatible with our earlier definition over Q and any other fields of char-

acteristic zero.

6.2 Integrality Conditions

6.2.1 Hopf Algebra

Lemma 6.5. The following hold with no extra conditions on mQ:

��.Z/ � HZ.m/;

��.HZ.m// � Z;

�.HZ.m// � HZ.m/:

47


Proof. The unit and antipode map conditions are automatically satisfied. The con-

dition for the counit map follow from zUZ.m/ ´ zUQ.m/ \ UZ.m/, which then

implies UZ.m/ D Z˚ zUZ.m/.

Definition 6.6. Next we present two further conditions on mQ:

� .UZ.m// � UZ.m/˝ UZ.m/; (�)

8JQ 2XQ.m/ W UZ.m/=JZ is a Z-module of finite type. (�)

And here JZ D JQ \ UZ.m/ for all JQ 2XQ.m/.

Lemma 6.7. If mQ satisfies .�/ then HZ.m/ is closed under multiplication.

Proof. Let a; b 2 HZ.m/ and u 2 UZ.m/ then by definition of the transpose map

we have:

��.a˝ b/.u/ D .a˝ b/.�.u//:

but �.u/ 2 UZ.m/ ˝ UZ.m/ by .�/ and a; b 2 HZ.m/, hence: .a ˝ b/.�.u// 2

Z.

Lemma 6.8. If mQ satisfies .�/ then HZ.m/ is closed under co-multiplication.

Proof. Let � 2 HZ.m/ � HQ.m/, since HQ.m/ is a Hopf algebra we already

have:

��.�/ 2 HQ.m/˝Q HQ.m/ � UQ.m/�˝Q UQ.m/

�

��UQ.m/˝Q UQ.m/

��D HOMQ

�UQ.m/˝Q UQ.m/; Q

�Since

��.�/�u˝ u0

�D �

��.u˝ u0/

�D �.uu0/; (6.9)

UZ.m/, as a subring, is closed under multiplication and � 2 HZ.m/ we have in

fact:

��.�/ 2 HOMZ�UZ.m/˝ UZ.m/; Z

�: (6.10)

48


On the other hand because � 2 HQ.m/, there exists a two sided ideal JQ 2XQ.m/

such that �.JQ/ D 0. Using (6.9) arrive at:

��.�/�JZ ˝ UZ.m/C UZ.m/˝ JZ

�D 0: (6.11)

From (6.10) and (6.11) we have:

��.�/ 2 HOMZ

�UZ.m/˝ UZ.m/

JZ ˝ UZ.m/C UZ.m/˝ JZ; Z

�D HOMZ

�UZ.m/=JZ ˝ UZ.m/=JZ; Z

�D HOMZ

�UZ.m/=JZ; Z

�˝ HOMZ

�UZ.m/=JZ; Z

�By .�/, UZ.m/=JZ is a Z-module of finite type, it is torsion free as well since JQ

is an ideal. Therefore

HOMZ�UZ.m/=JZ; Z

�� HZ.m/:

And from this we get

��.�/ � HZ.m/˝ HZ.m/:

6.2.2 Lattice

Lemma 6.12. If mQ satisfies .�/ then HQ.m/ D Q˝ HZ.m/.

Proof. If � 2 HQ.m/ then there exists a two sided ideal JQ 2 XQ.m/ such that

�.JQ/ D 0. Therefore we have a homomorphism of additive abelian groups:

� W UZ.m/=JZ ! Q:

However UZ.m/=JZ is a Z-module of finite type and so the image of � is fully

determined by its value on the finitely many generators of UZ.m/=JZ. By taking

the least common denominator of the image of this finite set of generators we see

that there exists 0 ¤ p 2 Z such that:

� .UZ.m// �1p

Z:

49

6.3. Integral Subalgebras

We therefore have p� 2 HZ.m/.

6.3 Integral Subalgebras

Theorem 6.13. If a subalgebra mQ � gQ satisfies .?/; .�/ and .�/ then it is inte-

gral.

Proof. This follows from Lemmas 6.7, 6.5, 6.8 and 6.12.

Using Theorem 6.13 in this section we examine whether the well known sub-

algebras of gQ are integral or not. However before doing so we find an equivalent,

but easier to verify, statement for .�/. We start with a definition:

Definition 6.14. Let YQ.m/ be the set of all UQ.mC a/-modules, E, such that:

(1) E is finite dimensional.

(1) E is a weight module for aQ.

(2) weights of E lie in P.

Lemma 6.15. .�/ is equivalent to:

8E 2 YQ.m/;8e 2 E W UZ.m/ � e is a Z-module of finite type. (��)

Proof. If JQ 2 XQ.m/ then UQ.m/=JQ 2 YQ.m/. .��/ implies that UZ.m/ �

.x C JQ/ is a Z-module of finite type for all x 2 UQ.m/. In particular UZ.m/ �

.1 C JQ/ is a Z-module of finite type. But since UZ.m/ acts on UQ.m/=JQ by

left multiplication and so UZ.m/ � .1C JQ/ Š UZ.m/=JZ which proves .�/.

Assume .�/ and let E 2 YQ.m/ be arbitrary. Now the set of elements e 2 E

such that UZ.m/ � e is a Z-module of finite type, is itself a UQ.mC a/-submodule

of E.

Lemma 6.16.

(1) If aQ CmQ satisfies .��/ then mQ also satisfies .��/.

(2) If tQ D aQ ˚mCQ then UZ.t/ D UZ.mC/˝ UZ.a/.

50

6.3. Integral Subalgebras

(3) If tQ D aQ ˚mCQ then tQ and mCQ both satisfy .��/.

(4) If aQ � mQ and UZ.m/ D UZ.m�/˝ UZ.t/, then mQ satisfies .��/.

Proof. Since UZ.m/ � UZ.aCm/ Lemma 6.15 implies (1) at once.

For each ˇ 2 Q: UZ.mC/ˇ D UQ.m

C/ˇ \ UZ.nC/ˇ . Since UZ.n

C/ˇ is a

Z-module of finite type, UZ.mC/ˇ is a direct factor of UZ.n

C/ˇ . Hence UZ.mC/

is a direct factor of UZ.nC/. From this and UZ.b/ D UZ.n

C/˝UZ.a/ we get (2).

Using part (1) we see that to show (3) it suffices to show that tQ satisfies .��/.

So let E 2 YQ.t/ and e 2 E, we can also assume that e is a weight vector and

therefore: UZ.a/ � e D Ze. Now using part (2) we may write:

UZ.t/ � e D�UZ.m

C/˝ UZ.a/�� e D UZ.m

C/ � .Ze/:

Since E is finite dimensional there exists a finite set ˙ �QC such that:

UZ.mC/ � e D

M˛2˙

UZ.mC/˛ � e:

Since UZ.mC/˛ are Z-modules of finite type this proves 3.

For (4) let E 2 YQ.m/ by (3), for every e 2 E the Z-modules UZ.m�/ � e and

UZ.t/ � e are of finite type. Now UZ.m/ D UZ.m�/˝ UZ.t/ implies (4).

Remark 6.17. The Lie algebras enumerated in Remark 5.18 are integral, we already

know that all these subalgebras satisfy .?/. For .�/ we note that it is enough to

show it for the generators of each subalgebra. Finally for .�/ we use the equivalent

formulation .��/ and Lemma 6.16. So we have the corresponding groups over Z:

AZ;BZ;B˙i;Z;Ci;Z;Li;Z;N˙Z ;Ni;Z;N˙˛i ;Z;Pi;Z:

Lemma 6.18.

51

6.4. The Arithmetic Group

(1) For each i 2 I we have the following isomorphisms:

Pi;Z D Ni;Z � Li;Z;

BZ D Ni;Z � Bi;Z;

Ci;Z D Ni;Z � B�i;Z;

BZ D NCZ �AZ:

(2) If mQ � m0Q are two subalgebras enumerated in Remark 5.18 then there

exists a natural morphism MZ !M0Z which is a closed immersion.

(3) For each i 2 I the natural morphisms

N�˛i ;Z � BZ ! Pi;Z

N˛i ;Z �Ci;Z ! Pi;Z

are open immersions.

Proof. The only delicate point is to show (2). More precisely, to show that BZ !

Pi;Z is a closed immersion. Using isomorphism of groups Pi;Z D Ni;Z�Li;Z and

BZ D Ni;Z � Bi;Z, we reduce to show that Bi;Z ! Li;Z is a closed immersion,

this is done by direct calculation. We show (3) with the same argument.

6.4 The Arithmetic Group

Definition 6.19. Now that we have the required group schemes, we define, � , the

split maximal Kac-Moody group over Z as the product of P1;Z; � � � ;Pn;Z and W Z,

amalgamated along their intersections. As before (see Definition 5.31) this group

itself is not a group scheme over Z.

52

Chapter 7

Structure Theory

In §1 we show that �GF acts on any integrable irreducible highest weight module

L.�/F. In fact we give an explicit construction for the representation map which is

based on the last paragraph of page 28 in [17]. Next we see how different subgroups

act and what their matrices look like in an admissible basis, most importantly we

show that the Chevalley lattice is stable under � (see Lemma 7.9).

In §2 we use the representations to show that �GF possesses a Tits system, this

proof is based on the representation theory of �GF and follows §5.12 in [23].

In §3 we prove an Iwasawa decomposition when F D R;C. This prove uses

the existence of the Tits system shown earlier, with Lemma 7.26 on the theory of

Tits systems being the crucial part of the proof.

In §4 we introduce the minimal group of Kac and Peterson, we will use this

group to show that the orbit of the highest weight vector is not the entire module.

7.1 Representation Theory

7.1.1 Constructing the Map

Let L.�/F be an integrable highest weight module for gF. We wish to construct a

representation: �GF! GL.L.�/F/:

By the construction of �GF it suffices to do so for each minimal parabolic subgroup

Pi;F. For every i 2 I , since UF.pi / acts on L.�/F, every v 2 L.�/F gives rise

to a map v W UF.pi / ! L.�/F defined by v.x/ D x � v. Since L.�/F is

integrable we see that all v have finite rank. But on the other hand we have a

53

7.1. Representation Theory

F-vector space isomorphism:

HOMfinF .UF.pi /; L.�/F/ Š UF.pi /

�˝F L.�/F; (7.1)

where HOMfinF .�; �/ is the set of all F-linear maps of finite rank. So combining

v 7! v with this isomorphism gives us a map:

L.�/F! UF.pi /�˝F L.�/F: (7.2)

We claim that the image of the map in (7.2) in fact lies in HF.pi /˝F L.�/F. To

see this we first recall the isomorphism of (7.1) can be explicitly written as:

˚ W UF.pi /! L.�/F

7!

1XiD1

x�i ˝ .xi /

where fx1; x2; � � � g is a basis for UF.pi / of our choosing. We take it to be the basis

afforded to us by the PBW Theorem with the ordering:

e�i < ˛_1 < � � � < ˛

_n < e1 < � � � < en < � � � :

If we show that the elements of the dual basis fx�1 ; x�2 ; � � � g all lie in HF.pi /, we

are done. But then we may write:

HF.pi / D HF.ni /˝F HF.li /

and we know that HF.ni / and HF.li / are the restricted dual of the corresponding

universal enveloping algebras. Hence the same has to be true for HF.pi /. So we

have a map:

L.�/F! HF.pi /˝F L.�/F:

From the theory of affine group schemes this corresponds to a representation of the

group:

Pi;F! GL.L.�/F/:

Remark 7.3. The above construction more than giving us a representation for �GF

has the virtue that it enables us to actually compute the representation for the

54


parabolic subgroups. Suppose we are given p 2 Pi;F then we may write its action

based on the following sequence using the fact that p 2 Pi;F is in fact a homomor-

phism of F-algebras: p W HF.pi /! F one can write:

L.�/F // HF.pi /˝F L.�/Fp˝1 // F˝F L.�/F

Š // L.�/F ;

So for a given v 2 L.�/F we may write:

p � v D .p ˝ 1/

1XiD1

m�i ˝ .mi � v/

!

D

1XiD1

p�m�i�˝ .mi � v/

D

1XiD1

p�m�i�.mi � v/

where fm1; m2; � � � g is a basis of our choosing for UF.pi / and fm�1; m�2; � � � g is its

dual.

7.1.2 Subgroups

Remark 7.4. Since 1� is the unit of HF.pi / and p is a F-algebra homomorphism

we have: p.1�/ D 1 for all p 2 Pi;F. So if we choose a basis such that m1 D 1

we can write the action in a slightly more useful form:

p � v D v C

1XiD2

p�m�i�.mi � v/:

Lemma 7.5. In any admissible basis for L.�/F the elements of AF are repre-

sented by diagonal matrices. More precisely a 2 AF acts on L.�/F;� by a�.

Proof. This follows from AF being the same as classically defined torus.

Lemma 7.6. In any admissible basis for L.�/F the elements of NCF are repre-

sented by upper triangular unipotent matrices. In particular: n � 1� D 1� for all

n 2 NCF .

55


Proof. Let v 2 L.�/F be a weight vector arbitrary, we have:

n � v D v C

1XiD2

n�m�i�.mi � v/

But when acting on L.�/F any non-constant monomial in UF.nC/ will lower

depth. So the terms in the sum (which are finitely many) are vectors in L.�/F and

all of their weight components have lower depth than v.

Corollary 7.7. AF normalizes NCF , in particular BF D NCFAF D AFNCF .

Proof. This follows from Lemmas 7.5 and 7.6.

Lemma 7.8. NCF acts faithfully on non-trivial integrable highest weight modules.

Proof. Suppose that there exists n 2 NCF such that n � v D v for all v 2 L.�/F.

Based on the action of n this means:

8v 2 L.�/F W

1XiD2

n�m�i�.mi � v/ D 0:

Now we can find a vector v2 2 L.�/F such that m2 � v2 ¤ 0 but mi � v2 D 0 for

all i > 2. This shows that n�m�2�D 0. By repeating this argument we see that

n�m�i�D 0 for all i > 2.

7.1.3 The Arithmetic Subgroup and the Chevalley Lattice

Lemma 7.9. For all p 2 Pj;Z we have: p � L.�/Z � L.�/Z.

Proof. Let v 2 L.�/Z be arbitrary, then we have:

p � v D

1XiD1

p.m�i /.mi � v/:

The series is in fact a finite sum we just need to show that each term belongs

to L.�/Z. Since mi 2 UZ.pj / we have: mi � v 2 L.�/Z, that leaves p.m�i /.

However p 2 Pj;Z is a ring homomorphism: p W HZ.pj / ! Z and since HZ.pj /

is a lattice in HF.pj / we can always choose a basis such that m�i 2 HZ.pj /.

56

7.2. Existence of a Tits System

7.2 Existence of a Tits System

Notation 7.10. For w 2 W , we use w to denote an element in W F which lies in

the coset wAF and under the quotient map W F!W goes to w.

Definition 7.11. For any real root � D w. j̨ / we define N�;F´ wNj̨ ;Fw

�1.

Lemma 7.12. Let Z D P1;F[ � � � [Pn;F[W F by definition there is a canonical

map Z! �GF. This map is injective.

Proof. When acting on L.�/F the image of Z under the canonical map acts as Z

itself would. So if we find a module VF on which the elements of Z act faithfully,

we are done. Let f$1; � � � ;$ng be a basis consisting of dominant integral weights

for P and set:

VF´ L.$1/F˚ � � � ˚ L.$n/F:

AF acts faithfully since f$1; � � � ;$ng generate P. By Lemma 7.8 we know that

NCF acts faithfully on each integrable highest weight module and therefore it acts

faithfully on VF. Given an admissible basis for each module, L.$i /F, AF acts

through diagonal matrices and NCF acts through upper triangular unipotent matri-

ces. Hence BF also acts faithfully. The faithfulness of the action of Pi;F follows

from this and the simplicity of PGL2;F.

This leaves W F, suppose there exists w 2W F that acts trivially on VF. Since

AF acts faithfully we can also assume that w ¤ 1. Therefore there exists � 2 �reC

such that w.�/ 2 ��. Since w acts trivially the action of wN�w�1 is the same as

N�;F butwN�;Fw�1D Nw.�/;F. Howeverw.�/ 2 �� which means the elements

of Nw.�/;F are represented by lower triangular unipotent matrices in GL.VF/.

This contradiction completes the proof.

Remark 7.13. Let N�;minF denote the subgroup that is generated by the collection of

the subgroups:˚N�;F W � 2 �

re�

. In an admissible basis for VF elements of BF

(resp. N�;minF ) operate through upper triangular matrices (resp. unipotent lower

triangular matrices). In particular we have:

BF \N�;minF D f1g:

57

7.2. Existence of a Tits System

We use the superscript to emphasize the fact that N�;minF ¤ N�F. In fact N�F is not

even a subgroup of �GF.

Remark 7.14. We identify the groups AF;NC

F ;BF;W F;Pi;F with their images

in �GF.

Lemma 7.15. BF and W F generate �GF.

Proof. This follows from the fact that each Pi;F is generated by BF and W i;F.

Lemma 7.16. BF \W F D AF.

Proof. We know AF � BF \W F. Let w 2 BF \W F be such that w ¤ 1. Pick

� 2 �reC

such that w.�/ 2 ��. Then:

wN�;Fw�1D Nw.�/;F � N

�;minF :

Since w 2 BF and N�;F � BF we have:

wN�;Fw�1� BF \N

�;minF D f1g;

which contradicts the faithfulness of NCF !�GF.

Lemma 7.17. r iBFr i ¤ BF.

Proof. This follows from: r iN˛i ;Fr i D N�˛i ;F � N�;minF and N

�;minF \ BF D

f1g.

Lemma 7.18. Pi;F D BF [ BFr iBi;F.

Proof. Recall that Li;F is a finite dimensional reductive group of semi-simple

rank 1 and therefore it has a Bruhat decomposition: Li;F D Bi;F [ Bi;Fr iBi;F.

Combining this decomposition with the isomorphisms: Pi;F D Ni;F � Li;F and

BF D Ni;F � Bi;F, which we have from Lemma 5.21 completes the proof.

Lemma 7.19. For all w 2W we have: r iBFw � BFwBF [ BFr iwBF.

Proof. We divide the proof in two cases:

58

7.3. Iwasawa Decomposition

� POSITIVE CASE: suppose � D w�1.˛i / � 0 then w�1N˛i ;Fw D N�;F �

NCF and we have:

r iBFw D r iAFNCFw

D r iAFr ir iNC

Fw

D AFr iNi;FN˛i ;Fw

D AFr iNi;Fr ir iN˛i ;Fw

D AF

�r iNi;Fr i

�r iN˛i ;Fw

D AFNi;Fr iN˛i ;Fw r iNi;Fr i � Ni;F

D AFNi;Fr iww�1N˛i ;Fw

D AFNi;Fr iwN�;F

� BFr iwBF:

� NEGATIVE CASE: suppose � D w�1.˛i / � 0, then .riw/�1 .˛i / � 0 and

we have:

r iBFw D .r iBFr i / r iw

� Pi;Fr iw

� .BF [ BFr iBF/ r iw Lemma 7:18

� BFr iw [ BF .r iBFr i / w

� BFr iw [ BFBFr i .r iw/BF Positive Case

� BFr iw [ BFwBF

� BFr iwBF [ BFwBF:

Theorem 7.20.��GF;BF;W F; fr1; � � � ; rng

�is a Tits system.

Proof. This follows from Lemmas 7.15, 7.16, 7.17 and Corollary 7.19.

7.3 Iwasawa Decomposition

Notation 7.21. In this section F D R;C.

59


Definition 7.22. Let �� denote the Hermitian conjugate with respect to positive

definite inner product f�; �g defined on L.�/F in §3.4. Based on the construction

of f�; �g we have: e�i D e�i for all i 2 I . A linear operator T W L.�/F ! L.�/F

is called unitary if T � D T �1. A group of linear automorphisms of L.�/F is

called unitary if all its elements are unitary operators.

Definition 7.23. Let K � �GF be the subgroup consisting of all the elements g 2�GF such that g� is defined and equals g�1, we refer to this subgroup as the unitary

form of �GF.

Remark 7.24. Note that K is non-trivial, it clearly contains Ki , the real maximal

compact subgroup of Li;F since L.�/F decomposes as a direct sum of irreducible

representations of li;F. Moreover since r i 2 Ki we see that W � K.

Lemma 7.25. BFr iBF D BFr iN˛i ;F.

Proof. First we note the following:

r iBFr i D r i�Ni;FN˛i ;FAF

�r i D Ni;FN�˛i ;FAF D Ni;FAFN�˛i ;F:

Then we can write:

BFr iBF D BFr iBF .r ir i /

D BF

�Ni;FAFN�˛i ;F

�r i

D BFN�˛i ;Fr i

D BFr iN˛i ;F

Next we need a Lemma from the theory of Tits systems:

Lemma 7.26 (Proposition 3.1 [14]). Let .G;B;N; S/ be a Tits system with Weyl

group W D N=.B \N/. If w1; w2 2 W satisfy `.w1w2/ D `.w1/C `.w2/, and

if X1; X2 are subsets of G satisfying:

(1) Bw1B D X1B with uniqueness of expression,

(2) Bw1B D X2B with uniqueness of expression.

60


Then: Bw1w2B D X1X2B with uniqueness of expression.

Proposition 7.27. �GF D BFK.

Proof. By Bruhat decomposition we only need to prove the claim for a Bruhat cell:

BFwBF. By Lemma 7.26 it is enough to do so for the Bruhat cells corresponding

to the fundamental reflections: BFr1BF; � � � ;BFrnBF. But using Lemma 7.25

we only need to show that r iN˛i ;F � BFK. But in Li;F we already have that

r iN˛i ;F � Bi;FKi D Li;F since Bi;F � BF this completes the proof.

Definition 7.28. Let ACF (resp. A1F) be the subgroup of AF consisting of elements

whose eigenvalues in L.�/F are positive real numbers (resp. are modulus one).

Then we have a polar decomposition: AF D ACFA1F.

Lemma 7.29. NCFACF \K D f1g.

Proof. When acting on L.�/F the elements of NCFACF \ K are represented by

unitary upper triangular matrices in any admissible basis. Therefore they have to

be diagonal so NCFACF \ K � ACF but the only unitary element of ACF is the

identity.

Iwasawa Decomposition. �GF D NCFACFK with uniqueness of expression.

Proof. The decomposition follows from �GF D BFK and observing that:

BF D NCFACFA1F D NCFACF.K \ BF/:

To prove the uniqueness suppose g 2 �GF has two decompositions:

g D nak D n0a0k0:

Then:

k0k�1 D a�1n�1n0a0

D a�1n�1n0�aa�1

�a0

D�a�1n�1n0a

� �a�1a0

�D n00a00

61

7.4. The Orbit of the Highest Weight Vector

Therefore k�1k0 2 NCFACF \ K D f1g. This implies: k D k0. But since we also

have NCF \ACF D f1g we can deduce that a D a0 and n D n0.

Notation 7.30. For g 2 �GF we use gNgAgK to denote its Iwasawa decomposition.

Remark 7.31. Let K0 denote the group generated by K1; � � � ;Kn, this is a subgroup

of K. However the proofs given in the section work with K0 instead of K as well

and one obtains �GF D NCFACFK0 with uniqueness of expression, which proves

K D K0.

Remark 7.32. The article [8] provides a different proof of Iwasawa decomposition

for split Kac-Moody groups.

7.4 The Orbit of the Highest Weight Vector

In this section we will look the orbit of 1� 2 L.�/F under the action of �GF. But

before doing so we need to introduce the minimal group:

For each i 2 I there exists a subgroup Gi;F � Li;F with Lie algebra:

gi;F D n�˛i ;F˚F˛_i ˚ n˛i ;F � li;F:

In fact we have an isomorphism i W SL2;F! Gi;F.

Notation 7.33. For convenience we will use the following notation: a b

c d

!j

´ j

a b

c d

!!2 Gj;F:

Then one sees that:

rj D

1

�1

!j

:

How the groups G1;F � � � ;Gn;F interact follows from the relations defining the

Kac-Moody algebra gF (see §2 in [14] for details):

Lemma 7.34.

(1) The torus, AF, is the product of the tori in G1;F � � � ;Gn;F.

62

7.4. The Orbit of the Highest Weight Vector

(2) For i; j 2 I and t 2 F�: t

t�1

!i

a b

c d

!j

t

t�1

!�1i

D

a tAij b

t�Aij c d

!j

:

(3) For i; j 2 I; i ¤ j and u; v 2 F: 1 u

1

!i

1

v 1

!j

D

1

v 1

!j

1 u

1

!i

:

(4) For i; j 2 I and t 2 F�:

rj

t

t�1

!i

r�1j D

t

t�1

!i

t�Aij

tAij

!j

:

The group generated by G1;F � � � ;Gn;F was first introduced in [14]. We will

refer to it as the minimal group and denote it by GKPF . In [19] it is shown that the

orbit of the highest weight vector in the projective space, P .L.�/F/, is given by

quadratic equations. More precisely:

Theorem 7.35. All v 2 GKPF � 1� satisfy the following in L.�/F˝F L.�/F:

.�j�/ v ˝ v DX

˛2�[f0g

x.i/˛ � v ˝ y.i/˛ � v;

wherenx.i/˛

oand

ny.i/˛

oare dual basis for g˛;F and g�˛;F with respect to the

standard invariant form on gF.

The minimal group is a subgroup of �GF, moreover when F D R;C we have

ACF;K � GKPF therefore the only difference between GKP

F and �GF is the unipo-

tent group, where the minimal group is missing all the positive imaginary roots.

However due to the structure of the highest weight modules the action of the two

unipotent groups and hence the two groups are the same.

63

Chapter 8

Reduction Theory

Notation 8.1. In this chapter we work only over the real numbers so we will drop

all the subscripts.

Let G be a finite dimensional real reductive group, V a representation for G

and a lattice VZ � V , reduction theory is concerned with vectors of minimal

length in g�1 � VZ as g varies in G. However using such an approach in the

symmetrizable indefinite case immediately runs into difficulties. First since V is

infinite dimensional there might not be a positive lower bound for the vectors in

g�1 �VZ, in other words one might have an infinite sequence of vectors in g�1 �VZ

whose length approaches zero (an explicit example is given in Remark 8.3 below).

So then the question becomes what is the right subset to consider, in other words

before proving anything we have to find the right question first.

In §1 we introduce four different subsets that have the necessary conditions to

possess a reduction theory.

In §2 we define �GAR which is the most natural choice from a geometric point

of view and contains a large subset of �G. However at present we don’t have a re-

duction theory for �GAR since it is not clear whether points on �GAR do have minima

on their � -orbits.

In §3 we define another subset, �GMO, the elements of which all have minima on

their � -orbits. Then Borel’s proof of reduction theory from the finite dimensional

case ([3] 16.6) works without any changes.

In §4 we show �GMO contains a large subset of the group: NC EXP.INT.T //K

(Theorem 8.23). The proof of this theorem is done in two parts: in §4.1 we give a

spectral characterization of NC EXP.INT.T //K. Then in §4.2, using this spectral

characterization, we show that NC EXP.INT.T //K is indeed a subset of �GMO. The

main idea of defining elements with decay (denoted by �G[) was inspired by the

proof of Lemma 17.15 in [12].

64

8.1. The Four Subsets

In §5 using direct calculations in the Kac-Moody group corresponding to a rank

2 GCM we show that �G[ is not � -invariant.

Finally in §6 we list a number of open problems.

8.1 The Four Subsets

Notation 8.2. Let � be the weight defined by˝�; ˛_i

˛D 1 for all i 2 I . We fix

L.�/ as a representation of �G where it acts from the right.

Since L.�/ is infinite dimensional having a positive lower bound does not en-

sure the existence of a minimum, moreover since � � 1� ¨ L.�/Z (see §7.4) one

can consider four different � -invariant subsets of �G as candidates to work with:

(1) inf 2� g�1 �1 � 1� > 0.

(2) g�1 �1 � 1� achieves a minimum as varies in � .

(3) infv2L.�/Z g�1 � v > 0.

(4) g�1 � v achieves a minimum as v varies in L.�/Z.

Let S1; � � � ;S4 be the corresponding subsets in �G. The relationship between these

sets are summarized in the following diagram where arrows indicate inclusion:2

S4 // S3

S1

S2

>>

2in the following simplified form it becomes easier to order the sets: let f W X ! R>0 be afunction and let Y ¨ X . Now compare the following statements: (1) f has a positive infimum onY , (2) f has a minimum on Y , (3) f has a positive infimum on X and (4) f has a minimum on X .

65

8.2. The Arithmetic Set

8.2 The Arithmetic Set

In this section we consider the set S3. One can formulate the definition of S3 in an

equivalent but more intuitive way: any element of �G defines a new metric on L.�/:

kvkg ´ g�1 � v :

We say g 2 �G is an arithmetic point if there is positive lower bound for the length

of the elements of L.�/Z under the metric k � kg . The set of all arithmetic points

is called the arithmetic set. From now we will use �GAR to denote the arithmetic

subset instead of S3.

Remark 8.3. This definition is based on the finite dimensional theory, in fact in

that case �GAR is the entire group (see [3] 16.2). This is not the case here, since

the torus, AC, has non-arithmetic points: let a 2 AC be such that a˛i < 1 for all

i 2 I . Consider the sequence˚1�k

1kD1

, where dp.�k/!1 as k !1, then: a�1 � 1�k D a��k D a��a�k D a�� a˛1�p1k � � � �a˛n�pnk� :Since ht.�k/!1 as k !1 and a˛i < 1 for all i 2 I we see that:

limk!1

a�1 � 1�k D 0:Lemma 8.4. Let g 2 �G and v 2 L.�/ with the weight space decomposition:

v D c11�1 C � � � C ck1�k ; cj 2 R:

Then: g�1 � v � g��j0A

ˇ̌cj0ˇ̌, where �j0 2 f�1; � � � ; �kg is of maximal depth in

that set.

Proof. g�1 � v D g�1A g�1N � v K is unitary

�

g�1A �

�cj01�j0

� choice of �j0

D

�g��j0A cj0

�1�j0

66


D g��j0A

ˇ̌cj0ˇ̌ 1�j0

g��j0A > 0

� g��j0A

ˇ̌cj0ˇ̌ 1�j0

D 1The following Lemma shows why the set of arithmetic points of the Torus,

which we will denote by AAR, is important:

Lemma 8.5. NCAARK � �GAR.

Proof. Let g 2 �G be such that gA 2 AAR we aim to show that g 2 �GAR. Since the

weight vectors˚1� W � 2P�

are a basis for L.�/Z in order to show that there is

a positive lower bound when g acts on L.�/Z we only need to show g has a lower

bound as it acts on these vectors. We have: g�1 � 1� D g�1A g�1N � 1� D g�1A �

�g�1N � 1�

� Now let v D g�1N � 1�, since elements of NC are represented by unipotent up-

per triangular matrices in GL.L.�// the weight space decomposition of v can be

written as follows:

v D 1� C c11�1 C � � � C ck1�k ;

where� has maximal depth among the weights f�; �1; � � � ; �kg. Using Lemma 8.4

with gA and v yields: g�1A � v � g��A :

In summary we have shown: g�1 � 1� � g�1A � 1� D g��A ;

However since gA 2 AAR the right hand side has a positive lower bound.

Remark 8.6. A converse to Lemma 8.5 does not hold, in other words it is possible

to have g 2 �GAR and gA …�GAR. First we need to do a calculation: g�1 � 1� 2 D g�1A g�1N � 1�

2D g�1A � 1�

2 C g�1A � v 2

67


D g�2�A C

g�1A �

�Xcj 1�j

� 2D g

�2�A C

Xc2j g�2�jA

gA … AAR therefore: g��A ! 0 as dp.�/ ! 1. However there are at least two

ways of avoiding a contradiction:

� there exists M 2 N such that n�1 � 1� has a weight component in L.�/�0with dp.�0/ < M for any weight vector 1� 2 L.�/�.

� the matrix coefficients of n�1 2 GL.L.�// grow exponentially with depth.

Because of Lemma 8.5 next we will try to compute AAR.

Definition 8.7. We define the following subset of positive roots:

�re� D

˚� 2 �re

C W � � � 2P�

:

�im� and �� are defined similarly.

Lemma 8.8. �� D �C.

Proof. We show this by proving �re� D �

reC

and �im� D �

imC

.

REAL ROOTS: Let � D w.˛i / be a positive real root, we define its correspond-

ing coroot as �_ D w.˛_i /. Now using Proposition 11.1 in [15] we have:

�reCn�

re� D

˚� 2 �re

C W˝�; �_

˛D 0

;

However we see that the set is in fact empty since � is dominant and integral and

since � is a positive root, �_ will be a positive coroot.

IMAGINARY ROOTS: Using Corollary 11.9 [15] �im� D �im

Cis equivalent to

.�j�/ ¤ 0 for all � DPpi˛i 2 �C, however (§5.2 [15]):

.�j�/ D

nXiD1

pi .�j˛i / D

nXiD1

pi�12j˛i j

2˝�; ˛_i

˛�D

12

nXiD1

pi j˛i j2 > 0:

Lemma 8.9. EXP.T / � AAR � EXP.CL.T //.

68

8.3. Points with Minima

Proof. Suppose EXP.´/ D a 2 AAR, using Lemma 8.8 we have:

0 < inf�2P�

a�� D inf�2��

a��a� D inf�2��

a� D inf�2��

eh�; í D inf�2�C

eh�; í

Which is equivalent to:

9L 2 R W 8� 2 �C W h�; í � L:

Now recall that if � 2 �imC

then n� 2 �imC

for all n 2 N. Therefore we can

conclude:

8� 2 �imC W h�; í � 0:

by Proposition 2.48 this means ´ 2 CL.T /.

Conversely let a D EXP.´/ 2 EXP.T /. Since the weight vectors generate the

Chevalley lattice so it is enough to show:

0 < inf�2P�

a��:

However based on Bardy there exist w0 2 W ; ´0 2 D such that ´ D w0.´0/,

therefore we have:

a�� D eh��;í D eh��;w0.´0/i D ehw�10 .��/; ´0i D e�hw

�10 .�/; ´0iehw

�10 .�/; ´0i;

where � 2 �C. Now since ´0 2 D we have h˛i ; ´0i � 0 for all i 2 I . Therefore

as long as w�10 .�/ 2 �C there is a lower bound for a��. Now the result follows

from ˇ̌w�10 .�C/ \��

ˇ̌<1:

Corollary 8.10. NC EXP.T /K � �GAR.

Proof. This follows from Lemma 8.5 and Lemma 8.9.

8.3 Points with Minima

The problem with the arithmetic set is that in order to prove the reduction theorem

one needs to first show the existence of minima. That is, for a given point g the

69


function ˚ achieves a minimum on the set �g. However since L.�/ is infinite

dimensional it is not obvious that an arithmetic point would achieve its minimum,

we might have a situation where the infimum is not in fact a minimum. Therefore

we consider the subset S2 instead.

Definition 8.11. First we define a function ˚ W �G ! R>0 given by: ˚.g/ D g�1 � 1� . Then S2 is the set of all elements g 2 �G, such that ˚ achieves a

positive minimum when considered as a function on the orbit: �g. More formally:

S2 Dng 2 �G W 9 0 2 � ;8 2 � W ˚. 0g/ � ˚. g/o :

From now on we will use �GMO to denote S2.

Remark 8.12. By design, �GMO is a � -invariant subset of �G. In fact if one defines:

� Dng 2 �G W 8 2 � W ˚.g/ � ˚. g/o ;

then: �GMO D ��.

Remark 8.13. It is not obvious from the definition that �GMO is in fact non-empty,

below we will show that it does contain a large subset of �G (see 8.23).

Definition 8.14. For � > 0 define:

A� D˚a 2 AC W 8i 2 I W a˛i � �

:

We also use �G� to denote the set NCA�K.

Reduction Theorem. There exists a real constant � > 0 such that for any g 2�GMO there exists 2 � with g 2 �G� .

8.3.1 Proof of Reduction Theorem

Remark 8.15. Since �GMO D ��, it is enough to show that� � �G� for some real

constant � > 0.

70


Remark 8.16. For each i 2 I we can write �G D NiAiLiK, where Ai D KER.˛i / �

A. In particular any g 2 �G has a decomposition:

g D ngag lgkg ;

where ng 2 Ni ; ag 2 Ai ; lg 2 Li and kg 2 K.

Lemma 8.17. ˚ is left-invariant under NC and right-invariant under K, in fact

we have: ˚.g/ D g��A .

Proof. Using Iwasawa decomposition, NC � 1� D 1� and the fact that K is unitary

with respect to k � k we see that ˚.g/ D ˚.NCg/ D ˚.gK/ and the first assertion

is proven. Now we can write:

˚.g/ D ˚.gA/ D g�1A � 1�

D g��A 1� D g��A

1� :

Now recall that k � k is normalized such that 1�

D 1.

Lemma 8.18. ˚.g/ D ˚.ag/˚.lg/.

Proof. Using the decomposition from Remark 8.16 and then applying Lemma 8.17

we see that ˚.g/ D ˚.ag lg/, now we use the definition of ˚ :

˚.ag lg/ D l�1g a�1g � 1�

D a��g l�1g � 1� D ˚.ag/˚.lg/:Lemma 8.19. If 2 � \ Li then: ˚. g/ D ˚.ag/˚. lg/.

Proof. 2 Li therefore it normalizes Ni and commutes with Ai :

˚. g/ D ˚� ngag lgkg

�D ˚

�n0gag lgkg

�D ˚.ag lg/;

now we use the previous Lemma.

Lemma 8.20. If g 2 � then: ˚.lg/ � inf 2�\Li ˚. lg/.

71

8.4. A Subset of �GMO

Proof.

˚.ag/˚.lg/ D ˚.g/ Lemma 8.18

� inf 2�

˚. g/ g 2 �

� inf 2�\Li

˚. g/ � \ Li � �

D inf 2�\Li

˚.ag/˚. lg/ Lemma 8.19

D ˚.ag/ inf 2�\Li

˚. lg/

Cancelling ˚.ag/ from both sides gives us the result.

Lemma 8.21. There is a real constant, " > 0, such that: inf 2�\Li ˚. lg/ � ".

Proof. This follows from Li being a finite dimensional reductive group of semi-

simple rank 1.

Lemma 8.22. If g 2 � then for all i 2 I we have:

l˛ig;A � "

�2:

Proof. Based on Lemma 8.21 and Lemma 8.20 we have:

" � inf 2�\Li

˚. lg/ � ˚.lg/ D l��g;A:

Now note that �jLi D12˛i .

Finally Lemma 8.22 combined with the following observation completes the

proof:

g˛iA D

�lg;Aag

�˛iD l

˛ig;A:

8.4 A Subset of �GMO

In this section we prove the following:

Theorem 8.23. NC EXP.INT.T //K � �GMO.

72


8.4.1 A Spectral Characterization of INT.T /

Definition 8.24. For a 2 AC and C > 0 define:

P�.a; C /´˚� 2P� W a

�� C

:

Definition 8.25. We say a 2 AC decays in L.�/ if P�.a; C / is finite for all

C > 0. Notice that if a decays in L.�/ then a� ! 0 as dp.�/!1 for � 2P�.

The subset of AC consisting of all such elements is denoted by A[.

Lemma 8.26. A[ � EXP.INT.T //

Proof. Let EXP.´/ D a 2 A[, then one has the following:

8C > 0 Wˇ̌P�.a; C /

ˇ̌<1$ 8C > 0 W

ˇ̌̌n� 2P� W a

�� C

oˇ̌̌<1

$ 8C > 0 Wˇ̌̌n� 2 �� W a

�� C

oˇ̌̌<1

$ 8C > 0 Wˇ̌̌n� 2 �� W a

�� a�C�1

oˇ̌̌<1

$ 8C 0 > 0 Wˇ̌̌n� 2 �� W a

�� C 0

oˇ̌̌<1

$ 8C 0 > 0 Wˇ̌̌n� 2 �� W e

h�; í� C 0

oˇ̌̌<1

$ 8K 2 R Wˇ̌˚� 2 �� W h�; í � K

ˇ̌<1

Taking K D 0 and using Lemma 8.8 implies ´ 2 INT.T /.

Lemma 8.27. A[ is invariant under the action of W .

Proof. Let EXP.´/ D a 2 A[ and take C > 0 and w 2W to be arbitrary:

P�

�waw�1; C

�D

n� 2P� W

�waw�1

�� C

oD

n� 2P� W EXP .w.´//� � C

oD

n� 2P� W e

h�;w.´/i� C

oD

n� 2P� W e

hw�1.�/; í � Co

D

n� 2P� W a

w�1.�/� C

o73


And this last set is finite because a 2 A[ and W permutes the set of weights.

Lemma 8.28. EXP .Dfin/ � A[.

Proof. Let ´ 2 Dfin then by definition ´ 2 FJ , where J � I is an arbitrary subset

of finite type. Take M´ to be the minimum of the set fh˛i ; í W i … J g, note that

M´ is a positive real number. For any � 2 �reC

one sees that h�; í � htJ .�/M´,

where htJ is defined as follows:

htJ�X

i2Ipi˛i

�D

Xi…J

pi :

Therefore for � 2P� one has:

h�; í D h� � �; í D h�; í � h�; í � h�; í � htJ .�/M´:

And so for a D EXP.´/ we can write:

a� D eh�; í � eh�; íe� htJ .�/M´ :

Let C > 0 be arbitrary, then:

C � a� � eh�; íe� htJ .�/M´ :

which implies htJ .�/ has to bounded. Therefore:

P�.a; C / D f� 2 �C W htJ .�/ is boundedg

So we can partition P�.a; C / into finitely many subsets in each of which pj ; j …

J are fixed while the rest of coefficients vary. However J is of finite type, hence

each partition itself has to be finite, hence a 2 A[.

Proposition 8.29. A[ D EXP.INT.T //.

Proof. Using Lemma 8.27 and 8.28 we have: EXP.INT.T // D EXP.W � Dfin/ �

A[. This combined with Lemma 8.26 shows EXP.INT.T // D A[.

74


8.4.2 Decay and Minima

Definition 8.30. Set �G[ ´ NCA[K, then the statement of Theorem 8.23 can be

rewritten as �G[ � �GMO. Note that this is a strict inclusion since 1 2 �GMO but

1 … �G[.

Definition 8.31. For a 2 AC and C > 0, let P�.a; C / be a subset of P� defined

as follows: if � 2P�.a; C / then � and all weights in P� of lower depth belong to

P�.a; C /.

Definition 8.32. For a 2 AC and C > 0 define:

L.�I a; C /´M

�2P�.a;C/

L.�/�;

�.a; C /´˚ 2 � W �1 � 1� 2 L.�I a; C /

:

Remark 8.33. L.�I a; C / is a NCAC-stable subspace of L.�/ for all a 2 AC and

C > 0.

Lemma 8.34. If a 2 A[ then for all C > 0, L.�I a; C / is finite dimensional.

Proof. a 2 A[ so P�.a; C / is finite for all C > 0. From the structure of the high-

est weight module it follows that P�.a; C / is finite for all C > 0 as well. Since

the weight spaces of L.�/ are finite dimensional, L.�I a; C / is finite dimensional

for all C > 0.

Lemma 8.35. If g 2 �G[, for all C > 0;˚ achieves a minimum on the set:

�.gA; C /g.

Proof. Let g 2 �G[, since ˚ is right invariant under K we may assume that g 2

NCAC. Take C > 0 to be arbitrary and consider the following set:

S D˚ �1 � 1� W 2 �.gA; C /

� L.�/Z \ L.�I gA; C /:

To prove the Lemma we need to show that g�1.S/ has an element of minimal

length. However L.�I gA; C / is NCAC-stable and therefore g�1.S/ still lies in

75


L.�I gA; C /. Since g 2 �G[, by Lemma 8.34 L.�I gA; C / is finite dimensional.

Therefore g�1.S/, as a discrete subset of the finite dimensional subspace, has an

element of minimal length.

Lemma 8.36. Let g 2 �G; v 2 L.�/Z and C > 0. If v … L.�I gA; C / then: g�1 � v > C�1.

Proof. Since v … L.�I gA; C /, based on the construction of P�.gA; C / we see

that among all the weights of maximal depth that appear in the weight decompo-

sition of v, there is at least one weight, �, such that � … P�.gA; C /. Therefore

� …P�.gA; C / as well and we have:

g�A < C: (8.37)

On the other hand Lemma 8.4 gives us: g�1 � v � g��A jc�j :

Since v 2 L.�/Z we have: jc�j � 1, which gives us: g�1 � v � g��A : (8.38)

Combining (8.38) and (8.37) gives the desired result.

Corollary 8.39. Let g 2 �G; 2 � and C > 0. If … �.gA; C / then: ˚. g/ >

C�1.

Corollary 8.40. Let g 2 �G; 2 � . Then 2 ��gA; ˚. g/

�1�.

Let g 2 �G[, by Lemma 8.35 ˚ achieves its minimum in �.gA; C / for all

C > 0. So pick an arbitrary element � 2 � and let 0 denote the minimum in

��gA; ˚.�g/

�1�. We claim 0 is the minimum on the entire orbit: �g. Suppose

2 � is arbitrary, there are two cases:

� 2 ��gA; ˚.�g/

�1�: then ˚. 0g/ � ˚. g/ follows from our choice of

0.

76

8.5. �G[ is not � -invariant

� … ��gA; ˚.�g/

�1�: then ˚.�g/ < ˚. g/ holds by Lemma 8.39. But

from Corollary 8.40 we have � 2 ��gA; ˚.�g/

�1�

and therefore: ˚. 0g/ �

˚.�g/.

Thus we have a global minimum on �g and therefore g 2 �GMO. This com-

pletes the proof of Theorem 8.23.

8.5 �G[ is not � -invariant

In this section we will show using indefinite GCMs of rank 2 that �G[ is not in fact

� -invariant. But first we need to carry on a few calculations in SL2.

8.5.1 Iwasawa Decomposition in SL2

Definition 8.41. For u 2 R and t 2 R>0 define:

��.u/ D

1

u 1

!; �.t/ D

t

t�1

!; �C.u/ D

1 u

1

!:

Lemma 8.42. The Iwasawa decomposition of SL2 can be explicitly written as

follows: a b

c d

!D �C

�ac C bd

�2

��.��1/

d��1 �c��1

c��1 d��1

!;

where � Dpc2 C d2.

Lemma 8.43. Let r be the non-trivial element of the Weyl group in SL2:

r D

1

�1

!:

Then its action on SL2=SO2 is given by:

r � �C.u/�.t/ D �C

��

u

u2 C t4

��

�t

pu2 C t4

�:

77

8.5. �G[ is not � -invariant

Proof. We use the Iwasawa decomposition:

r � �C.u/�.t/ D

t�1

�t �t�1u

!

D �C

��

u

u2 C t4

��

�t

pu2 C t4

�0@� upu2Ct4

t2pu2Ct4

�t2pu2Ct4

�upu2Ct4

1A8.5.2 The Counterexample

Let �G be the group corresponding to the rank 2 GCM

A D�Aij�D

2 �n

�m 2

!; m; n 2 N; n � m:

Let na D �1.u/�1.t1/�2.t2/ and consider it as an element of �G=K D NCAC.

We will calculate r1 � na 2 �G=K to show that �G[ is not � -invariant. Since the

minimal group, GKP (see §7.4) is a subgroup of �G and r1 � na 2 GKP we will

do the calculations in the minimal group where we have concrete generators and

relations for the whole group. First we introduce the following notation to simplify

our task:

� D

qu2 C t41 ;

D

qu2 C t41 t

�2m2 :

Now using Lemmas 8.42, 8.43 and 7.34 we have:

r1 � na D r1�1.u/�1.t1/�2.t2/

D

(�1

��u

�2

��1

� t1�

� �u��1 t21�

�1

�t21��1 �u��1

!1

)�2.t2/

D �1

��u

�2

��1

� t1�

��2.t2/

(�2.t2/

�1

�u��1 t21�

�1

�t21��1 �u��1

!1

�2.t2/

)

D �1

��u

�2

��1

� t1�

��2.t2/

�u��1 tm2

�t21��1�

t�m2��t21�

�1��u��1

!1

78

8.6. Open Problems

D �1

��u

�2

��1

� t1�

��2.t2/ �

�

(�1

�ut21 2

�t�m2 � tm2

��1

��

� �u �1 t21 t

m2 �1

�t21 t�m2 �1 �u �1

!1

)D �1

��u

�2

��1

� t1�

��2.t2/�1

�ut21 2

�t�m2 � tm2

��1

��

�D �1

��u

�2

��1

� t1�

��1

t�m2

"ut21 2

�t�m2 � tm2

�#!�2.t2/�1

��

�D �1

��u

�2

��1

t21�2

"t�m2

ut21 2

�t�m2 � tm2

�#!�1

� t1�

��2.t2/�1

��

�D �1

��u

2

��1

�t1

��2.t2/

Now in this calculation lets consider what has happened at the level of AC-component.

The initial point .t1; t2/ has been transformed into: t1q

u2 C t41 t�2m2

; t2

!:

First note that if one takes u D 0 then one recovers the simple action of the funda-

mental reflection on AC but if u ! 1 the first component approaches 0. Hence

even if a 2 EXP.INT.T // by taking u large enough in �1.u/ 2 N˛1 we can make

sure that r1 � na … NC EXP.INT.T //. In other words �G[ is not � -invariant.

8.6 Open Problems

Question 8.44. Find a relationship between �GAR and �GMO.

Question 8.45. Compute AAR and AMO (points in AC which also belong to �GMO).

Question 8.46. Compute the projection of �GAR and �GMO onto AC (This is a far

harder problem than computing AAR and AMO).

Question 8.47. Is NC=.NC \ �/ a projective limit of finite dimensional compact

spaces and hence compact?

79

Bibliography

[1] Ling Bao, Lisa Carbone, and Howard Garland. Integer forms of kac-

moody groups and eisenstein series in low dimensional supergravity theories.

Preprint.

[2] Nicole Bardy. Systèms de racines infinis. Mém. Soc. Math. Fr. (N.S.),

(65):vi+188, 1996.

[3] Armand Borel. Introduction aux groupes arithmétiques. Publications de

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1975 and 1982 French originals by Andrew Pressley.

[5] Lisa Carbone and Howard Garland. Infinite dimensional chevalley groups

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over finite fields. Commun. Contemp. Math, 5(5):813–867, 2003.

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Studies in Advanced Mathematics. Cambridge University Press, Cambridge,

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[10] H. Garland. Certain Eisenstein series on loop groups: convergence and the

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[11] Howard Garland. The arithmetic theory of loop algebras. J. Algebra,

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[12] Howard Garland. The arithmetic theory of loop groups. Inst. Hautes Études

Sci. Publ. Math., (52):5–136, 1980.

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dimensional groups. Astérisque, (Numero Hors Serie):165–208, 1985. The

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[15] Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University

Press, Cambridge, third edition, 1990.

[16] Shrawan Kumar. Kac-Moody groups, their flag varieties and representation

theory, volume 204 of Progress in Mathematics. Birkhäuser Boston Inc.,

Boston, MA, 2002.

[17] Olivier Mathieu. Formules de caractères pour les algèbres de Kac-Moody

générales. Astérisque, (159-160):267, 1988.

[18] Robert V. Moody and Arturo Pianzola. Lie algebras with triangular decom-

positions. Canadian Mathematical Society Series of Monographs and Ad-

vanced Texts. John Wiley & Sons Inc., New York, 1995. A Wiley-Interscience

Publication.

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sures ordonnées. In Preparation, 2012.

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French by John Stillwell.

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82

Index of Notation

L.�/C , 19

L.�/R, 22

L.�I a; C /, 75

M.�/C , 18

C�, 19

D, 16

Dfin, 16

�, 7

�im, 15

�im˙

, 15

�re, 15

�re˙

, 15

�˙, 8

FJ , 16

HF.m/, 45

HQ.m/, 35

HZ.m/, 47

� , 52

�.a; C /, 75�GAR, 66�G[, 75�GMO, 70�G� , 70�GF, 45�GQ, 45

P, 33

Pi , 33

P�, 20

P�.a; C /, 73

˘ , 4

˘_, 4

Q, 7

Q�, 40

Q_, 7

S, 20

S�, 21

T , 16

UZ.m/, 47

XQ.m/, 35

YQ.m/, 50

˛i , 4

˛_i , 4

�˙.u/, 77

dp, 20

�.t/, 77

EŒ��, 22

�, 20

ht, 7

htJ , 74

aC , 8

aR, 15

aZ, 28

bC , 8

b˙i;Q, 40

83

Index of Notation

ci;Q, 40

gC , 5

gR, 22

gZ, 28

gi;F, 62

n˙C , 8

n˙Z , 28

ni;Q, 40

pi;C , 8

� , 20

�0, 22

ACF, 61

AAR, 67

A[, 73

A� , 70

Ai , 71

AQ, 40

BQ, 40

B˙i;Q, 40

Ci;Q, 40

GKPF , 63

Gi;F, 62

K, 60

Ki , 60

Li;Q, 40

N˙Q , 40

N˙˛i ;Q, 40

Ni;Q, 40

Pi;Q, 40

W , 13

�, 70

!, 5

1�, 18

P�.a; C /, 75

�, 65

e˙i , 5

ri , 13

radi , 12

r�i , 12

84

Appendix A

Formulas in Associative Algebras

Let F be a field of characteristic zero. In this appendix we present a number of

formulas that hold in any associative F-algebra AF with a unit.

Notation A.1. For x 2 AF and n 2 N we set:

xŒn� Dxn

nŠ;

x

n

!Dx.x � 1/ � � � .x � nC 1/

nŠ:

Also we may denote ad.x/.y/ by Œx; y� when the latter is more convenient.

Lemma A.2 ([4] page 178). If x; y 2 AF and n 2 N:

1

nŠad.x/n.y/ D

XpCqDn

.�1/qxŒp�yxŒq�

Lemma A.3 ([4] page 178). Suppose ´; x 2 AF such that Œ´; x� D cx for c 2 F.

Then for all n 2 N, and all P 2 FŒX�, we have:

P.´/xŒn� D xŒn�P.Ć nc/:

Definition A.4. .x; y; ´/ with x; y; ´ 2 AF is called a S -triple if:

Œx; y� D ´; Œ´; x� D 2x; Œ´; y� D �2y:

Lemma A.5 ([4] page 70). If .x; y; ´/ is a S -triple in AF then we have the follow-

85

Appendix A. Formulas in Associative Algebras

ing: �´; xn

�D 2nxn�

´; yn�D �2nyn�

y; xn�D nxn�1.�´ � nC 1/ D n.�Ć n � 1/xn�1�

x; yn�D nyn�1.´ � nC 1/ D n.Ć n � 1/yn�1

Lemma A.6 ([24] page 9). If .x; y; ´/ is a S -triple in AF for p; q 2 N, we have:

xŒp�yŒq� D

min.p;q/XrD0

yŒq�r�

´ � q � p C 2r

r

!xŒp�r� (A.7)

86

Appendix B

Lie Algebras

Let R be a commutative ring with a unit element.

Definition B.1. An R-module LR, is called an R-algebra if one has a R-homomorphism:

� W LR ˝R LR ! LR.

Definition B.2. Given an R-algebra LR we define its opposite algebra, LopR

to be

the identical to LR as an R-module but with �op.x ˝ y/ D �.y ˝ x/.

Definition B.3. An R-linear map D W LR ! LR is called a derivation if it satisfies

Leibniz’s law: D.xy/ D D.x/y C xD.y/.

Lemma B.4. If D is a derivation and x1; x2; � � � ; xk 2 LR then:

D.x1x2 � � � xk/ D

kXiD1

x1 � � � xi�1D.xi /xiC1 � � � xk :

Definition B.5. For x 2 LR define ad.x/ W LR ! LR by ad.x/.y/ D xy � yx.

Lemma B.6. ad.x/ is a derivation for all x 2 LR.

Definition B.7. A R-Lie algebra is an R-algebra with the following properties:

1) The map � W LR ˝R LR ! LR admits a factorization:

LR ˝R LR !

2̂

LR ! LR;

lets denote the image of x ˝ y under this map by Œx; y� then condition be-

comes:

Œx; x� D 0; 8x 2 LR:

87

Appendix B. Lie Algebras

2) We have Jacobi’s identity:

ŒŒx; y�; ´�C ŒŒy; ´�; x�C ŒŒ´; x�; y� D 0

Definition B.8. Given a R-Lie algebra LR we define its opposite Lie algebra with

the following bracket:

Œx; y�op´ �Œx; y�:

Definition B.9. Let LR;L0R

be two R-Lie algebras, a map ' W LR ! L0R

is called a

Lie homomorphism if ' is R-linear and satisfies: ' .Œx; y�/ D Œ'.x/; '.y/�, for all

x; y 2 LR.

Example B.10. Let LR be an arbitrary R-algebra. One can equip LR with a Lie

algebra structure by defining: Œx; y� D 0 for all x; y 2 LR. Such a Lie algebra is

called commutative.

Example B.11. The set DER.LR/ of all derivations of an R-algebra LR is a Lie

algebra with the product: ŒD;D0� D DD0 �D0D.

Example B.12. Let LR be an R-algebra then Œx; y� D xy � yx turns LR into an

R-Lie algebra. We use LIE.LR/ when want to emphasize the Lie algebra structure

on LR. Note that the underlying sets of LR and LIE.LR/ are identical.

Theorem B.13. Let LR be an R-Lie algebra. For any x 2 LR define a map ad.x/ W

LR ! LR by ad.x/.y/ D Œx; y�, then:

1) ad.x/ is a derivation of LR.

2) The map x 7! ad.x/ is a Lie homomorphism of LR into DER.LR/.

Definition B.14. A universal enveloping algebra of an R-Lie algebra LR is a pair

.";UR/, where:

� UR is an associative R-algebra with a unit.

� " W LR ! UR is a Lie algebra homomorphism.

� HOMR�lie.LR; LIE.TR// Š HOMR�alg.UR;TR/.

88

Appendix B. Lie Algebras

Any Lie algebra possesses a universal enveloping algebra, moreover we have

the following functorial properties:

UR.L˚ L0/ D UR.L/˝R UR.L0/:

If we have a R-Lie algebra homomorphism: W LR ! L0R

then by universal

property:

U./ W UR.L/! UR.L0/:

is a homomorphism of R-algebras.

PBW Theorem ( [24] page 8). Let LR be an R-Lie algebra and .";UR/ its univer-

sal enveloping algebra. Then:

� " is injective.

� If LR is identified with its image in UR and if fx1; x2; � � � g is an ordered

R-basis for LR, then all monomials

xai1i1xai2i2� � � x

aikik;

in which i1 < � � � < ik and ai1 ; � � � ; aik 2 N, form an R-basis for UR.

89

Appendix C

Hopf Algebra

C.1 Definition

Let R be a commutative ring with a unit. A Hopf algebra over R is a 6-tuple,

.HR;�; �; �; �; / such that:

� HR is an R-algebra with �; � defining product and unit.

� HR is an R-coalgebra with �; � defining co-product and co-unit.

� These two structures are compatible, i.e. �; � are R-algebra homomorphisms.

� W HR �! HR is an R-algebra homomorphism, usually called antipode,

such that the following diagram commutes:

HR ˝R HR

1˝ // HR ˝R HR

�

%%HR

�99

� %%

� // R� // HR

HR ˝R HR ˝1

// HR ˝R HR

�

99

Given a Hopf algebra HR over R we can equip HOMR�alg.HR; R/with the structure

of a group scheme over R.

90

C.2. Enveloping Algebra

C.2 Enveloping Algebra

Let gR be a R-Lie algebra and consider the following Lie algebra homomorphism:

˚gR ! f0g

x 7! 0;

˚gR ! gR ˚ gR

x 7! .x; x/;

˚gR ! g

opR

x 7! �x:

Then we have the corresponding maps for enveloping algebras:

� W UR.g/! UR.f0g/ D R;

� W UR.g/! UR.g˚ g/ D UR.g/˝R UR.g/;

W UR.g/! UR.gop/ D UR.g/

op:

Consider gR as a subset of UR.g/ then for x 2 gR then: �.x/ D 0; �.x/ D 1˝ xC

x ˝ 1 and .x/ D �x.

Definition C.1. An element x 2 UR.g/ is called primitive if: �.x/ D x˝1C1˝x.

Lemma C.2.�UR.g/;�; �; �; �;

�is a co-commutative Hopf algebra, where:

� � is the multiplication,

� � is the unit,

� � is the co-multiplication,

� � is the co-unit,

� is the antipode.

C.3 The Dual of the Enveloping Algebra

The material in the section follows [9] §2.7.8.

Consider the transpose of the co-product:

�� W�UR.g/˝R UR.g/

��! UR.g/

�

91

C.3. The Dual of the Enveloping Algebra

combined with the restriction:

UR.g/�˝R UR.g/

��UR.g/˝R UR.g/

��we get a linear map: �� W UR.g/

� ˝R UR.g/� ! UR.g/

�. More explicitly for

f; g 2 UR.g/� the product is give by:

.fg/.u/ D ��.f ˝ g/.u/ D .f ˝ g/.�.u//:

The vector space UR.g/� is thus equipped with the structure of an algebra.

Notation C.3. Let Z1C

be the set of all infinite sequences of non-negative integers

and for � 2 Z1C

define:

x� D

1YnD1

x�nn 2 RŒŒx1; x2; � � � ��

e� D

1YnD1

eŒ�n�n 2 UR.g/:

Lemma C.4. For � 2 Z1C

we have:

�.e�/ DX

�C�D�

e� ˝ e�:

Proof. We have:

��eŒ�1�1

�D

1

�1Š��e�11

�D

1

�1Š�.e1/

�1

D1

�1Š.e1 ˝ 1C 1˝ e1/

�1

D

X�1C�1D�1

1

�1Š�1Še�11 ˝ e

�11

Hence:

�.e�/ D �

nYiD1

eŒ�i �i

!

92


D

nYiD1

��eŒ�i �i

�D

nYiD1

X�iC�iD�i

1

�i Š�i Še�1i ˝ e

�1i

D

X�1C�1D�1

��nC�nD�n

1

�1Š�1Š � � ��nŠ�nŠe�11 � � � e

�nn ˝ e

�1n � � � e

�nn

D

X�C�D�

e� ˝ e�

Theorem C.5. There is an isomorphism of the R-algebras: UR.g/� Š RŒŒx1; x2; � � � ��

given by the map:

f 7! sf ´X�

f .e�/x� :

Proof. Since˚e� W � 2 Z1

C

is a basis for UR.g/ the mapping f 7! sf is bijective

(in the same way that the dual of an infinite direct sum is an infinite direct product).

Moreover if f; g 2 UR.g/�, then:

sfg DX�2Z1

C

.fg/.e�/

D

X�2Z1

C

.f ˝ g/.�.e�//

D

X�2Z1

C

.f ˝ g/

0@ X�C�D�

e� ˝ e�

1AD

X�2Z1

C

X�C�D�

f .e�/˝ g.e�/

D

X�;�2Z1

C

f .e�/g.e�/x�C�D sf sg :

In particular, the algebra UR.g/� is associative and commutative. Its unity, 1�,

is a linear form such that Ker.1�/ D AR.g/ and 1�.1/ D 1. The unit map then

is the transpose co-unit map in UR.g/, that is we have: �� W R ! UR.g/�, where

93


��.1/ D 1�. Moreover the transpose of the principal anti-automorphism of UR.g/

is an automorphism of UR.g/� and is called the principal automorphism of UR.g/

�.

94

Appendix D

Amalgams and Tits Systems

The material of this appendix follows [21].

D.1 Direct Limits

Let .Gi /i2I be a family of groups such that for each pair we have a homomorphism

fij W Gi ! Gj . A group G and a family of homomorphisms .gi W Gi ! G/i2I is

called the direct limit of the family .Gi /i2I relative to .fij / if:

(1) gj ı fij D gi for all i; j 2 I .

(2) If H is a group with a family of homomorphisms .hi W Gi ! H/i2I such

that hj ı fij D hi for all i; j 2 I . Then there is exactly one homomorphism

� W G ! H such that hi D � ı gi

Proposition D.1. The pair consisting of G and the family of homomorphisms .gi W

Gi ! G/i2I exists and is unique up to unique isomorphism.

D.2 Tits Systems

A Tits System is a 4-tuple .G;B;N; S/whereG is a group, B andN are subgroups

of G, and S a subset of W D N=.B \N/ satisfying the following axioms:

(1) The set B [N generates G and B \N is a normal subgroup of N .

(2) The set S generates W D N=.B \N/ and consists of elements of order 2.

(3) BsB BwB � BwB [ BswB for s 2 S and w 2 W .

(4) For each s 2 S one has sBs�1 ¤ B .

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D.3. Tits’s Theorem

The groupW is called the Weyl group of .G;B;N; S/; the pair .W; S/ is a Coxeter

system, that is S and the relations:

.st/mst D 1; s; t 2 S and mst <1;

is a presentation for W . The group G is the disjoint union of double cosets

BwB;w 2 W . This is called the Bruhat decomposition.

If S 0 � S , let WS 0 be the subgroup of W generated by elements of S 0; then

PS 0 D BWS 0B . Then S 0 7! PS 0 is a bijection of the set of subsets of S onto

the set of subgroups of G containing B . PS 0 is then called the standard Parabolic

subgroup of type S 0.

D.3 Tits’s Theorem

Let G be a group, and let .Gi /i2I be a family of subgroups of G. We say G is the

product of Gi amalgamated along their intersections if G is the direct limit of the

system formed by the Gi , the Gi \Gj and the inclusions:

Gi \Gj � Gi ; Gi \Gj � Gj :

Theorem D.2 ([25]). Let .G;B;N; S/ be a Tits system; Then G is the product of

N and .Pfsg/s2S amalgamated along their intersections.

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