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TENSOR FACTORIZATION AND SPIN
CONSTRUCTION FOR KAC-MOODY
ALGEBRAS

By

Rajeev W alia .

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2007

ABSTRACT

TENSOR FACTORIZATION AND SPIN
CONSTRUCTION FOR KAC-MOODY
ALGEBRAS

By

Ra j eev W alia

In this paper we discuss the “Factorization phenomenon” which occurs when a
representation of a Lie algebra is restricted to a subalgebra, and the result factors
into a tensor product of smaller representations of the subalgebra. We analyze this
phenomenon for synmletrizable Kac-Moody algebras (including finite-dimensional,
semi-simple Lie algebras). We present a few factorization results for a general em-
bedding of a symmetrizable Kac-Moody algebra into another and provide an algebraic
explanation for such a phenomenon using Spin construction. We also give some ap—
plication of these results for semi-simple finite dimensional Lie algebras.

We extend the notion of Spin functor from finite-dimensional to symmetrizable
Kac-Moody algebras, which requires a very delicate treatment. We introduce a cer—
tain category of orthogonal g-representations for which, surprisingly, the Spin func-
tor gives a g—representation in Bernstein-Gelfand-Gelfand category 0. Also, for an
integrable representation Spin produces an integrable representation. We give the
formula for the character of Spin representation for the above category and work out

the factorization results for an embedding of a finite dimensional semi-simple Lie al-

gebra into its untwisted affine Lie algebra. Finally, we discuss classification of those

representations for which Spin is irreducible.

To my family

ACKNOWLEDGMENTS

It is a pleasure to thank the many people who made this thesis possible.

It is difficult to overstate my gratitude to my PhD. adviser, Dr. Peter Magyar.
With his inspiration. enthusiasm, and his great efforts to explain things clearly, he
helped me to effectively grasp the fundamentals of my field of research - Represen-
tation Theory. His patience, flexibility and faith in me in the past years, helped me
gain a good perspective of my goals and learn ways to realize them. Throughout my
thesis-writing period, he provided encouragement, sound advice, good company, and
lots of great ideas. I cannot thank him, enough.

I am thankful to all my committee members, specially Dr. Sagan and Dr. Kulkarni
for their time and suggestions on an early draft of my thesis.

This thesis gradually emerged amid the friendships that animated my MSU years
and provided their most lasting lessons. I would like to thank my friend Mohar who
patiently listened to my often partially formulated ideas regarding my research and
gave valuable suggestions to help me realize them concretely. I would also like to
thank my friend Pavan for lending his expertise in latex during Thesis-writing and
presentations.

Last, but not least, I thank my family : my parents, for educating me with aspects
from both arts and sciences, for unconditional support and encouragement to pursue
my interests ; my brother who has been the very first source of motivation and
inspiration for me to pursue Mathematics ; my sister for her constant support all

through.

TABLE OF CONTENTS

Introduction

1 Main results
1.1 Background for symmetrizable Kac-Moody algebras .......... '

1.2 Factorization Theorems .........................

1.3 Basic properties of the Spin functor ...................

1.4 Special cases for Finite-dimensional Lie algebras ............
1.4 1 Principal Specialization ..... y .................
1.4.2 Folding of Dynkin diagrams ...................

1.5 Factorization Theorems for affine Lie algebras .............
1.5.1 Affinized representations ................

1.5.2 Factorization Theorems for affine Lie algebras .........

1.6 Classification of coprirnary representations ...............

General Spin construction for augmented symmetrizable Kac-

Moody algebras
2.1 Finite dimensional case ..........................
2.2 General case ................................

Proofs

3.1 Proof of Lemma 1 .............................
3.2 Proof of Propositions 1-2 .........................
3.3 Proof of Theorem 1 ............................
3.4 Proof of Theorem 2 ............................
3.5 Proof of Propositions 3-4 .........................
3.6 Proof of Propositions 5-8 .........................
3.7 Proof of Propositions 9-10 ........................

BIBLIOGRAPHY

vi

\lO‘ICfl

10
14
14
15
16
17
18
19

21
21
23

27
27
28
39
39
40
43
52

6O

Introduction

The factorization phenomenon occurs when a representation of a Lie algebra 5 is
restricted to a subalgebra g, and the result factors into a tensor product of g-
representations:

vesaewsm

We will consider general embeddings of symmetrizable Kac-Moody algebras g C E.

This phenomenon has been widely studied. when E is an affine Lie algebra and
9 its underlying finite-dimensional subalgebra; see [FL1],[FL2],[KMN],[HK],[Ka} and
"[088]. In this case, Fourier and Littelman [FL1] have shown that every irreducible
fi-representation factors into a tensor product of infinitely many g-representations.
Their proof by character computations is essentially combinatorial. Our work aims
toward an algebraic framework in which factorization appears functorially and in a
more general context, treating finite and infinite dimensional Lie algebras simultane-
ously.

We define a large class of representations which exhibit tensor factorization. First
we give some motivation for this class in terms of the charaters of its representations.
For now, we consider embedding of one semi-simple finite dimensional Lie algebra
into another, but we will see later that the arguments also work for embeddings of
synunetrizable Kac-Moody algebras with some additional structure. We fix some

notation:

0 g C Q, an embedding of semi-simple finite dimensional Lie algebras.

p 2 half sum of all positive roots of E.

W = the Weyl group of g.

A,, = Ewew sign(w)e““’(“), the skew symmetrizer of e” with respect to W.

o R+ = the set of all positive roots of g.

o l/’()\) :irreducible representation with highest weight A.

V13 2 restriction of a E—representation V to 9.

Also, for an object a. associated to 9, we use ft to denote the corresponding object for
E. For example, [i denotes the half sum of all positive roots of E and V03) denotes
the irreducible representation of E with highest weight ,5.

Consider the character of V(/5). By \Veyl denominatoridentity:

AfizepH(1-e"a)

. deft

and W'cyl character formula:

A ~
QIharV(/\) = —*+—P.
Afi

we obtain:

Cheri/(p) = 8/3 H (1 + 8—0).
01613

This multiplicative form of the charater of V([)) suggests that the E-representation

V05) when restricted to 9, might factor into tensor product of g-representations.
Now, without loss of generality we may assume that Cartan subalgebra of g is

contained in Cartan subalgebra of E and positive roots of E restrict to positive roots

of 9. Then the restriction of the E—character Chart/([3) to 9 will in fact factor as

follows:

Cbar(V(/3)lg) = (Hp H (1+(3-al)> 6(fi1_l7) n (1+e-al)

aER agave
= Chat V(p) C(fil-p) H (1+ (3’01)
aER\R

where 1 denotes restriction from E to g and R is any subset of R. which on restriction to
9 forms the set of all positive roots of 9. Now, if we can find a g-representation whose
character is the second factor above, we can conclude that for any embedding g C E
of semi-simple Lie algebras, the irreducible E-representation V(fi), when restricted to
9, always factors into a tensor product of at least two g-representations, one of them
being V(p). '

The g-representation whose character is the second factor above is obtained using
Panyusliev‘s [P] reduced Spin functor Spino. We will define Spino in Chapter 2 (and
briefly in Section 1.3). Basically, Spin for a given Lie-algebra g is a functor from
the category of all g—representations which have a non-degenerate symmetric bilinear
form preserved by the action of 9 (called orthogonal g-representations) to the category
of all g-representations. By reducing multiplicities in the resulting representation we

obtain SpinO which has the remarkable property that:
SpinO(V1 {19 V2) E SpinO(V1) <53) SpinO(V2).

It is a well known fact that for any semi-simple finite dimensional Lie algebra g,
the representation V(p) can be realized as Spino of adjoint representation of g (which

is indeed orthogonal due to the invariant Killing form). Thus, for the Lie algebra E,

V(fi) = Spilloffil-

When we restrict to 9, it turns out that SpinO commutes with the restriction according

to :
Spino(ii)l ’5 2’" Spinofg'l),

3

where r is the number of positive roots of E which restrict to zero. Now, E E g 63 EJ-
as g-representation, where J. denotes the orthogonal complement with respect to the
Killing form. Therefore,

Wang a 2" Spin0(g e gi)

9— 2TlSpino(9) ® Spino(9i)l,

by the property of Spino mentioned above. So,

' Wang a Sol-10(9) e [2" Spinoeiil

‘ ’5 V(p) <3 {T Spino(9i)l,

as V (p) is isomorphic to Spin~0(g). Hence, we get the tensor factorization of the
restricted V(p). This is the content of Theorem 1 in Section 1.2 where it is extended
to embedding of symmetrizable Kac-Moody algebras with some additional structure.
The detailed proof is given later.

From this, using \Neyl character formula, we can obtain a tensor factorization of
the E-representation V(2fl -l- ,5) for any dominant weight [1, which forms the content
of Theorem 2 in Sectionl.2. In Section 1.3, we state some important properties of
the Spin functor. We describe some consequences of the above theorems for finite
dimensional semi-simple Lie algebras and untwisted affine Lie algebras in Section 1.4
and Section 1.5 respectively. In Section 1.6, for a subclass of orthogonal represen-
tations of untwisted affine Lie algebras, called affinized representations, we classify
those whose SpinO is irreducible. Following Panyushev [P] these are called coprimary

representations.

CHAPTER 1

Main results

1.1 Background for symmetrizable Kac-Moody al-
gebras *
An n X n matrix, A = (aij), is called a. generalized Carton matrix if:
1. (ti, 2 2 for all i : 1,2. - -- ,n.
2. aij is a non-positive integer for all i 75 j.
3. aij = 0 implies a}, = 0 for all i 76 3'.

For any n. x n matrix A = (aij) of rank 1, we define a realization of A as a triple
(h,II,IIV), where l) is a complex vector space, II = {011,02, - -- ,an} C [3* and IIV =
{(Y¥,(1.\2/, - -- ,er } C l) are indexed subsets in l)* and b respectively, satisfying the

following three conditions:
1. Both sets II and UV are linearly independent.
2. aj(O;/) = aij for all t,j = 1,2,-~- ,n.

3. dim(f)) = 2n — Z.

Two realizations (i), ll, UV) and (in, H1, HY) are called isomorphic if there exists a
vector space isomophism d) : l) ——> [)1 such that ¢(IIV) = Hi] and ¢*(II1) = II.
There exists a unique (up to isomorphism) realization of every n x n matrix. The
realizations of two matrices A and B are isomorphic if B can be obtained from A by
a permutution of the indexing set [Ka, Proposition 1.1].

An n x n matrix A is called symmetrizable if there exists an invertible diagonal
matrix D = diag(€1,€2, - -- ,cn) and a symmetric matrix B such that A = DB.
Definition (Symmetrizable Kac-Moody algebra) Let A = (aij) be a sym-
metrizable generalized Cartan matrix and let (l),II,IIV) be a realization of A. A
symmetrizable Kac—Moody algebra, 9, associated to A is defined as the Lie algebra

‘on generators Xi, (z. = 1, - - - .37).). all I] E t) with the following defining relations:
1. [111,112] : 0 for all HI, I12 6 l).

2. [H. Xiil = :L-rr,(h’)Xi,j for all i = 1, - ,n and‘H E I“).

3. [X,;,X_j] =5,,a;’ for all i,j= 1,--- ,n

4. attl(Xi,-)1—at1(XiJ-) = 0 for all 2',j = 1. - ~ .n.

Here, ad(X)() :2 [X , - ] and I) is called the Cartan subalgebra of 9.

Let g be symmetrizable Kac-lVIoody algebra. We define a non-degenerate synnnetric
bilinear form ( - , - ) on I) which can be extended (See [Ka, Thm 22]) to a non-
degenerate symmetric bilinear form on whole of 9 such that ( - , - ) is preserved by

the adjoint action of E, that is:
(PU/1,2) + (Y, [X,Zl) = 0

for all X, Y, Z E 9. Let A be a symmetrizable generalized Cartan matrix with a fixed
decomposition A 2 DB (See the definition of a symmetrizable matrix above). Let

I II I
l) :2 69?”:1 (Cay. Fix a complementary space I) to l) in I) and define:

(O‘.\-/, H) 1‘ a.,j(H)c,; V H E I);

//
(H17H2)=0 V Hi,H2Eb.

1.2 Factorization Theorems

We new state our main factorization results using Spin construction. We consider
embeddings, g C E, of synnnetrizable Kac-Moody algebras. Our analysis deals with
finite as well as infinite dimensional representations. For example, we consider in-
finite dimensional irreducible representations of an affine Lie algebra E with finite
dimensional weight spaces. If we restrictsuch a representation to a finite dimensional
Lie algebra g, the g-weight spaces noilonger remain finite dimensional. To avoid
this, we define below the notion of an augmented symmetrizable Kac-Moody algebra,
a cit-embedding of such algebras and d-finz'te representations so that any d-finite E—

representation V, when restricted to g, has finite dimensional weight spaces.

Definition (Augmented symmetrizable Kac-Moody algebra g) A Lie alge-
bra g is called an augmented symmetrizable Kae-Moody algebra if g has a certain

distinguished element (1 such that either:

0 g itself is a symmetrizable Kac—Moody algebra, d E l) and a,(d) E Z>0 for all

0,: E H,
0 or g = 91 $ (Cd where:

- 91 is a synnnetrizable Kac-Moody algebra.
- d commutes with the Cartan subalgebra [)1 of 91-

— (1 acts diagonally on the root vectors Xia of 91, that is : [(1, Xia] =
icaXia, for some constant on E Z>0, so that we can define a(d) := ca
which extends the action of roots (1 to U11 (1‘) (Cd). We call I) := In (1) (Cd

the Cartan subalgebra of g.

Example: Let 91 ”E 512C :2 (CHa $CXO$CX_Q with usual bracket relations. Define
[(1, H0] 2: 0 and [(1, Xin] := :tXia , so that a(d) := 1. Then, 9 := 91 63 (Cd is an

augmented symmetrizable Lie algebra.

Definition (rt—embedding) An embedding g C E of augmented symmetrizable Kac-
Moody algebras with distinguished element (1 and cf and Cartan subalgebras l) and E

respectively, is called a (l—embedding if :

o l’) C i) and

0 positive roots of g are restrictions of positive roots of E

Let g be an augmented symmetrizable Kac-Moody algebra with distinguished el-
ement d in the Cartan subalgebra t) and weiglit'lattice P. For a g-representation V
and A E P, let V(A) :2 {v E V : H(t') 2: A(H)v V H E b} denote the corresponding
weight space of V. A is called a weight of V if V(A) 75 {0}.

For the distinguished element d in the Cartan subalgebra of 9, we say a g-

representation V is d-fintte if :
o A(d) E Z—{O} for all non-zero weights A of V and
o $A(d)=k V(A) is finite-dimensional for each h E Z .

As we mentioned in the Introduction, the input for the Spin functor is an orthogonal
representation which we defined for semi-simple finite dimensional Lie algebras. The
same definition extends to the augmented symmetrizable Kac-Moody algebra too.
For an augmented symmetrizable Kac-Moody algebra g, a g-representation V is called
orthogonal if there exists a non-degenerate symmetric bilinear form Q on V, invariant

under the action of E, that is, Q(Xu,v) + Q(u,Xv) = 0 for all 11,1) 6 V and

X E E. For example, the action of a symmetrizable Kac-Moody algebra on iteself
by brackets, called adjoint representation, is orthogonal due to the invariant bilinear
form (see Section 1.1).

Next we define the adjoint representation of an augmented symmetrizable Kac-
Moody algebra in such a way that it is orthogonal, so that we can apply the Spin
functor to it (see the Introduction). Adjoint representation for an augmented sym-
metrizable Kac-Moody algebra g with distinguished element (1 is already defined if g
is itself a symmetrizable Kac-Moody algebra. So, let 9 = 91 3; (Cd. In this case, the
action of g on 91 by brackets is defined as the adjoint representation of 9. We can
show that the action of the distinguished element (1 preserves the bilinear form on g]
and thus 91 is orthogonal as a g—representation. i 3
Note. It is easy to show that for a (ll-embedding, g C E, the adjoint representation of
E is d-finite and orthogonal both as a E-representation and a g—representation.

Theorems 1 and 2 (given below) deSeribe a class of representations which exhibit
the factorization phenomenon (see the Introduction). We will follow the notations
used in the Introduction except that g C E will denote a d—embedding of two aug-
mented symmetrizable Kac-Moody algebras and p and p~ will denote the sum of all

fundamental weights of g and E respectively.

Theorem 1. For a d-embedding, g C E, of augmented symmetrizable Kac-Moody
algebras, suppose the adjoint representation of E decomposes into orthogonal g-
representations as: E g g @ p1 EB pg 613 . Then the E-representation V(p), when
restricted from E to 9, factors into a tensor product of g-representations as:

~

V(fiflfi g V(p) ® W1 3 11/2 ,8, . ..

with Wj = Spin0(pj) , where SpinO is the reduced Spin functor defined in Chapter 2.

In the finite-dimensional case, the Theorem is closely related to the results of Kostant

[K1, K2].

Theorem 1 leads to a large class of representations exhibiting tensor factorization.

Theorem 2. Let g C E be as in Theorem I, and let [i be a dominant weight ofE. Then
the E representation V(2fi + p), when restricted to 9, factors into a tensor product of
g-representations which include the same Wj as in Theorem 1. The other factor can
be expressed in terms of the irreducible decomposition of the restricted V(p).
That is. if we let:

mtg a 69 V(m).

. ,- ,
then: i A

i

“e. will prove these theorems in Chapter 3.

1.3 Basic properties of the Spin functor

We now describe the basic properties of Spin functor, reserving the more technical
discussion for Chapter 2. In the finite-dimensional case. the construction is quite
simple, and was examined by Panyushev [P]. For an n-dimensional vector space V
with a non—degenerate symmetric bilinear form, recall that the orthogonal Lie algebra
50( V) has a representation on the 2Ln/ 2]-dimensional space Spin(V) z: A'V+, the
total wedge space or exterior algebra of a maximal isotrOpic subspace V+ C V.

Let g be a semi-simple, finite dimensional Lie algebra. For an orthogonal g-
representation V, 9 acts by orthogonal matrices: that is, through an embedding
g C 50(V). Restricting the action of 50(V) makes Spin(V) a representation of g. If
the zero weight space of V has dimension r, it turns out that Spin(V) can be decom-
posed as the direct sum of 2l"/2J copies of a smaller representation, which we call
Spin0(V).

Now let 9 be an augmented symmetrizable Kac-Moody algebra. In Chapter 2,

we will define the g-representation Spin(V) = /'\'V+ for V in the category of all d-

10

finite and orthogonal (possibly infinite-dimensional) g-representations. We will prove
that the output, Spin(V), will be a d-finite g-representation in the category Oweak
(defined below). Category Oweak contains the Bernstein-Gelfand-Gelfand category C’)
and has similar properties. Further if the input representation V is root finite (defined
below) then we will prove, Spin(V) belongs to 0. If the zero weight space of V is
even, Spin(V) decomposes into direct sum of g-representation which we call half-Spin
representations /\e"e"VJr and AOddV+. We also denote these by Spineven(V) and
SpinOdd(V).

We define a partial ordering 3 called root order, on- the the weight lattice 'P of g
as follows: we say, 8 3 gr in the root order if 7 — t3 =- 20 can where a is a simple
positive root of g and (Ta 6 Z20 for all a.

For an augmented symmetrizable Kac—Moody algebra g and a g-representation V
define:

My :2 Set of all weights of V maximal in the root order.

Definition (Category Oweak of g-representations) Oweak consists of all g-

representations V such that for each weight B of V:
1. the weight space V(fi) is finite dimensional and
2. there exists A E M V such that B S A .

Remark: The morphisms in Oweak are g—representations homomorphisrns. Following
fact can be deduced, using [Ka, Thm 10.7], that for a representation V in Oweak which
is integrable (meaning simple root vectors act locally nilpotently), the isomorphism
class of V is determined by its character.

The well-known Bernstein- Gelfand- Gelfand category C’) of g-representations can be

defined as a subcategory of Oweak:

(9 := {V E Oweak : )1er is a finite set} .

11

It is worth noting that O can be defined to consist of g-representations V such that V
has finite-dimensional weight spaces and there exists a finite set F, a subset of weight
lattice P of 9, so that for each weight [3 of V we can find A 6 F with I} S A.

Let {cu-K; denote the simple positive roots of an augmented symmetrizable Kac-
Moody algebra g with distinguished element d. Let l)* be the dual Cartan subalgebra
of 9. Define the root cone C :2 {21:101'02' E l)* : a,- 6 R20 V i or a,- 6 R50 \7’ i}.
Definition (Root finite g-representation) We say a g-representation V is root-
finite if for every weight A of V, V(A) is finite dimensional, A(d) E Z — {0} for A at 0
and there are only finitely many weights of V“ in f)*— C. A

Propositions 1 and 2 give some basic properties of Spin(V).

Proposition 1. Let g be an augmented symmetrizable Koo—Moody algebra with dis-

tinguished element d. Let V, V1 and V2 be d-finite and orthogonal g-representations.
I. Spin(V) is d-finite and belongs to Oweak-

'2. V is integrable => Spin(V) is integrable and Spin(V) § Way" for some 9—
representation IV called SpinO(V). Here r 2 [mo / 2] where m0 is the dimension

of the zero weight, V(O) of V.

3. Let m,- := dim(Vi(0)), be the dimension of the zero weight space ofVi, fori = 1,2.

If at least one of ml or m2 is even, then:
Spin(Vl 8 V2) § Spin(Vl) 59 Spir1(V2).
If both ml and m2 are odd, then:
Spineven(V1 EB V2) 2 Spin(V1) 8' Spin(Vg) ’E SpinOdd(V1 69 V2).
and

Spin(Vl 33 V2) 2’ ( Spin(Vl) 8 Spin(V2) )“B2.

12

4. If V1 and V2 are integrable then
Spin0(V1 EB V2) § SpinO(V1) X SpinO(V2).
5. W is root-finite => IV is d-finite.
6. V is root-finite 42> Spin(V) E 0
7. For adjoint representation 9,

spilling) E’ V(p)-

Let

IO 7: Category of all d-finite and orthogonal g-representations ,
IR : Category of all root-finite and orthogonal g-representations.
Then, by Proposition 1(5),
IR C 10
and by Proposition 1(1) and 1(6) Spin(V) is a functor from the category To to the

category Oweak and also from category 1 R to category 0. Thus,

Spin
IO —_’ Oweak
U U
13 53-5“ o

The following Proposition gives the character of SpinO(V) in terms of the character

of V.

Proposition 2. Let V be an integrable g-representation in To . Let my be the

multiplicity of a weight B of V so that the character of V can be written as:

ChatV = Z m,3(e6+e_t3) + m0.
flld)>0 '

Then the g-representation SpinO(V) has the character:

CbatSpin0(V) =eA H (1+e—t3)m5.
ri(d)>0

13

Here A :2 23:1 0,-1\,j, where {1\,j}?:1 are the fundamental weights and the coefficient

0, is defined as follows:
1
(3i :: Z Enigffirry),

where the sum is over all weights .13 of V such that f3(d) > 0 and s,(,13)(d) < 0. Here
5,- denotes the reflection in the plane perpendicular to the simple root 0,. Because of

the (l-finitcncss of V, C, has finitely many nonzero terms .

Remark : When V is finite dimensional, the A simplifies to A :2 23((l)>0 $71135.

1.4 Special cases for Finite-dimensional Lie alge-
bras

We give some special cases of Theorems 1 and 2 when E C E is an arbitrary embedding
of finite-dimensional semi-simple Lie algebras. This embedding can be turned into
a (l—embedding of augmented synnnetrizable Kac-Moody algebras by appropriately

choosing a. d in Cartan subalgebra of 9.

1.4.1 Principal Specialization

We let 9 C E be the embedding of a principal three-dimensional subalgebra in the

special linear lie algebra: 512(C) C sln(C), defined as 5I2((.—') z: CA’ (l): (CY (l) Cl],

where
n—lt
X 3: 2252's“,
,,___
n—l _
Y 3= Zl('n—2lEi+l,i~,
1,:

n
H I: Z (n +1— 2013mm
i=1

Here, E'iJ denotes the n x n matrix which has 1 at (i, j)t‘h place and zero elsewhere.

14

The character of a sin-irreducible V(u) is the Schur polynomial S;,(:r:1, . . . ,xn) . Its
restriction from E to 9 corresponds to the principal specialization mils = qi—l, where
q = e” for a the simple root of 9. Theorem 2 implies the following factorization of

the specialized Schur function:

Proposition 3.
S2lt+l)(1~ (17(127 ' ° . 3 (171—1)

= (q(§l(1+q)s,,(1.q2,q4, - . - ,q2n‘2i) 'w1(q) - w2((1)-~wn—2(q),

where urk(q) = (1+q)(1+q2)~-(1+qk+1) , p = (n--1, . . . , 1,0) and all n — 1 factors

on the right-hand side of the formula are symmetric unimodal q-polynomials.

Definition (Symmetric unimodal polynomial) A polynomial, f (q)
Ziggy air/f, is symmetric unimodal if “NH 2 aM_',- for all i, and aN S S a K 2
0K+1 2 -- - _>_ 0.,” for some K.

Proposition 3 is a kind of multiplicative analog of a result of Reiner and Stanton

which states that for certain pairs A,,u., the centered difference S A(1, . . .,q""1)

q‘N .S'#(1, . . . .q""1) is symmetric unimodal.

1.4.2 Folding of Dynkin diagrams

Let E be a simple Lie algebra with Dynkin diagram D. A graph automorphism (b of
D, induces an automorphism, call it (1) again, on E. We let 9 be the fixed subalgebra,
under this automorphism, cf). Then the Dynkin diagram, D of g is called the folding

of D. For such an embedding, g C E, Theorem 1 implies:

Proposition 4
V(fi)l3 % V(p) ® [V(6(p+ps)+ps)® (a-2)V(0)l®“‘l,

where p3 is the half-sum of the positive short roots of g, a is the order of the auto-

. . i j . f“ .
morphism (b, and e is the number of edges o——o in D such that d erchanges i and

For example, the natural embedding 5021,,“ C Sign.” corresponds to horizontally
folding the diagram Agn to obtain Bn :

1 2 71—1 n n+1 n+2 2n—1 2n.
442” : .-———.'-- .——-.___—. . ... ._.

 

1 2 n—l n
Bn: o—o---o=>o

The automorphism is q')(i) :- 2n—i+1 of order a = 2 with a single folded edge e = 1 ,

~

so was a V(p) a vamp.) _

1.5 Factorization Theorems for affine Lie algebras

The most remarkable aspect of our construction appears when E = E is the untwisted

affine Lie algebra associated to a finite-dimensional semi-simple algebra g:
E = g®C[t,t—1] 9: (CK e) (Cd.

Here K is the central element and d the canonical derivation. We also let A0 be the
distinguished fundamental weight, and (5 the minimal in'iaginary root. Also, if a is an
object associated to 9, then 5: denotes the corresponding object for E.

Let d := pv + hd where h :2 2?:0 a,- is the Coxeter number. Here ai’s are the
numeric labels of the Dynkin diagram of E in [Ka, Page 54]) and {1V is the sum of all
fundamental co—weights of E, that is o(pV) = 1 for all simple positive roots at of 9.
Then g 6} (Cd C E is a d—embedding of augmented symmetrizable Kac-Moody algebra.

Let V 6 IR be an integrable E-representation of level zero, that is the center K
acts by zero. Then, even though the input E—representation V E I R has level zero, by
Proposition 2 the output Spin(V) is a representation of positive level in the Bernstein-
Gelfand-Gelfand category 0. That is, Spin is a functor from the category of graded
level zero representations in I R , a sub—category of the graded level-zero represen-

tations I examined by Chari and Greenstein [CG], to the positive-level category

(9.

16

Now we introduce a subcategory I A of I R . We will work out Theorems 1 and 2

for this Sim-category.

1.5.1 Affinized representations

For the remainder of this setion we will work with E—representations in a more re-
stricted class IA C I 1?. . the subcategory Of affinizations of finite—dimensional orthog-
onal g-representations. That is, for an orthogonal g-representation V, its affinization

is the E-representation

. V :2 ® th

keZ
where the loop algebra. acts as th- lk‘t.‘ :2 tk+l(X - r) for X E g, ’L' E V; the center
acts as 0; and the derivation (1 acts as till . This inherits a non-degenerate symmetric
bilinear form from V. Clmose a strictly dominant co-weight d1 in the Cartan sub—
algebra !) of 9 such that 13((11) E Z — {0} for all weights ,3 of V. Since V is finite
dimensional, for sufficiently large N, —N < [3((11), 9(d1) < N for all weights {3 of V
and highest root 6 of 9. Define d :2 Nd+d1. It can be verified that the weights of the
V are of the form A :: A15 + x3 for k. E Z and [3 a weight of V. Thus for all non-zero
weights A of V, A(d) E Z — {0}. Now, 9 (p (Cd C E is a d—embedding of augmented
symmetrizable Kac-lVIoody algebras. Also, it is easy to check that V E I R and is an
integrable E-representation.

For the representations V 6 IA, we refine Proposition 2 below to obtain the char-

A

acter of SpinO(V) in terms of the character of V.

Proposition 5. Let V be a finite dimensional orthogonal g-representation. Let T
be the set of all weights of V and mg the multiplicity of a weight ,8 E T so that the

character of V can be written as:

ChatV = Z 771,3(e-S+e_5) + mo.
,3(d1)>0

17

Then SpinO of the affinized E-representation V has the character:

€barSpi110(V) = MACAO H (1+e‘3)m/3 H(1+¢’5‘k5)mfi,
/3(d1)>0 k>0
BET
where l/ 2 iZ:3(d1)>0mBfi and c = 7323013)“) mfiB(0V)2, called the level of

Spin0(V). Here 6 is the highest root of 9.

1.5.2 Factorization Theorems for affine Lie algebras

If we restrict an affinized E-representation SpinO( V) to 9 EB (Cd and apply Proposition

5, we obtain:

Proposition 6. Spi110(V), when restricted from E to g Cd, factors into an infinite

tensor produri.’

Sl)i”()lvllg_,cd E SpinO(V) ‘3‘.) A'UV) (>3 ./\'(t2V) ;,;...7
Q't: '

Note. The E :3 (Cd-representation U k. :2 A’th contains a canonical one-dimensional
representation (C 1 = AOI.kV. The infinite tensor product above is the direct limit of
the maps:

U015?) U1 ('0 ' ' ' CX‘ U}; —-> U0 X U1 X ° - - 1?? “I: GO ”(k—Fl)
1.108%” 8‘-~®Uk t—> (1.0 ’8 ul ® ® ”Mk-31

where UO :: SpinO( V).
We now work out Theorems 1 and 2 for Affine Lie algebras.
Proposition 7. The E-representation V(p), when restricted to g {B Cd , factors into

an infinite tensor product:

A

r A g r: O N O 2 . ...
V(pHWm—VW 2 A (t9) m (t g) a .

Proposition 8. If we let:

H2

v< EBvoo.

- §
[0199}ng

18

then:

“ ‘ E 2 . ' ° ' 2
V(2;r+p)1g%(i _ ($V(2u,+p)) <29 /\ tg®/\ t g® .
2

1.6 Classification of coprimary representations
Motivated by Proposition 6, we ask: For which representations V is Spi110(V) irre-
ducible?

Definition. A g-reprr—rsentation V is coprimary if SpinO(V) is irreducible.

Panvushev P fives a com )lete list of co )Iimarv re )resentations V of a sim )le Lie
\ b V

algebra g and deduces the classification for a semi simple Lie algebra.

Proposition 9. Let V be an orthogonal representation of a finite dimensional simple
Lie algebra 9. Then V is coprimary i.e. Spinot’l’) is irreducible if and only if V is

itself irreducible and is one of the following :
1. V((I), for all E where SpinO(V) = V(p);
2. V(QS), for g E {502n+1C, spgnC, {4} where SpinO(V) = V(ps);
3, V(205), for g = 502n+1C (n. 2 1) where SpinO(V) = V(2p5 + p);

where 63 = highest short root of 9.
Note. For n = 1, we have 502n+1C E sIQC, and we take 65 z: 6.

The classification of coprimary affinized representations is as follows:
Proposition 10. For a representation V 6 IA of E obtained from a representation

V of a simple Lie algebra g,

V is coprimary 4:) . .
cases 1 or 2 of Proposztion .9

~ ( V is coprimary and belongs to )

19

Panyushev proves the irreducibility of SpinO(V) using the Weyl denominator iden-
tity for the Langlands dual of g, and analogously we can prove the irreducibility of
SpinO(V) in Proposition 10 using the Weyl denominator identity for the Langlands

dual of E a (possibly twisted) affine Lie algebra.

20

CHAPTER 2

General Spin construction for
augmented symmetrizable

Kac-Moody algebras

Next we give the construction of Spino for representations of augmented symmetriz-
able Kac-Moody algebras. This surprisingly delicate matter has been briefly studied
by Kac and Peterson [KP] and Pressley and Segal [PS, Chapter 12]. We will provide
a different and more detailed presentation. Also, we will do this in a more general
setting which is compatible with restriction of representations.

Let V be a vector space with basis {ez- : i E I} where the index set can be finite, I =
{m, . . . , 1,0, —1, . . . , —m} or {m, . . . , 1, —1 ..... —m} ; or infinite. I = Z or Z — {0}.

Define a symmetric bilinear form on V by Q(e,- ,ej) :2 diacj .

2.1 Finite dimensional case

First, let V be finite dimensional (I is finite). The orthogonal Lie algebra 50(V) is

defined to consist. of matrices which are. skew-symmetric with respect to the anti-

21

diagonal i.e.

50(V) :2 {A = (ai,j)i,j61 : am- 2 —a_j,_.i}.
Thus, 50(V) has a basis {Zm- :2 Ei,j — E_j,_,- :i,j E I, i > —j} where EM are the
coordinate matrices. We define the Clifford algebra C(V, Q) as the associative algebra
with 1 generated by all v E V with defining relations e_,-ej = —eje_i + 260' V i,j E
I. There is an embedding of Lie algebras defined by :

op : 50(V) —> C(V,Q)
Z‘i,j *—-) 211(8—i8j - €j'€_l').

Now, the Clifford algebra has an action on a wedge space Spin(V) 2: /\°V+, which

on generators {el- : i E I} of C(V.Q) is as- follows : Define I+ =1 {i E I : i > 0}.

Then A'V+ has a basis {cg :2 ejl /\ej2 /\ . . . AeJ-h} for 0 _<_ l; g [1+] and J :
{jl > jg > > jk} C 1+. Here [1] z: #(A). For i 6 1+ define

e(i,J) e(J_{,-}) if i E J

821(61) 3: 61 ABJ- J 7'é f}; 8—170?th { 0 if [$1 ,

where C(t, J) :2 2(—1)|{j€'lij>i}|. Also, 1(eJ) :2 eJ, e,(e{} = 1) = ez- and 80(8J) :=
(—1)|J|€J if 0 E I. Finally, due to the embedding (9p defined earlier, this action of
(7(V, Q) induces an action of 50(V) on /\°V+ which is called Spin representation of
orthogonal Lie algebra 50(V).

As described in Section 1.3, for an orthogonal g-representation V, g C 50(V) and
9 acts on Spin(V) by restriction. It is easy to find the character of Spin(V)(see [P])
as a g representation. It turns out that if m0 is the dimension of the zero weight
space of V then Spin(V) is isomorphic to the direct sum of 2]"‘0/2J copies of another
g-representation which is defined as Spin0(V). The character of Spin0(V) is given in

Proposition 2.

22

2.2 General case

Now, let V be infinite—dimensional (I is Z or Z — {0}) with only finite linear combina-
tions of {e,: : i E I} allowed. First, we naively extend the above definitions with the
following modifications: The orthogonal Lie algebra, now denoted by 5000(V), consist
of skew-symmetric matrices with respect to anti-diagonal (as before) which have only
finite number of non-zero entries in each column (skew-symmetry implies the same
on the rows too), so that 5000(V) is closed under commutator. Clifford algebra is
allowed to have infinite sums of finite products of (e, : i E I}. The map (pp is still an
embedding of Lie algebras. The infinite wedge Spin(l') :2 A'V+ is now an infinite
dimensional vector space («insisting of finite linear coml'iinations of finite wedges of
{el- : i 6 1+}.

The action of {6, : i E I} on /\'V+, as defined in the finite case, does not ex-
tend to Clifford algebra nor induce an action of 50,x( V). For example, for Y 2:
2215250 ZZZ-J41 E 50960"), op(Y) = ZieZ>0 (3%5— does not act on 1 E A'V+ as
it leads to an infinite sum. Also, for H := Ziel+ ZN, an infinite diagonal matrix in
5090(V), oF(ll)E(..’(V,(2) does not act on 1 E A°V+ as ¢F(Zi’i)(1)=1/2.

In order to resolve these two issues, next we suitably modify 5000(V) and (,7') F and
define a smaller Lie algebra 50(V) and a map 96 so that the image of 50(V) under 4'»
does act on /\'V+.

The Lie algebra 50(V) consists of matrices A = (“i.jliajEI such that:
1. A is skew-symmetric with respect to the anti-diagonal: aid- 2 —-a_j7_,- .
2. Each column (a’lj)iEI has finitely many non-zero entries.
3. The blocks (0-i,—j)i,j>0 and (a_i1j)i,j>0 have finitely many non-zero entries.
Define the map:
95 : S0(V) —-* C(V, Q)

1
Zn] r——+ —2eJ-e_,-.

23

Now referring to the matrices Y and H defined earlier, note that the matrix Y E
5000(V) does not belong to 50(V) and even though H belongs to 50(V), o(H) does
act on 1 E /\'V+. Further, we can verify that image((,2>) C C (V, Q) does act on A’V+.
In exchange, the map <25 is not a Lie algebra map and image((,/)) is not closed under

brackets in C(V. Q). But the central extension of the irnage(¢):
5~0(V) :22 {96(A) : A E 50(V)} 69 Cl C C(V,Q),

is a Lie algebra which also acts on /\°V+ (as (C acts diagonally). The Lie algebra

5700/) is a central extension of 50(V) also by one dimensional center (Cl due to the

following exact sequence of Lie algebra maps :

O——> (C —> 5~o(V) L 50W) ———> 0
1r——> 1 i—> O

€j€_i *—+ —2Zi7j.

This can be verified using the following cormnutator relations in 5~0(V) and 50(V).
[6:16-13 (ESQ—7‘] Z 26l.3€j6—T — 26i~_r€j€3 + 26j.‘3€—re‘-’l — 26].,7‘838-4: .

[ZZij,QZr3] = —46i,sZr,j + 46.i‘_,.Z 4d

J,—SZi,-T + 46j,rZi,s-

_ s, j

“78 can prove that this extension does not split when V is infinite dimensional.
Thus 573(V)-representation /\"V+ which we call Spin representation does not induce
an action of 50(V). Therefore, the orthogonal Lie algebra 50(V), when V is infinite
dimensional, does not have a Spin representation, but its central extension §6(V)
does.

The above construction can also be carried out when V is finite dimensional.
There the extension splits as (image(qbp) EB C1) because (pp is an embedding and
it o q‘) F = 150(V)’ Thus, 5~0(V)-representation /\'V+ does induce an action of 50(V)
and the resulting representation coincides with the Spin representation of 50(V) de-
fined earlier. Thus, the above construction is the general construction of Spin(V) for

a finite or infinite dimensional vector space V.

24

Now let 9 be an augmented symmetrizable Kac-Moody algebra with distinguished
element (1 and V a d—finite orthogonal g-representation (as in Section 1.2). Orthog-
onality and (fl-finiteness of V leads to a map E —» 50(V). Once we have the map
E —> 50(V), using it : 570(V) —> 50(V) defined earlier, we get an induced map
E —> 5~0(V) for any augmented symmetrizable Kac-Moody algebra due to the fol-

lowing lemma.

Lemma 1. Let g be an augmented symmetrizable Koo-Moody algebra with Carton
- u r
subalgebra f). Fix a complementary subspace f) to l) := [1:1 (3an in f), where each

of is a simple co-root, so that:

I, I, .
92f) :9!) $912.

’ II
where 91? is the space spanned by all roots. F i;r. ll! 6 (f) (b ER)* such that 153(9)?) =

0. Then for any Lie algebra map a : g——>50(V) there exists a unique lifting [r :

g—>£;V0(V) such that the following diagram commutes:

p

9 —”—> 50(1 )
&\ I it
5700/)

and d : (.5 o o + if) on f)” ”p 9!?- Recall that 5~0(V) was defined as ¢(50(V)) 33 CI.
Note. The conclusion of the above lemma is also true if the quadruple
(so(V),sb(V),¢,rr) is replaced by (5,35,?) satisfying following conditions : s is
any Lie algebra and E is a central extension of s by one dimensional centre C1 due
to a surjective Lie algebra map f : 3 —+ 5 with kernel C1. The map a : s —+ E is

a vector space map which maps Cartan subalgebra of 5 onto Cartan subalgebra of 3,
satisfying 3 = 5(5) 63 (Cl and it“ o 5 : Ids.
Now since s~o(V) acts 011 Spin(V), that is:
g —"-* 50(V)
6 \, T rt
570(V) —+ Endc Spin(V),

25

for a given orthogonal d—fiuite g-representation V, we may define the g-representation
Spin(V) . In Section 1.3, we defined Spin0(V) and some basic properties of Spin(V)

and Spino(V) are listed.

26

CHAPTER 3

Proofs

3.1 Proof of Lemma 1

Let Xii- i = 1 - - - n be the simple root vectors of g and {d} , (lg. - -- ,d]} be a basis of
t)”. Then, Xiis and d j’s generate g as a Lie algebra.

Note that by the comnuitativity of the diagram and due to the map if), the map & is
uniquely defined on the generators of 9. Now, to be able to extend this map 6 to whole
of 9, we need that 6(Xii) and 5(dj) in §J(V) satisfy the defining bracket relations
of 9. But since 0(th') and (7(dj) in 50(V) satisfy the defining bracket relations of g
(as o is a Lie algebra map), and it : 50(V) ——> 50(V) is a Lie algebra map, mapping
6(Xii), Er(d_,-) to 0(Xii), 0(dj) in 50(V), we can prove that 6(Xii), &(dj) also
satisfy each defining bracket relation of 9 up to a constant because ker(rr) = (C. We
show that this constant is zero for each relation.

Let f) and fi be the cartan subalgebras of 50(V) and 570(V) respectively so that
I) = 1r"1(f)). Then the constant term in 6(Xii), when expressed in standard basis
of 5~0(V), is zero and 6(dj) E E as 0(dj) E f). Set 6(a)“ :2 [6(Xi),&(X_,-)] for
i = 1, - - - , n. Then clearly, (3(02’) E E for all i. This defines 5' from f) intofi (may not

be injective). For any H E b we can easily conclude that [&(I-I), Q'(Xi.i]—(1’i(H)&(Xii)

27

is a constant. This constant must be zero because constant term in 6(Xii) is zero and
6(H) E 6. Now for i 5A j, X := [6(X,),&(X_j)] must be constant and for all H E I),
&(H) acts diagonally on X with eigenvalue (or, - (rj)(H). This implies that X = 0.

Similarly, the generators satisfy the last bracket relation also (see Section 1.1).

3.2 Proof of Propositions 1-2

First we obtain the character formula for Spin(V) as {0(V)-representation. As in
Chapter 2, let V be the vector space with basis {6,- : i E I } We can check that
E :: $1.6 1+ (Ce,e_,- G3 (C1 is a Cartan subalgebra of 5730".) Consider the dual basis

of {1, 12:23: :j E 1+} i.e. L,- E 6* defined as:

Zo<1>=1 Zo(93;l)=-0 261+
Zl(l)=0 Ede €__ )2: 5” i.j€1+.

-)' ".- .. o +" . + .. -—. - -
A basis of weight VBCtOIb of /\ V is {eg . J C I }, where eJ .— €11 AeJ2/\ AeJk’
for .I : {j1,j2, - -- :IA-l- Observe that :
1(&1) == €J

e-e_- __ 6"] if jEJ
L748") _ [ 0 if jeJ.

So, (’J has the weight L0 + Z Lj. Therefore,

jEJ
(L0+_Z Lj)
ChatSpin(V)=€bat(/\’V+) = Z c 36" ,
Jg1+ ~
2 eLO n (1+eLi).
iE I+

7

as 57)(V)-representation. Due to the map, E L 50(1 ), (described in Chapter 2
Lemma 1), Spin(V) becomes a g-representation. Then, L, 06, i E (I+ U{O}), are the
weights of Spin(V) as a g-representation. Due to the commutativity of the diagram

in lemma 1 we can show that L- o c} = —L.- o o i E I+ where L’s are the wei hts
2 l r a 2.

28

of the defining representation V of 50(V). Let ,8,- := L,- o o, i E 1+ and A := L0 0 0.

Thus, as a g-representation,

Chat Spin(V) = eA H (1+ e‘fii) .
iE 1+

Without loss of generality, we may assume fi,(d) 2 O for all i E 1+. When V is finite

dimensional (see [P]), A : 22i61+ /)’,-.
Proof of Proposition 1.

Now we. prove parts (1) - (7) of Proposition 1.
Proof of {1). '
We show the following:
V is d—finite
=> Spin(V) is d—finite and the set {7((1) : 'y is a weight of Spin(V)} is bounded above.
:> Spin(V) E Oweak- . l I

Let V be d-finite. Result is obvious if V is finite dimensional. So, let V be a
infinite (llll’lEIlSl(.)I'lal so that I = Z, 1+ = Z>0 and the set of positive weights are
{13, : i E Z>0}. Any weight of Spin(V) is of the form : A — a where a = 2,16%“) aid,-
for a, = 0 or 1 for i E Z>0 where a,- = O for all but finitely many i’s. Let’s call such
a sequence, (a,),;eZ>0, an (a)-sequence.

For d—finiteness of Spin(V), it’s enough to show that the above character when
restricted to (Cd has finite coefficients. This is equivalent to: For each N, there are

finitely many (a)-sequences such that (A — a)(d) = N. Define

Lk 32 f2 E Z>0 3 flifd) = k},

29

Lk is a finite set due to d-finiteness of V.

(A-a)(d) = N

=> A(d)—N = Z a, k.

kEZZO ZELk
=> A(d)—N = Z bkk, . ‘ (3.1)

where bk 2: Z ,6 L A: (1.1. In the above equatimi l: E Z30 because [3,] d) 2 0 for all
i and the sum is a finite sum as (1,: 7t 0 only for finitely many i’s. Clearly, bk.
is finite. Thus each (a)-sequence (a-ilieZ>0 satisfying equation (3.1) gives a non-
negative integral partition of A(d) — N where each nonsnegative integer is repeated
bk times. Conversely, for every such partition given by (bklkezzo with all but finite
number of bk to be zero, there exists only finitey many (a)-sequences, (ailiez>0 such
that bk 2 2,61% a.,-_. Since there are finitely many such partitions of A(d) — N, there
are finitely many (a)-sequences satisfying equation (3.1). Thus Spin(V) is d—finite.

Finally, {7(d) : 7 is a weight of Spin(V)} is bounded above by A(d) as 'y = A — a
for a = 2,762») aw,- and ,Hz-(d) Z 0. That proves the first implication.

To prove Spin(V) E Oweak, we show that if W is a d—finite representation, such that
{7(d) : '7' is a weight of W} is bounded above then W E Oweak- First of all weight
spaces of W are finite dimensional by d—finiteness of W. So, let ’P(W) denote the set
of all weights of W. For fl E ’P(W), let 7 E St? :: {'y E ’P(W) :B _<_ 7}. That is,
'7 — ,6 = 2 can, for some ca E Z20 where the sum is over simple positive roots of g.
This implies 7(d) —,13(d) 2 0 as a(d) > 0 by definition of d. As 7(d) is bounded above,
K g 7((1) g N where K = [3((1). So 3;; is finite, otherwise ®K37(d)SN IV“) will

be infinite dimensional contradicting the d—finiteness of IV (see Section 1.2). Now the

30

set of all maximal weights of the non-empty finite set 83 is non-empty and intersects

M W non-trivially. Thus IV E Owe-ak-

Proof of (2).

The result is obvious if V is finite dimensional. Assume that V is infinite dimensional.
Let X be a positive or negative root vector of 9. We will Show that X acts locally
nilpotently on Spin(V) if it acts locally nilpotently on 50(V) (see Chapter 2).

First assume X to be a positive root vector. We denote the matrix of the action
of X on the representation V given by the map E ——> 50(V) by X only. Recall, that
50(V) is defined with respect to a [polarization of V = ’V+ {B V". Fix an ordered
basis {- -- > e2 > el > e_1 > 8-2 > } of V, where Vi = EBieZi Cei. We may
assume that X is a strictly upper triangular matrix in 50(V) with respect to above
basis. We can write X =2 Z + F such that Z(Vi) C Vi, F(Vi) C VZF. Then
by definition of 50(V), F(V) is finite dimensional. Since X is a upper triangular,
F(V+) = {0}. Thus, let F := 21.3, br,in.—t, a finite sum, where {ZN} forms a basis
of 50(V) defined in Chapter 2. Referring to Lemma 1 in Section 2.2, the image of X,
say X, in 50(V) can be written as: X —_— Z + F, where Z = ¢(Z) and F = e5(F) in
573(V) (see Chapter 2). Here P = —1/2Z,.J br,te_te_,~. Now action of Z and F on
Spin(V) :2 /\'V+ is defined as:

Z(e,-,1/\e,:2 /\---/\e,-k) :=Ze,-1/\e,-2 /\~-/\e,-k + (2,1 AZei2A---Ae,k +
where 2 denotes the transpose of Z. Thus,

~ ’! A A. A
Zp(€i A6; /\°--/\€z")= E -—I)-+-Zple.i AZp2e, /\°-'/\Zpk€i .

2111(1

~

F(€i1A"'AeT/\"'AetA"'/\eiklZZZCT‘JeilA"'/\éT/\"'Aét/\"'Aeik-
rt

where e,- denotes the absence of e, and em = i2br,t depending on positions of er and

31

et . Also, Cnt = 0 if e,— and/or et do not occur in the wedge. Let
LIE71.:C€1$C82®"EBC6m .

for m E Z>0 and let

Z.Vyn :2 ® ZAk(Vrn) .
kEZZO

By definition of Z, above is finite direct sum as Zk(Vm) = 0 for large k, say q.
Lemma 2. For any given K E Z>0 and a matrix Z in 501(V) satisfying Z(Vi) C Vi
and Zq(Vm) = 0 for some (1,ZP[/\k(Z'Vm)]= Oifor all k s K and P = K(q— l)+ 1.

Proof. Leta :2 (2,1 /\ (1,7,) /\ - - - /\ (ilk E /\k(Z'Vm).

1. pg . . .
ZP(a)= Z .-——————'qu8,1./\Zq2e,2A---/\que,jk.

.i... ,_
qi+~-+qk=P qr qr.

Since, for 1 S s S k, (2,3 E Z'Vm, => Zq3(e,js) E Z’(qu9Vm). Now, if qs S q -- 1 for
all s then P :2 ZEZI qs _<_ k(q-1) g K(q -— 1) = P f1. So, P g P —1=><=. Thus,
qs Z q for some 8. And, Zq3(e,-S) E Z'(qu Vm) = 0 for some 3 meaning ZP(a) = 0
for all a E Ak(Z°Vm) since a is arbitrary. Hence, Z P [Ak(Z ' Vm)] = 0 which is Lemma

2.

We will prove that for a given 3 E Z>0, we can find N, depending on s such that:
(Z + f‘)N(/\3Vm) = 0, (3.2)

which will prove that X = Z + P acts nilpotently on Spin(V) = A'V+.

Set l := [3], P :2 s(q — 1) + 1 and N := (l + 1)(P — 1) + l + 1. Consider:

(2+ mam/m) = 69 a; 2P1r2P2...r2Pk—H (Asvm).
k=0,N ZPizN—k
Verify that for any p,k E Z>0, Zp(/\k(Z°Vm)) C Ak(Z'Vm) and P(/\k(Z'Vm)) C
Ak‘2(Z'Vm). Thus,

(1 ;= 2P1 E‘ZP2 . - - PZPkHMSi/m) c AS-2k(2' ’m).

32

where 1ight hand 18 defined as zero for s < 21:. Let k < [3 J— — 1. Since Z,- P: N— k,

P, 2 P for some 2' for a similar reason as in Lemma 2. Let PJ- 2 P. Now,
FZPJ+1-.-F2Pk+1(/\Sv;n)c AS‘2("-J+ll(z"vm).

By Lemma 2,
AP‘. ~~P-+1 ~~P. 8 y _—
Z J(FZ J ---FZ A-+1)(/\ l/m)—0

=> (.12: 21’1F...FZPJ(FZPJ+1 ---.FZPk+1)(ASVm) = 0
=> (Z+F)N(/\ ,,.)_0

which proves X : Z + F is locally nilpotent on Spin(V) = A°V+.

Note that we have not used the integrability of V yet. Now. for X a negative root
vector of 9. whose matrix corresponds to a strictly lower triangular matrix, the proof
will require the integrability of V. Most of the analysis is same but is significantly
different at few places.

As befme X— — Z + F. This time F(V =,{0} F: :Zr,tbr-tZ‘-T,t and F =
——1/2 2,}, bmeter.

~

F(eilAeg/M- Ac“) 2:—§Zb,tetl\er/\eil/\-~Aeik.

rt

It is enough to prove equation (3.2) for all m 2 Max {r,t : br,t # 0}. In the previous
case, we got for free the condition that Z q(vm) = 0 for some q, beacuse Z was a
strictly lower triangular matrix with Z (Vi) C Vi. In this case, we use the fact that
X is a locally nilpotent matrix. So, there exists a q such that (Z +F)q(V,,_, ) = 0 where
V": :2 (36-1 EB Ce_2 €- - - - EB (Ce--m . Since X is skew-symmetric, this is equivalent to
(Z+F)q (Vm)— — 0 where F 18 the transpose of F. Since, F(V )— — {0}, F(V+)= {O}.
111 particular, F (Vm) = {0} which leads to Z q(Vm) = {0} as required.

Let Z'Vm C V3, for some R depending on m. We modify 1 := [Ea—SJ P :=
(s + 2l)(q — 1) + 1 to define N :2 (1+1)(P — 1) +1 + 1 as before. Also,

F<A*‘(2°vm)) c Ak+2(z"vm)

33

for all k E Z>0 so that
(1;: 2P1 F'ZP2 - . . FZPk+1(/\8Vm) c AS+2k(Z°Vm),

and is zero if s + 215 > R and the proof goes through.
The fact about Spi110(V) follows from the character formula given before the proof

of Proposition 1.

Proof of (3’).

Let the chosen basis of weight vectors of V1 be {6, : 2' E [1} and that of V2 be
I

{Ci : z' E [2}. If at least one of ml or 7712-18 even, (V1 69 V2)+ = V1+ {E V;. De-

fine the map: ,
Spin(Vl :1} l2) —p-> Spin(Vl) >3 Spin(l/TZ)
I I
e] ACK +—> €J QIQEK
1 +-—-> 1 >3; 1
for J C [1+ and K C I “L, where at least one of them is not. empty and e” 2: 1 and
I ~ ~ . .
e{} z: 1. Here, 1) is an isomorphism of 50(V1) EB 50(V2)-1nodules as it can be easdy
verified that. X o p = p o X for X E £§B(V1) and X E £70052).
Now let both ml and 7112 be odd, so that the chosen bases of V1 and V2 are
I I I I ,
{- -- ,62,81,€0,e_1,6_2, ~ - -} and { -- ,e'2,el,eO,e_l,e_2, - - - } respectively. Define
_ I _ I . .
u :2 (e0 + ze0)/\/2, v 2: (co — zeO)/\/2 so that 11,11 are paired non-degenerately With

respect to the bilnear form. Now, (V1 69 V2)+ = V1+ ea V2+ 6}} (Cu. Define the map,

Spin( ’1) (>3 Spin(Vg) i» Spineve‘“(V1 {3- V2)

I _1t1t2 I .
6.11®€.12 ._. $_il[2__e,,1I\eJ2/\(1—t+t%)
1631 1——+ 1

where, tk :2 IJklmod 2, k = 1,2 and t := (IJ1|+|J2|)mod 2 and 2' 2: \/—1. Again,
p is an isomorphism of 5~0(V1) EB 50(V2)-modules as it. can be easily verified that
X o p = p o X for X E §0(V1) and X E 56(V2). A similar isomorphism can be given
for SpinOdd(V1 @ V2).

Proof of (4). SpinO(V1 EB V2) 2’ SpinO(V1) 63> SpinO(V2).

34

Using Proposition 1 (3), let at least one of ml or m2 is even say, m1. So, let m1 = 2].]

and 7712 2 2kg + e, where 6 = 0 or 1. Now using Proposition 1 (2) we get:
2k1+k2 SpinO(V1 '8 V2) 2 2/‘71 Spi110(V1) 63 2k2 Spi110(V2),

which gives the desired result.

Now let ml 2 2k1 + 1 and "12 2 2kg + 1 then as before we get:
2I‘71H‘7‘3H—l Spi110(V1 63 V2) § 2"] SpinO(V1) 03 2k2 Spi110(V2),
which again gives the desired result.

Note 1. : Proposition 3 is also true for an infinite direct sum V 2: V1 $1 V2 39 - - - if V

is d—finite. Also. we define the infinite tensor product on the right hand side as: Let

(7/.- 1: SPiIIOUi) ® (9 Spillofl’i:) ‘3 Spi11<1fl1~+1'5‘Tl"'k+2P)

There is an isomorphism U k -—> Uh.“ by Proposition 1 (4) The infinite tensor prod-

uct, Spi110(V1) >9 Spinal/’32) >3 - - - , is defined as direct limit of maps Uk —+ Uk+l-

Proof of (5).
Let 1V be root finite. So, there exists only finitely many weights of W out-
side the root cone C defined in Section 1.3 and PV has finite dimensional weight
spaces. Hence, for (if-finiteness of W, it is enough to prove that the sets S: :=
{/3 E ’P H Liz/300211?) D {/3 E P(W) 0 (Ft : [3((1) = k}, is finite for each [C E Zi,
where 73 denotes the weight lattice of g, F(W) are the weights of W and Cat :=
{ZS-1:1 ejaj : cj E 1Ri U {0}}.

Let PR be the root lattice, ’P/PR = {771. P2, - - - }. There exists finitely many i’s
for which ”P,- flC+ %{},sayi=1,---,m. For 1 g i S m, let pi E P,- 00+
be a coset representative of P,- such that for each 6 E ”P,- O C+ can be written as

g = I); + Zle cjaj with cj E Z20. Then

71 Tl
{/3EP,HC+:/3(d)=k} = pi+chaj : p,(d)+:cjaj(d)=k, CjEZZO ,
j=1 1:1

35

which is clearly a finite set using the definition of d. This leads to finitness of 8;.

Similarly we prove that S I.— is finite.

Proof of (6).

Let V be (if-finite and orthogonal. To prove:

V is root-finite <=> Spin(V) E (9.

(=>)
Use the character formula given before the proof of PrOposition 1,
Chat Spin(V) = 6A H (1+ 6—317).
y z'E 1+
Refer to Proof of 5 .for the definitions of C, C7+,C_,’P,PR, pj,’PJ-,j 7- 1---m and
define the sets:
11:2{2'E 1+:r3ie-PnCJt},
[22={'iEI+ZfiIjEp—C}.
Note that [1 U [2 2 1+ as ,8, E C _ because by assumption ,L3,(d) > 0 for all z' E 1+.
By root finiteness, 12 is a finite set.
Now let M denote the set of all minimal weights (in the root order defined in
Section 1.3) of the finite set {$235.12 )8,- : J2 C [2} . Thus, M is finite. Define the set

of weights in 'P:
T:={A—Pj—’Y : j=1---m, ’yEM’},

which is a finite set and pj’s are coset representatives defined earlier. We will show
that elements of T “cover” all weights of Spin(V) in the root order.

Due to the character formula above, any weight of Spin(V) is of the form A —
ZieJfii = A — 2,6,1 13,: — 21'ng .32" for some set J C 1+ and J1 := J n [1 and
.12 2: .1 fl [2. Let /\ be any such weight of Spin(V). For the sum ZieJl [3,: in A,

each [3,- can be written as pj + bj for some j and bj E PR 0 C+. Now any finite

36

sum of coset representatives pj’s can be written as pk + a for a E PR 0 0+. So
2231152 : pk + a + 23nd bj = pk + b for b E PR 0 0+. Write c := 22'ng [3,.
Therefore, A = A — (M.- + b) — c. Choose an element 7 E M such that 'y S c so that
c—yEPRflC+. Sett:=A—pk—'y ET. Tlielit—)\=b+c—7EPRflC+ so

that A S t. This proves that Spin(V) E (9.

(=>)

Suppose V is not root finite. Refer to the character formula given at the beginning
of the Proof of the Proposition 1. Since V is not root finite, there exists an infinite
set J such that /3i E P — C for all 2 E J (See definitions of P and C' in Proof of 5).

Extend the collection of simple positive roots, {aL.}Z:1, of g to a basis of f)*, say:

{(11) ' ' Human+1f ' ,(YP} '
For 2 E J, let
n
.32 = Z CLICO}; = 132,1 + 132:2
k=1
where (3,31 lies in the real space spanned by {ab - - - ,o'n} and 413,32 lies in the real space
spanned by {an+1, - -- ,ap}. Because 8,: E P — C, for each 2 E J, either fli’g # 0 or

[3,32 = 0 and there exists a Is: such that 1 S k S n for which ‘5th < 0.

Thus, we may find an infinite set J1 C J and a k such that if n + 1 S k S P then
02,]: is of same sign for all 2' E J1 and if 1 S k S n then CiJc < 0 for all 2' E J1.

Define 7K 2: ZiEK ([3,. for K a finite subset of J1. Also, let AK 2: A — ’YK- Then
AK is a weight of Spin(V) for each K C J1. If n+1 S k S P then A K when expressed
in terms of above basis, will have coefficient of (1;, unbounded (above or below) when
K varies over finite subsets of J1. If 1 S k S n then this coefficient will be unbounded
above. Thus these weights of Spin(V) can not be bounded above in root order by

finitely many weights from the weight lattice. Thus Spin( V) E (9.

Proof of (7). V(p) ’5 Spin0(g).

For the distinguished element d of g, the character formula given at the beginning of

37

the proof of Proposition 1 leads to:

Qif)c1tSpin0(geA H(1+e 0),
aER+

where the positive roots R+ corresponds to the d—positive weights {a E R : a(d) > 0}
of adjoint representation 9. Also, A = ELI ciAi, for A, the 2th fundamental weight
of g. The formula above is the character formula for V(p) if c,- = 1.

Restrict the adjoint representation to the 4—dimensional subalgebra D,- := 5,: EB Cd C
9, where 5,- 91 512((C) corresponds to the simple root 0:,- and decompose the adjoint
representation into finite dimensional D,--orthogonal-irreducibles: g 12 @k Vk. Here

1 denotes the restriction to 0,. Thus, by Proposition 1(4):

Spillofg l) ”E Spino(V1) Q.3‘S})1110(V2)§§1 - .-

Using the character formula for SpinO for finite dimension representations, [P], the
distinguished weight M, of Spin0(Vk) is the half sum of d—positive weights of Vk.
Consider a highest. weight vector v = EB aflXfi of Vk for [3’s positive roots of g. If
{3 = :0 baa for a’s simple roots of Q, then ad(X -01.)!(X‘3) is a positive root vector
or zero for all 1 unless [3 = 02,-. This leads to Ago!) 2 0 if 2; ¢ 9(02') and Ak(a;/) = 1
if v E gmi). Thus, c,- = 2k /\k(a;/) =1 for all 2'.

Proof of Proposition 2.

Refer to the character formula given before the proof of Proposition 1 which leads to:
CbarSpiI10(g eA H (1 +6
{i(d)>0
where we can write, A = 2:21 c.,-A,-, for A,- the ith fundamental weight of g. The
weights ,8 of V such that Md) > 0 are called d—positive weights of V. If V is finite
dimensional then by [P], A = 2:301)” imgfi. When V is infinite dimensional, this is

an infinite sum but as in proof of Proposition 1(7), we may restrict V to D,- := SiEEKCd C

38

g and decompose V13 69k Vk into finite dimensional Di-orthogonal-irreducibles Vk.
Then the distinguished weight M, of SpinO(Vk) is the half sum of d-positive weights
of Vk. But for a weight [3 of V if both '13 and ”(13) are d—positive, that is, Md) > 0
and 3213((1) > 0 they will not contribute to Ak(aly) because ,13(a,V) + 36(an = 0. By

Proposition 1(4), c, = 2k Ak(o;/), and the result follows.

3.3 Proof of Theorem 1
Proposition 1 (7), says:
Spinofii) '5 V(I3)

\V'e restrict the adjoint. representation 6 to g and apply Spino with respect to 9. Using,

Proposition 1 (1) we. get the following commutative diagram:

~ SpinO , ~

9 -—2 V (P)

l , e l
Spino

921a P1 3 pg ‘39 - -' —+ V(p) (Ci) 1V1 18> W2 '8) ' '-
where vertical arrows denote restriction. The diagram commutes for the following
reason. Fix a d—finitc, orthogonal fi-representations V (such as adjoint representation
of 5). Clearly, Spin commutes with restriction, 1% when acted on V. Express, Spin
in terms of Spino using Proposition 1(2). Since 9 C E is a d—embedding and V is
a d-finite representation, the non-zero fi-weights of V restrict to non-zero g—weights.
Thus the dimension of the zero weight space does not change upon restriction. Hence

Spino also connnutes with 13 when applied to V.

3.4 Proof of Theorem 2

Let X). :2 Chat V(/\). For any X = ZAEI cAeA, I C P the weight lattice, define X0) =

2: A6 I eye”. Weyl character formula for character of an irreducible g-representation

39

with highest weight A, says:

14A+

___, P

X)‘ ~ A ‘
p

 

where the skew-svmmetrizer, A z: , si 11 to em“) and W is We 1 rou .
. p uEW g Y S P

#2“ ‘ = _...._.__
\#+P A~

 

X2fi+fi : _T)
. (2)

Thus. X2fi+15 2 ’Xfi- Xfi (3.3)

Let XI? 1: Z, \m, where 1 denotes restriction from E to 9. Also, let W 2: W1 <20
W2 to - - - , where H-"k's are defined in Theorem 1.

2.
X2fl+fii = rifll Xpl

k
= (Edi-”(Xe Charm); by Theorem 1.
i=1

ll

k
(2: xii)xp) €bdt(W)
i=1

k
Therefore, mg“; l = (Z X2/"i+/’) €hat(l'V) for the same reason as for equation (3.3).

i=1
which proves Theorem 2.
3.5 Proof of Propositions 3-4

Proof of Proposition 3

We will use SA(131,1‘2,' . - ,rn) to denote irreducible characters for E = sInC and Xk

for irreducible character of principal g = 512C with highest weight kw. The restriction

40

(n+1—22')/2

in this case corresponds to setting 313,-, in SA as q for q = e“, and a the root

of 9. We fix the following notation:

SA 1:: 5',\((I("—1)/21GUI—By2 ' ,q—(n—1)/2),

3

and
~(2) . 2 2 2
LSA 0: SA(-E1’JT2,-.. ’17:).

Then character of adjoint representation of 6 when restricted to 9 gives:

801:12 + X4 +~-12,,._1,‘.

where 6 is the highest root of E. We apply SpinO 011 corresponding representations,
and use the character formula for Spino from Proposition 2. Then using Theorem 1
we obtain:
5/11: X1 - U1 'U'z ° ”Um—2

where 22 '— UNIV”;2 + “1(2) and — (n — 1 n -- 2 1 0) Now Theorem 2

' A‘ '_ jzl I q ( p — 1 1 1 ~ - 1 1
leads to:

~2,u+p l— (11 1X1) '“1 '“2 ' ' ' “72—2

where all n -— 1 factors on right hand side are characters of g = 512C. In order to

translate this in language of principal specialization, we observe the following identity:

~. ,— N
b,\(11Q1q21m1qn 1):q SAL

n—l

T!
where N = —2— Z /\.,-, for /\ = (A1, A2, - - - ,An). Using the above formula, we obtain:
i=1

82p.+p(11(l~(l2r ' ' ' ,qn—l)
n.
= (q(3)(1+<I) 511(11q21q4,---,(12”‘2)) -wi(q) - 102((1)---wn—2(II).

where wk(q) = (1+q)(1+(12) - - - (1+qk+1), where all (n — 1) factors 011 the right are

symmetric unimodal as they have been obtained from slgC-characters.

Proof of Proposition 4

41

We list all graph automorphisms of Dynkin diagrams, D, of simple Lie algebras
E and the corresponding fixed subalgebras 9. We obtain the factorizations using
Theorem 1. Let (9 and (5 be the highest roots of g and E repectively so that V(6)
and V(é) denote their adjoint representations. Let V(é) 13% V(6) 93 p1 $ pg 9 - - - ,
so that V(fi) lg§ V(p) <81 W1 (>3 W2 Q3 - - -. We specify p,’s and W53. The fact that

SpinO(pi) T—1 W", will be proved in Proposition 9.

1. Graph automorphisms of order 2.

(b) 5 = Dn+l : 50271+2C-
9

[in = sanHC

n 49> n + 1, interchanges two forked nodes and fixes others.
V(d) 1g”; V(6) EB V(OS), where 63 is the highest short root of g.
V(I'I) 15’5 W) e V(ps).

(C) 5 = E6-
9 = F4.

1 «6—51 5 2 £1 4 and fixes others.(See page 53 [Ka] for ordering of nodes)

V(d) 13% V(O) EB V(63), where 65 is the highest short root of 9.
HF) 13% We) ® V(Ps)-
((1) 5 = A212 = 5[271+1C-
9 Z 3n 2' 5°2n+1C-
2' £1 2n + 1 — 2'.

V(é) 13% V(Q) 69 V(265), where 65 is the highest short root of g.

42

~

We) 13% V(p) a V(p + 2103)-

2. Graph automorphism of order 3.

(a) 5 = D4-
9 = 02.
d) cyclically pemutes the three outer nodes and fixes the middle node.

V(é) 132 V(6) EB V(Os) EB V(63), where 63 is the highest short root of 9.

Va) 1'32 V(p) e (ms) + v<011e<v<ps>+ we».

3.6 Proof of Propositions 5-8
Proof of Proposition 5.

We will prove Proposition 5 using Proposition 2. Recall that —N < ,U( (l 1). 6((11) < N
for all weights B E T of V. Now all weights of V are of the form k6+13 with multiplicity
2113 for A“ E Z and 13 a weight of V. Thus, for d = Naf+d17 (k6+,{3)(d) = kN+/}((ll) > 0
if and only if k > 0 or k = 0, [3((11) > 0 by the definition of N. Proposition 2 leads
to:

€batSpi110(V) = e" H (He—fife I] (1+e’3’k6)m3 ,

13(d1)>0 k>0
[JET

where A 2 21:0 ciAi, c,- as defined in Proposition 2. It is easy to verify that for
2' = 1 - - - n, c,- = E %m [1,8 (022’) summing over weights 3 of V (as opposed to V) such
that ,13(d1) > 0 and 5,-(13)(d1) < 0 because 3,-(k5 + B) = k6 + sl-(fi). Futher, since V is
finite dimensional, we may drop the condition 3,-(fi)((11) < 0 and sum over all weights

[3 of V such that r3011) > 0 because May) + 5,-(B)(a;’) = 0. Thus:

1 .
c, = Z yup/3012’), 2: 1, - -- ,n

flfd1)>0
For 2? = 0 case, (M +,/3)((16’) : —/}(()V) as (r0 = K — 19V. Also, TILko‘iB = mg.

Therefore by replacing ,8 by —fi, we get (:0 = Z %n1.,13,8(6v), summing over all k E Z

43

and 13 E T such that (M + B)(d) > 0 and 300:6 + B)(d) < 0 which simplifies to the

inequality:

[3((11)
N

80.13 ((11 )
N

 

< k < s(ev) + (3.4)

where .39 denotes reflection corresponding to the highest root 6 of 9.
Now, by definition of N, 1892 and w are fractions. So the inequality (3.4)
implies that {3(6V) Z 0. Since (:0 involves summing émfiewv) over inequality (3.4),

we may sum over {3(0V) > 0. Consider the following cases:

Case 1 : 13(6V) > 0. 21(111) < 0 (=> 39,3011) < 0).
Case 2: (3(6V) > 0, [3(d1) > 0 and '39,/3(d1) > 0.
Case 3 : ,/3(6V) > 0, 13(d1) > 0 and 593(011) < 0.

In Case 1, inequality (3.4) :> 0 S A: S 13((IV) — 1. In Case 2, inequality (3.4)
=> 1 S k S 8(6V) and in Case 3, it implies l S k S (3((9V) — 1. Thus, we get:

1 , - , 1 , . 1 ,, . -
c0 _—_ Z éylnyfizr'xowh Z 57n.213,13(9V)2+ Z 5m/3(/1(6V)2—-/}(0V))
Case] Case2 Case3

1 1 a 1
= Z 2m’3/3(6V)2— Z E’n'13»‘3(gv)-

Case 1,2,3 Case 3
Now, in the first sum the union of the three cases leads to the case [3(0V) > 0 and
due to the square in the sum, it is equivalent to summing over [3(d1) > 0. In the
second sum over Case 3, we may drop (,(3(6V) > 0) as it is implied by 6(d1) >
0 and 39/3011) < 0. Further, we may also drop (393011) < 0) for the same reason

which led to the expression for c,, 2' = 1 - - - n. Therefore:

1 1 ,
c0 = Z §m-,6/3(9V)2- Z Ema/NOV),

,d(d1)>0 o(d1)>0
1
c, = Z 57725/3(a2-’) i=1,2-~n,
ff(d1)>0

and A 2 23:0 ciA, which leads to Proposition 5 using A,- 2 w,- + a2’A0 and 0V =

2‘21 o2’o2’. Here, 37,- is the 2'th-fundamental weight of g.

44

Alternate proof of Proposition 5.

Now we will use a different machinery involving computations in 53(V) to prove
Proposition 5. Refer to the characer formula in the Proof of Proposition 1. Now,
the Lie algebra under consideration is E (in place of 9) obtained from affinization
of finite dimensional Lie algebra g. The representation under consideration is V (in
place of V) obtained from affinization of finite dimensional g-represntation V. The
fi-representation V is orthogonal, so we have the map

A a .

g —> 50( V)
corresponding to the g-representation V given by the map

9 —0—> 50(V).

Recall that in the Proof of Proposition 1, for the comutative diagram given by Lemma

1:
5—1» sow)
c} \ T71“
an?)

we did not calculate A = Z0 0 5 which we do now in the present setting.

First we do this for E = 571(V), g = 50(V) when V is the defining representation
of 50(V). Recall from Section 2.1, the definition of 50(V) for a vector space V with
the following basis {em > ...,e1,eo,e_1,...,e_m}. Let [m] :2 {—m....,0,...,m}.
To distinguish objects associated to 50(V) and %(V) with corresponding objects of
g and a, we will put a I on top.

A basis of the cartan subalgebra of diagonal matrices of 50(V) is:
{Hi :=Zi’,: 22:1...112}

and let the dual basis be:

{L.,-:2'=1...m}.

45

The set of positive roots are:

p-11} {1:- r} .
i " J i>j20U '+ J i>j>0

The highest root:

Define a bilinear form 13 on 50(V) as I3(X,Y) 2: éTrace(XY). The highest

4

root vectors X

0 and .X

are Zn,_(,,_1) and Z_( respectively. Note that

_5 n-l),n

3(X5, X_é) = 1. Let @(V) be the corresponding affine Lie algebra with central

element K and derivation (I. So

and: {B C(rtz,,,)tcf{ecd

keZ,i>-—j
with cartan subalgebra
EB (CH, 62 (CK ea Cd
i=1...m

where H,- := Z“. Let the dual cartan subalgebra be

® CL,‘ [P CAO (P Cd.

2'=1...m
Let V be the corresponding affinized representation of V of 50(V) as defined in Section

1.5.1. Denote this representation by a : 515(V) —1 50(V). Also, recall the map

A A

7r : 571(V) ——> 50(V) defined in Section 2. Then by Lemma 1, with If) = 0, we have a

Lie algebra map :1, : 55(V) ——1 571(17) such that the following diagram commutes:

A

50(V)
/_~1 or
row) 1» €007).

Lemma 3.

.. ~ 2 1'". .
A2=L007=§(ZL¢)+AO

221

Proof of Lemma 3.

46

Recall that for the basis element tkej of V we write 6(ko')‘ So for 50(V) and 50(V)
the indexng set, instead of Z, is ( Z x [112]). But for the index (0, j ) we will write 3' .

A A

According to Section 1.5.1 , 7 : 5 (V) —> 50(V) is defined so that,

ifzm') = Zi,j + Z (Z(k,1i),(k,j)—Z(k,—j),(k,—i))a

keZ+
1021.1) = Z (Z(k+1,i),(k.j)_Z(k+l,—j),(k,—z‘))v
kEZZO
“IQ—1214) = Z (Z(k,i),(k+1,j)_Z(Ic,—j),(k+l,—i))-
156220

A

According to Lemma 1 with If) 2 0, SI : 5 (V) —> $~O(V) is defined so that for 2' # 3',

~ , 1 1
“1(ZI'J) : *7Z(’j(’._i — Z 15(()'(k,j)(f(*-k,—i)— c(k.*-I)(i(—k,j))’
Ike 2*
:, .. -_ 1 1 . , .
.'(fZl,_]) — — Z 2f“‘(k.j)‘(—k—1,—2)'”‘*(k.—'i)"(—l:-~1,j))1
FEE/€20
~ —1 _ 1 ,
W Zn) — ‘ 2f€<r+1.11‘v(v—k.—zz1‘€<k+1.~-i1‘31.-k.11)-
A6220

Using fil'(Z,:,J-), 2 753' we can also find 5(Hi) as follows:

\.

17,-: 2,3,: = [220, Zoy] => MHz-i = [3(Zi.0)a3(ZO.i)l

and using HZ”) for 2 75 j, we can show that:

~ v I 8i€__,j I r
”If/’2') = (5 — 2 > — Z 5((f(1:,2)‘*(—1:,—i) — “(k,—zt)"(—I.~,i))- (3-0)

ke 2+

 

Now, we find 3(K). Let X5 and X4) denote highest root vectors and 5V the highest

coroot in 53(V). Then,

V —1".,.._. .. .. stews
[tX_0,t X0] _ ]X_0-,X0] + B( 0,X_0)K.
[tX_6«,t—1Xé] = _év + éTrace(U'9~,X_é)K.

Applying 5/ both sides we get:

[i’ftx_5),‘i(t_lxg)l = are) + irracedgxsntli').

In 573(V), the highest root vectors X5 and X45 are Zn,_(n_1) and Z_(n_1),n respec-
tively and 5V is (Zn_1,n_1 + Zn”). So,
,1", U __ 1
r(l‘X_g) — - Z 2(€(k,n)€(—k—1,n—1) — €(k,n—1)e(—k—1,n))v
kEZZO
~. —1 V _ 1
7(t X5) - - Z: 2(€(k+1,—n+1)€(—k,—n) — €(k+1,—n)€(—k,_n+1)),
kEZZO
5(9’V) = 1_ en—le—n+l _ 6718—71
l .
'- Z 2(8(k,n-1)€(—k,—n+1) — €(k,—n+1)€(-k,n—1)
k6 1+

+ 6(k1n)€(_ka—n) — C(kv_n)e(_kan))l
Now, it is easy to verify that:

[a<t.>2_,>,a<z-lxg>]

] /
“ 2V5]: —l(3——n+l + (inc-Tl)
1 , . .
+ Z 2("(m—1)"(—k,—n+1) ‘ “(k,——n+1)“(-—k,n—1) + V(km)‘ (—k,---n} ‘ ('(k_—n.)‘-J'(—k,n)):

1051+
= —~}(HV) + 1.
Using
1 v - _ v - v 1 v v . '-,
[wxwu 1295)] = we) + §Trace(Xg- 225)va ),
and knowing that $Trace()u(6’, X49) : 1, we get :
w?) : 1. (3.6)
In Lemma 1 we choose 1,» = 0 so that ,
(Constant term in fi(d)) = O. (3.7)

Recall, Z0 is zero on cartan subalgebra of 573(V) except on constants with Z0(1) = 1.
Therefore, using equations 3.5, 3.6 and 3.7, we conclude that:
~ 1 m ., U
L0 0 i = 5(2 L,-) + A0.
z=l
which proves Lemma 3.
For the general case, let V be an orthogonal representation of a finite dimensional

Lie algebra 9 given by, a : g -—r 50(V) and V be its affinized representation of the

48

corresponding affine Lie algebra E (as described in Section 1.5) given by 6 : E —*
50(V). Then as in Lemma 1, we have the map 6 : E ——> 570(V) such that following

diagram commutes.

a :1» saw)

6 \ T 71'

5~0(V)

As before, let V has a basis of weight vectors, {cm,...,e1,eo,e_1,.. . ,_m} ((20 ap-
pears if V is odd dimensional). Let weight of ej be ,Bj, j > 0. Assume that the
killing form 8 on g is normalized so that B(Xg, X_g) = 1, where X0 and X-g are
root vectors for the highest root 9 of 9. Let 19V 2 coroot of highest root 0 of 9. Then,

Lemma 4.

AI: Z005=V+CAO

"7. T"
v ‘ . j — 1 I . | -— l i . V 2
“hue I/ — 2 E 1U, and r. — 2 E 1/3, (0 ) .
l: 1:

Proof of Lemma 4.

In the connnutative diagram

33» an?)
6 \, T7r
an»)

we. relate 6 and 6 with A} and A} described in commutative diagram in Lemma 3,

A

50(V)
/‘/ T7r

-
A

50(V) —7—+ 570(V).

Since both a and 53(V) act on V via 6 and '9 where action of a is obtained by
affinization of the action of g on V by matrices in 50(V) and action of %(V) is
obtained by affinization of defining action 50(V) on V, image of 6 lies in image of
6/. Note that, due to the map & : %(V) ——+ 50(V) with kernel CR, €0(V) is a
central extension of the image of 3. Thus using the note after Lemma 1 with if! = 0,

there exist a map of Lie algebras 7/ : E —-+ §0(V) such that the following diagram

49

 

commutes. .
A 0’ "
g -—-> 50(V )

01 /‘i
2??)(V)

Combining this with the commutative diagram

50(V)
/ ‘7 T7r
530/) L. gem,
we have the commutative diagram:
E L) 50(V)
721 / “7' TTF

s?o(V) i) 5720/).

Comparing above diagram with

a L see?)
6 \, Tvr
5~0(V)

we get 6 :— :y o 77 because :1, o 7} satisfies the conditions of Lemma 1 and 6 is unique.
Next we compute I), so that we can compute Z006 = Zoofion. For X E g, 77(th) =
IkU(X ), where (T : g -——+ 50(V) gave the finite dimensionl orthogonal representation
V of g. In particular, 7}(H) = GUI) for H E b. Calculating bracket of 0th simple root

vectors, tX_9 and t’ng in E where B(X__g, X9) = 1 by assumption, we get,

[tX__g,t”'1Xg] —_—- [X_9,X9] + B(X_6.X9)K.
[tX__0.t’1X0] = —0V + 11".

Applying 7) both sides:

[nt(X—9)~nt‘1(Xo)l = -n(9v) + MK),
[ta<X-o>.t-la<xo)] = -n(9V) + 77%”)-

Calculating [/U(X_9),f—10'(Xg)] in 533(V),

V

3(0X—9mx0)’ = —n<6V) + we.
[3

between + ‘ .
+ (0X_9,0'X0)K : -—7)(6V) + 77(K).

—TI(H0)

”— Tl—Ifi

This shows that:

71(K) = B(6X-9, 0X9)R (3.8)
7}(H) = 6(H) for H 61) (3.9)
77((1) = d (3.10)
Since 6 = 7y o 7}. ~ ~
[1006 = Loofior},
= (LOOilon;
1 m V V
: (2 2114+ A0) 0 r),
1,:

(Using Lemma 3)
= $155132: 0 n) + (510 077)-
Now, using equations 3.8, 3.9, 3.10 and 6(H)(e,-) :2 [3,-(H)e,- for H E I), we can show
that L,- o n = a, and A0 0 n = B(6X_g,6X9)A0. Now, 9 embeds in 50(V) via
0' : g —-—> 50(V). Thus invariant bilinear form 3 on 50(V) is invariant over 6(9) also.

Therefore,

U

B(6X_g,6X9) = —{,B( 6X_9, [0X9.06V] )
— 15% [ X X 6"
— 2 (I 0,0 —0l 30 )
= 590%, 09V)
2 fli'I‘race(66V - 60V)

( By definition of [3 )

m.

= 211 . Z fldavlz

z=—m
( As 66V is a diagonal matrix )

1 m v 2
= 2 Z 52'“? l
i=1
which proves Lemma 4.

Now we can write the character of Spin(V), when V is obtained by affinization of

a representation V of g ( 0' : g ——+ 50(V) ). Recall from Section 1.5.1, the action of

E on V, 6 : E —> 50(V). For the chosen basis of V, {tkei E 63“.,” : k E Z,i E [171]}:
H)(tke,;) = tk6(H)(e,j) =,13,(I—1)tke,- for H 6b

11301563,) 2 O

6(d)(tke,j) = ktke, .

Thus, weight of tke, : €(k.i) is ,8,- + 156. Using Lemma 4 in the character formula

given in the Proof of Proposition 1, we get:

, m _,, '
Chat Spin(V) = e” n (1 + ei dz) emf) H (1+ e—_B,-—k6)
i=1

k>0,i€[m]
1 m 1 m
where I/ = Q Z 13,: and c = 2 Z [3,:(6V)2.
i=1 i=1
In case of Affinized representations, (zero weight space of V) C (zero weight space

of L) Then from Proposition 1 (‘2), Proposition 5 follows directly.

Proof of Proposition 6.

Exactly same as that of Theorem 1 using property of SpinO given in Proposition 1(4).
Proof of Proposition 7.

Direct consequence of Propositions 6 and 1(7).

Proof of Proposition 8.

This is just Theorem 2 for affine Lie algebras where we make use of Proposition 7.

3.7 Proof of Propositions 9-10

We will use R, R5 and R1 to denote the set of all roots, short roots (if any) and long
roots (if any) of a finite-dimensional semi-simple Lie algebra g with distinguished
element (1 = pV, the sum of all fundamental co—weights of 9. Similarly 12+, R: and
171+ will denote the set of all positive roots, positive short roots and positive long

roots.

52

Proof of Proposition 9.

This classification was done by [P] Here we give proofs of the facts about Spin0(V)
for each of the 3 cases.

Proof for Case I is given in Proposition 1(7) earlier which uses Weyl denominator
identity. We use it again to prove other cases also.

Proof for Case 2.

Let y := Char Spino V(BS). By Proposition 2:

X =' 6’03 H (1+e‘0‘).

06R:

\V'eyl denominator identity is:

.1,, 2 ep H .(l—e‘a) H (1-- 8—0),

age: aeaf

,\ rip = cps-HO H (l—e—Qa) H (l-e-a)

aER: OER;

= e98“) H (1 — 6—0). (3.11)

as (2RjuRf)
If for g, ( ||61||2 / ”63“?) = 2 (as in Case 2) then (2R3 U R!) forms the root system of
the dual algebra, denoted by E, of 9. Then the half sum of positive roots 6 of E is

given by ,6 = p3 + p. Thus by Weyl denominator identity for E and equation (3.11):

~

X AP = J/‘fi : 4'1p8+p.

Since, 9 and E have the same Weyl group and A” is the anti symmetrizer of n w.r.t.

~

Weyl group, Ap5+p = Aps+p-

A
=> x = if“) ZQ‘W‘W/Jsl
p

53

1..”

'..'.1

Proof for Case 3.

For n 2 2, Panyushev [P, Prop. 3.8] showed that the set of all nonzero weights of
V(263) is S = 213,; U R, U R] with multiplicity of each nonzero weight as 1. For n = 1,
define R; := R, R: 2: R+ and R1 := (LR? := {}. Also define Hoe” f(a) 2: 1

for any function f. Let X 2: Chat SpinO V(263). Thus by Proposition 2:

x :2 1999+” I] (l-Fe'2a) [I (la-9'“) II (14-6—0)

06R: . OER: OER?
Ap = 4’ H (1—e‘0) H (1—e—0‘).
aeR: aenf
X. Alp : €2ps+2p H (1 __ 8‘40) H (1_ 9--20)
OER: oER+

l
(2 .44), = ‘42(ps¥p) as p, = p for 71:1)

: em”) [I (1—e-20) (Forn22)
06(2njonl)

= r12); (By comparing with A); of dual algebra E for n 2 2)
= "I12(/)8+P) (As ,6 = p, + p as in case 2)
/l(

=> X = w = CbatV(2p3 +p) (For 7121).
/)

Proof of Proposition 10.

Proof of (=>).
Let SpinO(V) be irreducible and denote its character by x. Also let S denote the set
of all non-zero weights of V and q :2 ed. Then by Proposition 5:
X 2 Chat SpinO(V) eCAO H (1+ e—Bq—k)mr3 H (1 + q’k)mO
k>0,13€S k>0
where c = 2:36er irrigx‘iwv)? Suppose that Qibat SpinO(V) 2 23:1 XVi where XVz'

is the irreducible character with highest weight Vi. We first show that s = 1 meaning

54

V is coprimary. Weyl denominator identity for E is :
* _ +hVA ,— ,— —k6 4.5
1i,,_ef’ 0 H(1—(. 0) H (1—(.0 )H(1—e )”
a€R+ k>Q0€R k>0
where hV is the dual Coxeter number. Using Weyl denominator identity for 9, namely,

Ap = e” “0612+ (1 — 1?“) and writing q = e", we can say:

: hVA —- —k —k
Apzxipe 0 H (1-e “q ) “(l—q )n
k>Qo€R k>0
Mutiplying X with 11/), we get:

A

s V , _ _
\"Ap = (Z 1w. A11) “(CM M” + (---)q l + (---)(I 2 +
i=1

3 . v . _ _J
: (Edi/a1“) €(r+h M0 + (---1q 1 + (--)<1 2 +
1:1

X ' if, will contain the term e”i+p+("+hV)AOf—= eui +CA0+15 for each 2' = 1 ..... s where
V, is a dominant weight of 9. By character of Spi110( V), all its weights of are of the
form: 2213€S+ (Id/3 for some «1113 g (1.3 s 111.15. Since c = $21,365.; 111.]313(6V)2,
V,(6V) g c. Also, since 12,: is a dominant weight of 9, this shows that u, + CA0 is a
dominant weight of E for all 17 = 1 . . . .3. Hence X contains irreducible XVH'CAO for each
i in its decomposition into irreducibles. So, .3 must be 1 because X 2 Chat SpinO(V)
is irreducible.

Next, we show that when V = V(263) for g = 502n+1C then SpinO(V) is not
irreducible. By [P, Prop 3.8], the set of all nonzero weights of V(263) is S = 2R3 U
R, U R, with multiplicity of each nonzero weight as 1. This holds for n = 1 also, if
we define R3 := R, If] :2 {} and R: :2 l{+, R: :2 {}. Let “oak/(Q) :2 1 for
any function f and S + = 2R: U R: U Rf. Now by Proposition 5:

CbarSpirio(V)=62p3+p+CAO [I (He-'5) H (1+e-B-k6)1'[(1+e-k5)mo.
[365+ 100,568 k>0

We calculate the level c of the representation SpinO( V) as follows: For 11. 2 2, R: =
{Li}, Rf = (L,- j: LJ- : i < j} and dual positive roots {2H,} U (H,- i HJ- :i< j}

r

05

where {H,- : 1 g i g n} is the dual basis of {Li : 1 g 2' g n}. Then 0 = L1 + L2 and

(1V = H1 + H2 = H...l + Hm, + 2 2:311“, Thus:

0 _ 2n+3 for11_>_2
_ 10 forn=1.

Also, the multiplicity of zero weight space, 1110 = n, the rank of g, by Panyushev

A

[P, Prop 3.8]. Thus, Ghar SpinO(V) reduces to:

X 2: ChatSpinO(V) -_— €2Ps+p 8010 H (1+e7f3) H (1+e—13—k6) H(1+e—k6)n.
565+ k>0,13eS k>0

The highest dominant weight appearing in X is A :: 2/13 + p + (:AO. To prove that X
is not an irreducible character of E, it is enough to produce another dominant weight
appearing in x, say, A, such that '(A—Al) can not be expressed as non-negative integral
linear combination of simple positive roots of E. For in 2 2, take A, = 2x13 + p+ 2L1 ~—
6 + CAO which corresponds to the term in x, 6295+” eCAO its—(3‘1“S for 13 = ~2L] E S
and k = 1. Clearly, A, is dominant as A, := 2p; + p+ 2L1 is dominant weight of g and
A'wv) = 2n+2 g c = 2n+3. A—A' = -—2L1+6 = 6—(L1+L2)—(L1—L2)= 00—01.
For n. = 1, take A, 2 2p; +p+ (2a —6)+CA0 = 7p—6+10A0. Then AI(6V) = 7 S 10.
Here, A : 3/)+10A0. So, A — A' = —4/) + 6 = —2('r + 6 = (5 — ()7) — a = (110 — (11.
Proof of (4:).

Using Proposition 1 (7), it is enough to prove :

A

v = V(Qs) => Spin0(V) = V(fis).

where, ,5, :2 p3 + fig/A0, hg’ :2 Z,- of, 1’s corresponding to short simple roots of E.
In the proof of Case 2 of Proposition 9, we dealt with finite dimensional Lie algebras
g and its dual algebra E. In the same spirit, here we will deal with the affine Lie algebra
E for g E {$02n+1C, 513an, f4} and its Langlands dual (obtained by reversing the
arrows of the Dynkin diagram of E) denoted by E. So if R denotes an object associated

to 9 (say R = the set of roots of 9) then the. corresponding object (the (multi-)set

56

of roots) associated to E, E or E will be denoted by R, R and R respectively. Let

A

x = Chat SpinO(V). In order to completly adapt the proof for Case 2 of Proposition
9, we will define multisets associated to roots of E, namely R: and R? and show the

following Facts:

1. X2065 U (1+e‘a).
aefii

2. R: U R? = R+ the multiset of all positive roots of E.
3. 2R;1L U R?” = R+ the multiset of all positive roots of E.
4- 15 2 {38 + P-

First we introduce the folldwing notation: Forany set A. define Afk} :2 a multiset
consisting of elements of .4 with each element appearing 11: times. Further, the multiset
Ail} will be written as A.

Proof of Fact 2

Now, the multiset of all positive roots of E:
R+ 2: 12+ o {a+k6:a eR,keZ>0} o {k5;kez>0},,
For g E {502n+1C, 5132,,C, f4}, define:
13;“ := a: o {a+1.-6 ; a 6 R3,}; 6 z>0} 11 {M :1. e z>0},,3
where 115 1: number of short simple positive roots of 9. Similarly,
I? := Rf o {0+k6 : a e R1,}: 6 z>0} 11 {k6 ; k e z>0},,_,,s.

Clearly, 2 is true.
Proof of Fact 3
2 2 _ -
g E{502n+1C15p2nCs 14} => (H91 119.11 ) — 2 mg
A 1 l l V 2 2 2
=> 9 E {81(1):ch ).F4( )} => 9 e {A;,)_,.Df,,3,,Eg l}.

57

Note that, E is a twisted affine Lie algebra which contains the finite dimensional dual
algebra E of g. The set of all positive roots of E is 2R: U Rf. Now 3 is obvious for
real roots or one can check directly using [Ka, Prop. 6.3]. We only need to check
the multiplicities of imaginary positive roots. We observe that, in 2R;F U Rf the
multiplicity of 2k6 is 113 + (n — 11.5) = n and multiplicity of (2k + 1)(5 is n - 11.3 which
matches with multiplicities given by V. Kac [Ka, Corollary 8.3] for twisted affine Lie
algebra E.

Proof of Fact 4

Let Us UH) = the set of all simple positive short and long roots of E. Similarly,

let 2.5 = the set of all simple positive coroots corresponding to short roots of E and
let 2, = the set of all simple positive coroots corresponding to long roots of E. Then
(2 HS U H1) and (% 25 U 21) forms the sets of simple positive roots and coroots of
E respectively. From this we can conclude that p z: ,3. + p.

Proof of Fact 1

Refer to Proposition 5 for expression for x :2 ¢har SpinO(V). It’s easy to check that
set of all nonzero weights of V : V(HS) is S = R5. Since multiplicity of zero weight

space is as, Proposition 5 leads to :

X : (fl/+CAO H (1+e—O)
aefij

. ._ 1 _ __ l V 2
where, 1/ .— Q 2061?: a — p3 and c — 2 Zoe]? 0(6 ) .
We will show that 1/ + cAO = [33. Since, 19 is always a long root for g E

{502n+1C, stagnC, f4}, (1(19V) is O or 1 for all a E R: due to following Lemma.

Lemma 5.

a E R+\{6} => 01(6V) = 0 or 1.
Proof. Verify the following facts about any finite root system R = R+ U R‘.

1.01ER+ =¢ a+0¢R.

2.016R’ :> a—9¢R.
3. F0r(.1E/1’.+, (11—19611? => a—REIf— => (i—26¢R.

Consider the restriction of the adjoint representation, V(B), to the 512C corresponding
to 0, 50 2: (CXO :rB (CX__9 e1) CQV. Using above facts, we conclude; for a fixed oz 6
R+\ {t9} :

o If a — 6 5? R, then CXQ is a trivial irreducible component of V(0) as an 50-

representation, and thus 0(6V) = 0.

o If a —— 6 E R, then (CXQ €13 (CXa—Q is an irreducible component of V(6) as an

50-representation, and thus 0(6V) z 1.

I

This proves Lemma 5.

Thus, (1(9‘“) is 0 or 1 for all a E R: => 0(6)")2 2 01(0V). Therefore, c 2 p3(6V) =
11)! as [13 = the sum of all fundamental weights of 9 corresponding to simple short
roots and 6V 2 Z];1a;/a;/. We get, l/ + CAO = p, +‘hXA0 = ,63 which proves the
Fact 1.

Finally we get the proof by replacing p, p3, R: and Rf by [1,63, R: and Rf re-

spectively in Proof of Case 2 in Proposition 9.

59

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