kazhdan-lusztig conjecture for generalized kac-moody ... · introduction this paper is the second...

TRANSACTIONS of theAMERICAN MATHEMATICAL SOCIETYVolume 347, Number 10, October 1995

KAZHDAN-LUSZTIG CONJECTUREFOR GENERALIZED KAC-MOODY ALGEBRAS. II:

PROOF OF THE CONJECTURE

SATOSHI NAITO

Abstract. Generalized Kac-Moody algebras were introduced by Borcherds in

the study of Conway and Norton's moonshine conjectures for the Monster spo-

radic simple group. In this paper, we prove the Kazhdan-Lusztig conjecture

for generalized Kac-Moody algebras under a certain mild condition, by using

a generalization (to the case of generalized Kac-Moody algebras) of Jantzen's

character sum formula. Our (main) formula generalizes the celebrated result

for the case of Kac-Moody algebras, and describes the characters of irreducible

highest weight modules over generalized Kac-Moody algebras in terms of the

"extended" Kazhdan-Lusztig polynomials.

Introduction

This paper is the second part of our work on the Kazhdan-Lusztig conjec-

ture for symmetrizable generalized Kac-Moody algebras. Here, as in our pre-

vious paper [N], a generalized Kac-Moody algebra (GKM algebra for short) is

nothing but a complex contragredient Lie algebra g(A) associated to a certain

real square matrix (called a GGCM) A = (a,;),;e/ indexed by a finite set7 = {1, 2, ... , n} (see [K, Chapter 11] and §1 below). (This definition is dueto Kac, and slightly different from the one by Borcherds [Bl].)

In [N], we proved that the multiplicity [V((w, ß) o A) : L((w', ß') o A)]

((w,ß), (w', ß') £ W x sé (A)) of the irreducible highest weight g(A)-module L((w', ß')oA) with highest weight (w', ß')oA in the Verma module

V((w, ß) o A) with highest weight (w, ß)o A is independent of the choice of

the dominant integral weight A £ P+ . Furthermore, we conjectured that, under

the condition on the GGCM A = (a¡j)ijeI that a„ ^ 0 (i £ I), the multi-plicity [V((w, ß)o A) : L((w', ß') o A)] is equal to P{w,ß),{w> ,ß<)(l), where

P(w,ß),(w' ,ß')(q) is the extended Kazhdan-Lusztig polynomial in q for Wxs/ .

In the present paper, we show that this conjecture is true. Namely, we provethe following theorem.

Theorem I (Theorem 6.2.2). Let q(A) be a GKM algebra. Assume that theGGCM A = (aij)ij€¡ is symmetrizable, and satisfies the condition that a„ ^ 0

Received by the editors September 4, 1994.

1991 Mathematics Subject Classification. Primary 17B67, 17B65, 17B70; Secondary 17B10.Key words and phrases. Generalized Kac-Moody algebra, character formula, Kazhdan-Lusztig

conjecture.

Partially supported by Grant-in-Aid for Scientific Research (No. 05740015), The Ministry ofEducation, Science and Culture, Japan.

© 1995 American Mathematical Society

3891

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

3892 SATOSHINAITO

(i £ I). Then, for a dominant integral weight A £ P+ and (w, ß) £ Wxsrf(A),we have

chV((w,ß)oA)= ¿2 P{w,ß),(W>,ß')(l)chL((w',ß')oA).

(w1 ,ß')eWxsn?(A)

Equivalently, for (w, ß) e W x s/(A), we have

chL((w,ß)oA)= Y, (-i)(Kw')+Wß'))-(H«>)+Wß))

{w' ,ß')CiWxsif(A)

•Q{w,ß),{W',ßn(l)chV((w',ß')oA),

where Q(W,ß),(w',ß')(Q) ((w>> ß') £ IV x sé'(A)) are the extended inverse

Kazhdan-Lusztig polynomials in q, such that for (w, ß), (w', ß') £ W x sé ,

(y,y)Ç.Wxs?

= ^(w,ß),{w' ,ß')-

Here ch F denotes the formal character of a (weight) g(A)-module V, i(w) is

the length of w £ W, and ht(jff) ¿s the height of ß £s/ .

Since GKM algebras obviously include Kac-Moody algebras, our result above

is a generalization of the celebrated result (for Kac-Moody algebras) due to

Kashiwara and Tanisaki [Ka, KT], or to Casian [C].

Our main tool is the following generalization (to symmetrizable GKM alge-

bras) of Jantzen's character sum formula for a quotient of two Verma modules

(cf. [J] and [RW]).

Theorem II (Theorem 4.3.12 and Proposition 5.2.1). Let q(A) be a GKM alge-bra associated to a symmetrizable GGCM A = (ajj)jjei satisfying the condition

that an ¿0 (i £ I). Let a = w(af £ W ■ 77"" with w £ IV and a, £ 77"",and let X £ b* be such that 2(X + p\a) = (a\a). Then the quotient module

N(X) := V(X)/V(X- a) has a Q(A)-module filtration

N(X) = N(X)o D N(X)X 3 N(X)2 D • • •

such that

(1) N(X)/N(X)X^L(X) asa g(A)-module,(2) the following holds:

y chN(x)¡ =y y Chv(x-iß)!>1 ß€A+ /ez>,

2(k+p\ß)=l{ß\ß)

-¿2 ¿2 chV(X-a-my)-a(X)chV(X-a),yeJ+ m€Z> i

2(k-a+p\y)=m(y\y)

where a(X) £ Z, the roots ß £ A+ and y £ A+ are taken with their

multiplicities.

In particular, in the case where X = w(A+p)-p for A £ P+ with (A\otj) = 0,

the constant a(X) above is equal to 1.

The proof of Jantzen's character sum formula for symmetrizable Kac-Moodyalgebras by Rocha-Caridi and Wallach [RW] can be adapted with only minor


GENERALIZED KAC-MOODY ALGEBRAS 3893

modifications to the case of symmetrizable GKM algebras g(A) if we assume

that the GGCM A = (Oij)i,Mi satisfies the condition that a„ ^¿ 0 (i £ I). Sowe shall only sketch the proof, pointing out where the modifications are needed.

This paper is organized as follows. In §1 we recall some elementary facts

about GKM algebras and the notation used in [N]. In §2 we review the notions

of the extended Bruhat ordering on W x sé and the extended Kazhdan-Lusztig

polynomials for W x sé introduced in [N]. In addition, we define the extended

inverse Kazhdan-Lusztig polynomials for W x sé . In §3 , following [K] and

[DGK], we give a brief account of fundamentals of the representation theory of

GKM algebras that we need in this paper. In §4 we study two kinds of Jantzen's

character sum formulas for symmetrizable GKM algebras. One correspondsto a single Verma module, and is proved more generally for symmetrizable

contragredient Lie algebras by Kac and Kazhdan [KK]. The other corresponds

to a quotient of two Verma modules, and plays an essential role in the proof of

the Kazhdan-Lusztig conjecture (Theorems 6.1.2 and 6.2.2).

In §5, using the former character sum formula, we show that, for A £ P+ ,

w £ W , and a¡ £ sé (A), the multiplicity [V((w , 0) o A) : L((w , af o A)] is

equal to Pw,w(l) = 1 under the assumption that a„ ^ 0 (i £ I). We then use

this result to obtain further information (= the second assertion of Theorem II)

about the latter character sum formula. In §6 we establish our main theorem

(Theorem I) stated above.

I would like to express my sincere thanks to Professor Kiyokazu Suto for

many valuable discussions and helpful suggestions.

1. Preliminaries and notation

In this section, we recall some elementary facts about generalized Kac-Moody

algebras from [Bl] and [K, Chapter 11], and also fix notation.

1.1. Notation. Since this paper is a continuation of [N], we follow its notation.

Throughout this paper, all the vector spaces are over the field C of complex

numbers, unless otherwise specifically stated. For a finite-dimensional vector

space V, we denote by (• , •) a duality pairing between V and its algebraic

dual F*:=Homc(F,C).For a Lie algebra a, U(d) denotes its universal enveloping algebra.

We denote by R the field of real numbers, and by Z the ring of rational

integers. For an integer aaa £ Z, we put

Z>m := {ac £ Z | k > aaa} , Z>m := {ac e Z | k > m}.

1.2. Generalized Kac-Moody algebras. Let / = {1,2,...,aa} be a finite

index set, and let A = (a,^),-je/ be a real nxn matrix satisfying the following

conditions:

(Cl) either a„ = 2 or a¡¡ < 0 for i £ I ;(C2) ai} < 0 if I# j, and fly £ Z for ; / i if au = 2 ;(C3) aij = 0&aji = 0.

We call such a matrix a GGCM (= generalized generalized Cartan matrix).

For any GGCM A = (fly)f,ye/j we have a triple (b, 77 = {q;},€/, 77v =

{q7},6/) satisfying the following (see [K, Chapter 1]):

(RI) b is a finite-dimensional (complex) vector space such that dimc f) =

2aa - rank A ;


3894 SATOSHI NAITO

(R2) n = {a,},e/ c b* is linearly independent, and 77v = {af}ie¡ c b islinearly independent, where b* := Homc(f), C) ;

(R3) (ctj, of) = a¡j (i, j £ I), where (• , •) denotes a duality pairing be-

tween b and b* .

The above triple is called a realization of A .

From now on, we assume that the GGCM A = (flyO/jg/ 1S symmetrizable,

i.e., there exists a diagonal matrix D := diag(ei, e2, ... , e„) such that detD ^

0 and D~XA is symmetric. Then we may and do assume that e, > 0 for alli£l.

The generalized Kac-Moody algebra (= GKM algebra) q(A) associated to a

symmetrizable GGCM A = (tttj)tjèj is the Lie algebra (over C) generatedby the above vector space b and the elements e¡, f (i £ I) satisfying the

following relations (see [Bl], [K, Chapter 11]):

{[h, h'] = 0 for h, h' e b,

[h, e¡] = (a,, h)e¡, [h, f] = -(a¡, h) fi for h £ b, i£l,

[e¡, fj] = ôija) for i,j£l,

(F2) (adei)x-a»ef = Q, (ad f})l~a'J fj = 0 if a,, = 2 and f ¿ i,

(F3) [ei,ej] = 0, [f,ff] = 0 if a,,, aj}< 0 and ai}¡ = 0,

Then we have the root space decomposition of g(A) with respect to the Carian

subalgebra b :

a(A) = b © E® 0« © E® 0« 'a€¿I+ a€d-

where zl+ (c ß+ := X},e/Z>0a:/) is the set of positive roots, ¿f_ (= —A+)is the set of negative roots, and gQ is the root space corresponding to a root

a £ A = A+ U zL C b*. Note that gQj = Ce¡, a_a, = Cfi for / e /, and that

mult(a) := dimc fla = dimc fl-a < +oo for a £ A+ .

Put n+ := ¿Zîej+ fla , n_ := £«6j+ ß_a , and b := h © n+ .

1.3. Imaginary simple roots. We put Ve := {i £ I \ an = 2}, /"" := {/ e

1 I <2„ < 0}, and nre := {a, e II \ i e Ire} the set of real simple roots,

nim := {a, £ n \i £ I'm} the set of imaginary simple roots.

For a,, a7 e 77"" , we say that a, is perpendicular to af if a,7 = 0. (Remark

that an imaginary simple root a, e 77"" is perpendicular to itself if a„ = 0.)

For X £ b* and a¡ £ 77"" , we say that a, is perpendicular to X if (X, af) = 0.

Now fix an element A £ P+ := {X £ b* \ (X, af) > 0 (a e /), and (A, of) £Z>o if a« = 2}. Then we define a subset J^(¿1) (resp. sé (A) ) of h* to be theset of all sums of distinct (resp. not necessarily distinct), pairwise perpendicular,

imaginary simple roots perpendicular to A . Note that sé :=sé(0) contains the

set {0} U 77"" U {maj \ m £ Z>2, o; £ 77"" with a¡f = 0} by definition, whilemaf (aaa > 2) do not belong to 5? := ^(0). For an element ß = 2~^¿g/«n kta¡

(ac, £ Z>o), we put ht(j8) = S¿6/,m /c(, and call it the height of ß .

1.4. Weyl group, real roots, and imaginary roots. For i £ Ve, let r¡ be the

simple reflection of h* given by: r¡(X) = X - (X, ct^a, (X £ h*). The Weylgroup W of q(A) is the subgroup of GL(b*) generated by the r, 's (i £ Ire).

Note that (W, {r¡ \ i £ Ire}) is a Coxeter system. For an element w £ W,

t(w) denotes the length of w .



Let Are := W • IIre (the set of real roots), and Aim := A \ Are (the set of

imaginary roots).

Here we recall the following characterization of positive imaginary roots: for

an element a = £¿6/Ac,a, £ Q+\{0} , we define supp(a) to be the subdiagram

of the Dynkin diagram of A = (a¡f)jj€i corresponding to the subset {i £ I \Ac, > 1} of /. Then we know that

AimnA+ = [J w(K),

where K := {a £ Q+\ {0} | (a, aY) < 0 (i £ I), and supp(a) is connected}.

In particular, the set Aim n A+ of positive imaginary roots is IP-stable.For a real root a = w(a¡) (w £ IV, a¡ £ nre), we define the reflection ra

of h* with respect to a by: ra(X) = X - (X, av)a (X £ b*), where av :=

w(aY) e h is the dual real root of a. Note that ra = wr¡w~x £ IV.

1.5. Invariant bilinear forms. Since we have been assuming that the GGCM

A = (a¡j)ijei is symmetrizable, there exists a nondegenerate, symmetric, in-

variant bilinear form (•!•) on q(A) . Note that the restriction of this bilinearform (•!•) to the Cartan subalgebra b is also nondegenerate, so that it induces

(through a linear isomorphism v: b —* b* ) on h* a nondegenerate, symmetric,

H^-invariant bilinear form, which we again denote by (-|-). Moreover, we have

v(af) = e,a, (1<a'<aa), (a,|o/) = e"1 • ai} (I < i, j < n).

We remark that a root a £ A is an imaginary root if and only if (aja) < 0.

1.6. Verma modules over GKM algebras and their irreducible quotients. For

X £ b*, we denote by V(X) the Verma module U(g(A)) <8>u(b) C(A) with highestweight X over the GKM algebra q(A) . Here C(X) is the one-dimensional b-

module with weight X, on which n+ acts trivially. As is well-known, the Verma

module V(X) contains a unique maximal proper g(^)-submodule V'(X). We

define L(X) to be the quotient a(^)-module V(X)/V'(X), and hence L(X) is

the irreducible highest weight g(y4)-module with highest weight X.

2. BRUHAT ORDERING AND KAZHDAN-LUSZTIG POLYNOMIALS

2.1. Extended Bruhat ordering. Here we recall the notion of the Bruhat or-

dering on the direct product IV x sé of the Weyl group W and sé = sé(0)

introduced in [N].

Definition 2.1.1 (Bruhat ordering). Let wx, w;2 £ W. We write Wi <— u;2 ifthere exists some y £ Are n A+ such that wx = ryW2 and l(wx) = ¿(wf) + 1.

Moreover, for w, w' £ W, we write w ^ w' if w = w' or if there exist

wx, ... ,wt£lV such that

w^wk<--,r-wtfrrw\

Definition 2.1.2 ([N, Definition 2.3]). Let ßx, ß2 £ sé . We write ßx *- ß2if there exists some af £ 77"" such that ßx = ß2 + a¡. Moreover, for ß =

T,k€iim mk<*k. ß' = T,ice¡"" m'kak € sé , we write ß ^ ß' if mk > m'k for all


3896 SATOSHI NAITO

Definition 2.1.3 ([N, Definition 2.4]). For (wx, ßx), (w2, ß2) £ W x sé , we

write (wx, ßx) «- (w2, ß2)

if wx <— W2 and ßx = ß2 or if wx = w2 and ßx <— ß2.

Moreover, for (w, ß), (w', ß') £ IV x sé , we write (w, ß) ^ (w', ß') if

w ^ w' and ß > /?'.

2.2. Extended Kazhdan-Lusztig polynomials. Let W be the Weyl group of

the GKM algebra q(A) . Recall that (W, {r¡ | i £ Ire}) is a Coxeter system.

Then the Hecke algebra %?(W) of W is the associative algebra over the Lau-

rent polynomial ring Z[q¿, q~^] (in the indeterminate q? ) which has a free

Z[qi, q~^]-hasis {Tw}w€w with the following relations:

(HI) TWTW, = TWW, if l(ww') = l(w) + l(w') (w,w'£W);

(H2) (Tn + l)(Tn - q) = 0 (i £ Ire) ■

Let i be the involutive automorphism of the ring %?(W) defined by: i(q^) =

<7_i, i(Tw) = (Tw-i)~x (w £ IV). Then we know the following proposition

due to Kazhdan and Lusztig [KL1].

Proposition 2.2.1 ([KL1, Theorem 1.1]). For each w £ IV, there exists a unique

element CW£%?(W) having the following properties:

(1) i(Cw) = Cw;

(2) Cw = (-l)tlwW-¥'2Zy<w(-lY{y)<l-tiy)i(Py.v>(<l))Ty, where Pw,w(q) =

1, and Py<w(q) £ Z[q] is a polynomial with integer coefficients in the

indeterminate q of degree < (l/2)-(l(w)-i(y)-l) for y ^w , y j^w .

Moreover, the elements Cw (w £ IV) form a free Z[q$, q~¿]-basis of ^(W).

The above polynomials Py,w(q) £ Z[#] (y < w) are called the Kazhdan-

Lusztig polynomials. We set PyyW(q) := 0 unless y < w .

Now, for ß, ß' £ sé = sé(0), we define a polynomial Pß:ß'(q) in q by

(I if ß'Zß,

PßM9)-=\0 otherwise.

Furthermore, for (w, ß), (w', ß') e W x sé , we put

P(w,ß),(W',ß')(q):=Pw,wi(q)-Pß,ß'(Q)-

We call the polynomials P(W,ß),{W',ß')(Q) ((™, ß), (W, ß1) £ W x sé) the

extended Kazhdan-Lusztig polynomials for IV x sé .

It is also known (cf. [KL2], [KT]) that there exist the inverse Kazhdan-Lusztig

polynomials Qw<y(q) (w ^y£ W) for the Coxeter system (W, {r, | /' £ Ve}),

such that

Y. (-^)e(y)~e{w)Qw,y(Q)Py,w'(g) = Sw,W' (w^w').

w^.y^W

We set Qw,y(q) '•= 0 unless w < y.

We put Qß,ß>(q) := Pß,ß>(q) for ß, ß' £ sé , and then

Q(w,ß),(w',ß')(q) ■= Qw,w(q) • Qß,ß'(q)



for (w , ß), (w', ß') £ W x sé . We call the polynomials Q(W,ß),(w',ß')(q)

((w, ß), (w', ß') £ IV xsé) the extended inverse Kazhdan-Lusztig polynomi-

als for W xsé .

3. Basic representation theory of GKM algebras

In this section, we briefly review some elementary notions and results of the

representation theory of GKM algebras.

3.1. Category cf. Before defining the category cf, we introduce a partial

ordering on h* by:

X >p (X, p £ b*) if X-p £ Q+ = ^Z>0a,-.i€l

Now we define the category cf of g(^)-modules as follows. The objects of

cf are g(^)-modules V satisfying the following (see [K, Chapter 9]):

(1) V admits a weight space decomposition

v = Y?vT

with finite-dimensional weight spaces Vx ;

(2) there exists a finite subset {Xx, ... , Xs} of h* such that the set P(V)

of all weights of V is contained in a finite union (js¡=x D(X¡), where

D(Xi) := {X £ b* | X < Xi} (1 < i < s).

The morphisms in cf are a(^4)-module homomorphisms.

We note that the category cf is closed under the operations of taking sub-

modules, quotients, finite direct sums, and (finite) tensor products.

Obviously, highest weight g(;4)-modules, such as V(X) and L(X) (X £ h*),are in the category cf.

3.2. Formal characters and multiplicities. Here, from [K, Chapter 9] and

[DGK], we recall the definitions of the formal character of a module V in

the category cf and the multiplicity [V : L(p)] of L(p) in V for p £ b*.First we define the algebra I? as follows: the elements of I? are series of

the form 2~^T€Xr cre(r), where c,eC and cT = 0 for x outside a finite union

of sets of the form D(X) = X - Q+ (X £ b*). Here the elements e(t) (x £ h*)

(called formal exponentials) are linearly independent, and are in one-to-one

correspondence with the elements x of b*. The multiplication in f is defined

by e(xx) ' e(x2) := e(xx + x2) (xx, x2 £ b*). Thus, f becomes a commutative

associative algebra over C with the identity e(0).

For a module V = Y!%t,* *7 in the category cf, we define the formal char-

acter ch V of V by ch V := 2^Tefi*(^^mc *7MT) • Clearly, we have ch V £ %

for a module V in the category cf.

Furthermore, a family {fx = Z7e(). cxre(x)}xex of elements in 'S indexed

by an arbitrary set X is said to be summable if

(1) there exist Xx, ... , Xk £ b* such that, for each x £ X, cXT = 0 for x

outside \J*=xD(Xi),(2) for each x £ b*, cx% = 0 for all but finitely many x £ X .


3898 SATOSHI NAITO

In this case, we define J2xeXfx to be the element TffJXç.i].Ç2lx€xcxx)e(x) of W ,

and call it the saaaaa of the fx 's.

Then we know the following.

Proposition 3.2.1 ([DGK, Proposition 3.4]). For a module V in the category cf,

there exists a unique set {a^}^^ of nonnegative integers such that the following

equality holds in the algebra S :

aßchL(p).uet>'

Definition 3.2.2 ([DGK, Definition 3.5(b)]). The above integer aß is called themultiplicity of L(p) in V, and is denoted by [V : L(p)].

Remark 3.2.3. It is clear that the above definition of the multiplicity [V : L(p)]

is identical with the one in [N], or [K, Chapter 9].

We note that for a module V £ cf and p £ b*, [V : L(p)] ^ 0 if and onlyif L(p) is an irreducible subquotient of V.

3.3. Some module-theoretic results on GKM algebras. Here we collect some

results about the irreducible subquotients and embeddings of Verma modules

over symmetrizable GKM algebras established in [N], which we shall use fre-

quently later.We choose and fix an element p £b* such that (p, af) = (1/2) • a,, for all

a £ I, or equivalently, (p\a¡) = (1/2) • (a¡\a¡) for all i £ I. From now on, we

shall use the notation

(w , ß) o A := w(A + p- ß)- p

for (w, ß) £ IV x sé and A £ P+ .For the following results, the symmetrizability assumption on the GGCM A

is essential.

Theorem 3.3.1 ([N, Theorem 3.5]). Let A £ P+, (w, ß) £ IV x sé (A). 77aé>aaany irreducible subquotient of the Verma module V((w, ß) o A) is isomor-

phic to L((w', ß') o A) for some (w', ß') £ W x sé (A) with (w', ß') >

(w, ß). Conversely, for any (w1, ß') £ IV x sé (A) with (w', ß') ^ (w, ß),

L((w', ß') o A) is isomorphic to an irreducible subquotient of V((w, ß) o A).

Theorem 3.3.2 ([N, Theorem 3.7]). Fix A £ P+. Let (wx, ßx), (w2, ß2) £IV x sé (A). Then we have

dimcHomg{A)(V((wx,ßx)oA),V((w2,ß2)oA)) < 1.

Note that any nonzero g(v4)-module homomorphism between two Verma

modules is injective. So we may write

V((wx,ßx)oA)cV((w2,ß2)oA)

when the equality holds in the above theorem.

Theorem 3.3.3 ([N, Theorem 3.9]). Let A £ P+, (wx, ßx), (w2, ß2) £ W xsé (A). Then

V((wx,ßx)oA)cV((w2,ß2)oA)

«=> (wx, ßx)^(w2, ß2)

*=* [V((w2,ß2)°A):L((wx,ßx)oA)]?0.



4. Generalization of Jantzen's character sum formula

In this section, we obtain a generalization to GKM algebras of Jantzen's

character sum formula for a quotient of two Verma modules.

The original formula was proved by Jantzen [J] for finite-dimensional semi-

simple Lie algebras (over C), and was generalized to the case of symmetrizable

Kac-Moody algebras by Rocha-Caridi and Wallach [RW].We now generalize this formula to the case of symmetrizable GKM algebras

2(A) associated to a GGCM A = (a,7),)je/ satisfying the condition that a¡¡ / 0

(i £ I). (This condition seems essential to us.) As mentioned earlier, under

this condition, we may apply the proof (for the case of Kac-Moody algebras)

by Rocha-Caridi and Wallach to the case of GKM algebras by making minorchanges. So we shall give only a sketch of the proof, which, however, occupies

the whole of §4.3 .

4.1. Shapovalov form and its determinant. Here we recall the definition of

the Shapovalov form on the universal enveloping algebra C/(n_) of the Lie

subalgebra n_ = Yla&A 8-<* OI> fl(^) > ana the result of Kac and Kazhdan [KK]

about the determinant of the Shapovalov form restricted to each weight space

£/(n_)_, for n£Q+.Let q(A) be the GKM algebra associated to a symmetrizable GGCM A =

(aij)ijej. Then the universal enveloping algebra U(2(A)) of q(A) can be

written as U(2(A)) = C/(n_) ®c U(b) <8>c U(n+), corresponding to the triangular

decomposition: g(A) = n_ © h © n+ . In particular, we have the following

decomposition as a vector space:

U(2(A)) = i/(h) © (n-U(s(A)) + U(g(A))n+).

We denote by p: U(2(A)) —► U(b) the Harish-Chandra projection of U(2(A))onto U(b) parallel to the second summand of the above sum.

Then the Shapovalov form F(- , •) with values in U(b) on U(2(A)) (see [S])

is defined byF(x,y):=p(a(x)y) (x, y £ U(9(A))),

where a is the involutive anti-automorphism of U(g(A)) determined by cT(e¡)

= fi, o(fi) = e¡ (i £ I), and a (h) = h (h£b).Clearly, F(- , •) is a symmetric bilinear (i/(h)-valued) form, such that

F(zx, y) = F(x, o(z)y) (x, y, z £ U(g(A))).

In particular, we have

F(U(B(A))m , U(q(A))„2) = 0 if nx ¿ n2 £ Q = £ Za,,i€I

where U(g(A))n is the weight space of weight n £ Q (c b*) of U(g(A)) under

the (extended) adjoint action of b.

Furthermore, on the Verma module V(X) (X £ b*), we define a symmetric

bilinear (C-valued) form (• , -)a (see [J]) by

(giv2 , g2vk)k := X(F(gx, g2)) (gx, g2 £ U(q(A))) ,

where v2 = 1 ® 1 £ V(X) is the canonical generator. Here X £ b* is natu-

rally extended to U(b) (^C[h*])by: X(hxh2 ■ ■ ■ hk) := (X, hx)(X, h2) ■ ■■ (X, hk)(hx,h2,... ,hk£b).


3900 SATOSHI NAITO

One can easily check that the form (• , -)a is well-defined, and satisfies thefollowing properties:

(1) (gvi, v2)x = (vi, o(g)v2)k (g£ U(q(A)), vx,v2£ V(X));

(2) (V(X)T¡,V(X)T2)2 = 0 (xx¿x2£b*);

(3) the radical of (• , -)a is a unique maximal proper g(^)-submodule ofV(X).

This form (• , -)2 on the Verma module V(X) is also called a Shapovalov form.

For r\ £ Q+ , we denote by Fn the restriction of the Shapovalov form F to

the weight space C/(n_)_, := i/(n_) n U($(A))-n. Note that dimc U(n-)-v =K(n) < +00, where K(n) (n £ Q+) is the (generalized) Kostant partition

function (see [K, Chapter 10]).We know the following theorem, due to Kac and Kazhdan [KK].

Theorem 4.1.1 ([KK, Theorem 1]). We keep the above notation. Then, up to a

nonzero constant factor (depending on the basis of Í7(n_)_,), we have

OO fc~( tl_j' R \

det/7,= nn^'^+w»-^^)ß€J+j=l

where the roots ß £ A+ are taken with their multiplicities.

Remark 4.1.2. In [KK], the above theorem is proved in the more general setting

of contragredient Lie algebras.

4.2. Jantzen's character sum formula for a single Verma module. Here we

review the theory of Jantzen's character sum formula for a single Verma module

(cf. [J]). This was generalized to the case of contragredient Lie algebras by Kac

and Kazhdan [KK], by using Theorem 4.1.1 above. However, we prefer to use a

slightly different derivation of the filtration of a Verma module given by Rocha-

Caridi and Wallach, which may be thought of as a special case of the filtration

given in §4.3 (see [RW] and [W]).Let g(A) be a symmetrizable GKM algebra. Recall that the universal en-

veloping algebra £/(n_) of n_ is isomorphic to Verma modules V(X) (A eh*)

as a vector space via the map Í7(n_) —> V(X) given by y h-> y <g> 1 = yvk, where

v2 = 1 <8> 1 € V(X) is the canonical generator. Then, via the above map, we can

regard Í7(n_) asa g(,4)-module with an action of q(A) denoted by n2. (Here

we remark that for y e C/(n_)_, (n £ Q+) and h £ h, we have n2(h)y =

(X - n, h)y.) Moreover, again via the above map, the Shapovalov form (• , -)2can also be regarded as a bilinear form on C/(n_) (depending on A G h*), such

that (n2(g)x, y)2 = (x, nk(o(g))y)2 (g £ U(q(A)) , x, y £ C7(n_)).

Now we look upon C[r] ®c f (n_) = Y^€q+ C[a] <8>c ^(n-)-, as the space of

all polynomials in one variable / with values in U(n-). We fix an element C

of b* such that (Qn) # 0 for any tj€Q+\ {0} .Then we have the following.

Theorem 4.2.1 (see [J], [KK], [W]). We keep the above notation. Set V(X)¡ :=

{f(0)v2 £ V(X) | f£ C[t]®c t/(n_), (f(t),x)x+ir £ t¡C[t] for all y £ C/(n_)}for a > 0. Then the filtration

V(X) = V(X)o D V(X)X D V(X)2 D ■ ■ ■



defines a $(A)-module filtration of V(X) such that

(1) for any n e Q+, V(X)2-n n V(X)¡ = 0 for some i,(2) V(X)/V(X)X<=L(X) asa s(A)-module,(3) the following equality holds in the algebra I? :

YlchV(X)i=Y E chV(X-lß),É>1 ß<EJ+ Aez>,

2{X+p\ß)=l(ß\ß)

where the roots ß £ A+ are taken with their multiplicities.

The above filtration V(X) = V(X)0 D V(X)X D V(X)2 D ■■■ is calledJantzen's filtration of V(X), and the assertion (3) is called Jantzen's character

sum formula.

4.3. Generalization of Jantzen's character sum formula for a quotient of two

Verma modules. The aim of this subsection is to sketch the proof of a general-

ization (to GKM algebras) of Jantzen's character sum formula for a quotient of

two Verma modules (= the first assertion of Theorem II in the Introduction).

The proof is based on a number of lemmas below.

From now till the end of this paper, we assume that the GGCM A = (fl¿_/)¿,ye/

is symmetrizable, and satisfies the condition that a,, ^0 for i £ I. This

assumption is crucial for the proofs of some lemmas, and of course, for the

main result (Theorem 4.3.12) of this subsection.

First we recall the following useful lemma.

Lemma 4.3.1 (see [N, Lemma 3.3 and Remark 3.4]). Let p £ P+, w £ W, andy £ A+ . Then the following are equivalent:

(1) 2(w(p + p)\y) = m(y\y) for some m £ Z>x ;(2) we have either of the following two cases:

(a) y £ Are and i(ryw) > ¿(w),

(b) w~x(y) £ 77"" and (w~x(y)\p) = 0.

Moreover, in case (a), we have ryw ^ w and m = (w(p + p), yv). In case

(b), we have m = 1.

Now we introduce some notation, following [RW]. We set

T:= U (w,0)oP+,

and then, for each i £ Ire ,

r+:={X£r\ (X + p,af)>0} and r~ :={X£r\ (X + p^^KO}.

In addition, for a £ A and ac € Z>i , we put

b*a,k:={X£b*\2(X + p\a) = k(a\a)}.

If a £ W -nim (cA+nAim), we put

m(a) := min{£(w) | a = w(af (w £ W, j £ I"")}.

Here we note that wx(afl) = w2(a.j2) (wx, u>2 £ W, jx, j2 £ I"") impliesah = ah > since (ajx, af), (a72, af) < 0 (i £ Ire) (cf. the proof of [K, Propo-

sition 3.12]). We also note that, for a = w(aj) (w £ W, a¡ £ /7""), ka £ A


3902 SATOSHINAITO

if and only if Ac = ± 1, and dime fla = dime 0a, = 1, since the root system A

is invariant with their multiplicities under the action of the Weyl group W.

Remark 4.3.2. Let a £ W • 77"" , and let j £ Iim and w £ W be such thata = w(af with £(w) = m(a). Then A £ P+ n h* k implies that

k(aj\aj) = 2(A + p\aj) = 2(A\af + (e*;|a;),

so that

0 < 2(A\aj) = (k - I) • (afaf < 0.

Thus, P+nh* k is Zariski-dense in h* k if and only if Ac = 1, since (afaf <

0.The proof of the following lemma is essentially the same as that of [RW,

Lemma 3.2], except that we use Remark 4.3.2 above.

Lemma 4.3.3 (cf. [RW, Lemma 3.2]). If a £ W-nim is not an imaginary simple

root, then there exists some i £ Ire such that

(1) m(n(a)) < m(a),(2) Fj n h* j is Zariski-dense in b*a ,.

We now record (for later use) the following formula, which holds for an

arbitrary i £ I :

[et, ft] = kfï~x(ay - (k - ij - an/2) (i £ I).

Theorem 4.3.4 (cf. [S, Lemma 1] and [RW, Theorem 3.3]). Let a = £i€/ miai e

W ■ nim. Then there exists 0a>1 e C/(n_ © h)_a = £/(n_ © h) n U(a(A))-a

satisfying.

(1) {St, 0a,i] £ U(3(A))(u~x(a) + (p\a) - (1/2) • (a\a)) + U(0(A))n+ fori£i;

(2) da,i= fxm[ f2m2 ■ ■ ■ fnmn + ¿Z,aib,, where a, £ t/,(n_), b, £ U(b) with

p < Yliei mi ■ P^ere {Up(n-)}p>o « the canonical filtration of f/(n_).

Sketch of proof. The proof is by induction on m(a). If m(a) = 0, then ob-

viously a = a; for some j £ /"". In this case, we can clearly take f¡ as

®a¡, i •

Suppose that aaa(o:) > 1 . Then, by Lemma 4.3.3, there exists some i £ Ve

such that m(r¡(a)) < m(a) and such that T~ nf)* , is Zariski-dense in fj* ¡ .

Now, for each X £ /]~nh* [, we can construct an element Qa% X(X) £ f/(n_)_Q

satisfying the conditions i'), ii'), and iii') in the proof of [S, Lemma 1] with

B replaced by F~ n h* , (and La>i by h* ,) as follows.

We write X = (w, 0) o A for w £ W and A £ P+ . Put f := (r,, 0) o X =

(r¡w, 0) o A . Since X£b*aX, i.e.,

(q|q) = 2(X + p\a) = 2(w(A + p)\a),

we get a = w(ak) for some ak £ W" with (A\ak) = 0, by Lemma 4.3.1.

Furthermore, since X £ F~ , i.e.,

0 > (w(A + p), a)) = 2(w(A + p)\aI)/(ai\ai),



we get w ^ r¡w again by Lemma 4.3.1. So we have

X-a = w(A + p)- p- w(ak)

= w(A + p-ak)-p

= (w,ak)oA,

and

ip - n(a) = (rtw)(A + p)-p- nw(ak)

= (r¡w)(A + p-ak)- p

= (r¡w, ak)oA,

with w ^ r¡w .

Therefore, by Theorems 3.3.2 and 3.3.3, we have two series of inclusions:

V{X-a)^V(X)^V(y/),

V{X-a)^V(ti/-ri(a))^V(y/),

(where the corresponding composite maps V(X - a) <-► V(tp) are identical.

From this, using the induction hypothesis (since m(r¡(a)) < m(a)), we obtain

the desired element 0Q> X(X) £ c7(n_)_Q, as in the proof of [S, Lemma 1] (we

take s = a¡).

Now the rest of the argument in the proof of [S, Lemma 1] goes through

without any serious change. D

Let X- n- —* C ,be the linear map defined by x(f) — 1 (/ € /), andX([n- , n_]) = 0. We extend / to the algebra homomorphism from £/(n_) to

C, which is again denoted by x ■For X £ b*, let (• , -)2 be the Shapovalov form on the Verma module V(X),

which we also regard as a bilinear form on the universal enveloping algebra

Í7(n_),asin §4.2.Let 31 := {X £ b* \ V(X) is irreducible} . Then, by results of Kac and Kazh-

dan [KK], we have

31 = {X £ b* | 2(X + p\a) # k(a\a) for any a £ A+ and Ac e Z>i}.

Before proceeding further, we prepare the following notation: for the Verma

module V(X) = £® ,,. V(X)X, we put V(X)[n_] := nT€h* V(^ (direct product

of weight spaces). Then the h-module structure on the vector space V(X)[n_]

can be extended in the natural way to a g(^)-module structure.

The proofs of the next Lemmas 4.3.5 and 4.3.6 are essentially the same as

those of [GW, Lemma 4.1] and [GW, Corollary 4.3], respectively. However,

we give some detail for the convenience of the reader, since our setting differs

considerably from that of [GW].

Lemma 4.3.5 (cf. [GW, Lemma 4.1] and [RW, Lemma 3.5]). For each p £ Q+ ,there exists a rational function X i-> n-ß(X) from b* into U(n-)-ß with the

following properties:

(1) n0(X) = l;

(2) if X is in 3, then n-ß is defined at X, and we have (y, n-ß(X))2 = /(y)

for y £ U(n-)-p\


3904 SATOSHINAITO

(3) for i £ I, there is a rational function X h-> n'_ß ¡(X) from b* into

U(xi-)-n+a¡, whose singularity set is contained in that of n-ß, such that

for X£3,

[ei,n-ß(X)\ = n-ß+ai(X) + n'_ß<i(X)(av - (X, <#)).

Sketch of proof. If V(X) is irreducible, then the Shapovalov form (• , -)x on

V(X) is clearly nondegenerate. Moreover, we can naturally extend (• , -)x to a

pairing (again denoted by (• , -)x ) of f7(n_) with V(X\n_], such that (yX, v)x

= (X,a(y)v)x (X,y£U(n_), v£V(X)[n_}).In this case, the map v h-> (., v)x defines a linear isomorphism </>: V(X\n_x- —►

Homc(i7(n_), C). Since x is an element of Homc(í/(n_), C), there exists

a unique element v0 £ V(X)[n_] such that 4>(v0) = X, ie., (y, v0)x = x(y)(y £ U(x\-)). Here we have, for x £ U(n+), y £ c7(n_),

(y, xv0)x = (a(x)y, v0)x = x(o(x)y) = x(o(x))x(y)

= x(o(x))(y, vo)i = (y, x(o(x))vQ)k,

so that xvq = x(a(x))vo ■Because v0 is an element of V(X\n_i-, we can write

v0= J2 n-ß(X)vx,peQ+

where v2 is the canonical generator of V(X), and for p £ Q+, n-ß(X) £

U(n-)-ß is uniquely determined by the property

(y,n-ß(X))x = x(y) (y£U(n-).ß).

To show that ñ-M(X) depends rationally on A, we define a positive definite

inner product (• , •) on t/(n_) in a natural way, using a basis of f/(n_) con-

structed from root vectors by the Poincaré-Birkhoff-Witt theorem. Then, for

p = J2i€¡ m¡ai £ Q+ , there exists a polynomial map X >-> xß(X) from b* into

Endc(í7(n_)_/Í) such that

(y,y')x = (y, V(A)y') (y,y'£ U(n-).ß).

If X £ 3, i.e., V(X) is irreducible, then (• , -)x is nondegenerate on U(r\-)-ß ,

so detr^iA) ^ 0 for all p £ Q+. Furthermore, in this case, we see from the

above argument that n-ß(X) = xM(X)~X fxm> f2m2 ■ ■ ■ f™".

Finally, the assertion (3) follows from the fact that xv0 = x(o~(x))v0 (x £n+) by the same argument as in the proof of [GW, Lemma 4.1]. Thus, we have

completed the proof of the lemma. D

In order to derive a formula for aa_/j(A) , we prepare some more notation.

For p = Y,iei miai e Q+ \ {0} , let ht(p) := £j€i m, and set m := ht(p). Let

3(p) := {(ii, ... , im) | 1 < ij < n and Y!J=X ai¡ = ß} ■ For J 6 J(p) and Ac

(1 < Ac < w), we set ph := Y!f=k <*i, = ß - ¿ö ah > fj '■= A •'//«» andej := e¡m ■ ■ ■ eix . Define, for A e b*, p £ Q+\ {0}, and J £ l(p),

m

Pj(X, p) :=W(2(X + p) -pjk\pJk).k=\



In addition, we set

3' := {X £ b* | Lß(X) ¿ 0 for all p £ Q+ \ {0}},

where LU(X) := (2(X + p) - p\p). (Note that 3P c 3 .)Then we have the following.

Lemma 4.3.6 (cf. [GW, Corollary 4.3], [RW, Lemma 3.6]). Let p = Y,ieI AW,a,£ Q+\ {0}, and suppose that X £ 3'. Then we have

n-ß(X) = 2h^- ¿2 Pj(*,fi)-lfj.

JeJ(fi)

Sketch of proof. Since A £ 3' c 3!, the Verma module V(X) is irreducible. In

the proof of Lemma 4.3.5, we have shown that the element Vq = 2~2ßea+ n-ß(X)vx

£ V(X)[n_] satisfies xvq = x(°~(x))vo for x £ n+ . Therefore, it is obviousthat the vectors n-ß(X)vx (p £ Q+) are annihilated by the derived subalgebra

[n+ , n+] of n+ .

Here we recall from [K, Chapter 2] that the (generalized) Casimir operator

Q is given by

Q = 2u-X(p) + Y,uiui+ EEW'i aëJ+ i

where {«,}, and {«'}, are dual bases of h with respect to (•!•), and for a £ A+ ,

{e(a}i and {ei^}, are bases of Qa and g-a , respectively, such that (ei'V-a) =

Let v £ V(X)x_ß be an [n+, n+]-invariant for p £ Q+ . Then, clearly we have

ei''v = 0 for a £ A+ \ 77. Therefore, from the fact that the Casimir operator

Q acts on V(X) by the scalar (A + p\X + p) - (p\p), we can easily deduce that

Lß(X)v = '52ieI2fjeiV. Then, by the same argument as in the proof of [GW,

Lemma 4.2], we obtain

v = c(p). J2 PA^ß)-lfjVx

for some constant c(p) £ C. Furthermore, the constant c(p) is determined by

2ht^exm'---e^v = c(p)vx.

Now we can apply this argument to the vector n-ß(X)vx, since the vector

n-ß(X)vx £ V(X)X-ß is an [n+, n+]-invariant, as remaked above. In this case,

we can show that c(p) = 2ht(^ , arguing as in the proof of [GW, Corollary 4.3].

This completes the proof of the lemma. D

For notational simplicity, we put

Z>i • A+ := {y £ Q+ \ y = fa for some j £ Z>x and a £ A+}.

Set for ^(MIO}',

*„:= nyez>rJ+

y<ß


3906 SATOSHI NAITO

Lemma 4.3.7 (cf. [GW, Lemma 4.4] and [RW, Lemma 3.7]). Let p £ Q+ \ {0} .Then the function S-ß(X) := Rß(X)n-ß(X) extends to a polynomial map from b*

into U(n_)-ß.

Sketch of proof. We put for p £ Q+ \ {0} ,

Pß := {A € h* | Lß(X) = 0}.

First, we show that, for 0 ^ p £ Q+ \ Z> ] • A+ and y £ Z> i • A+, we havePß^ Py. Here it is obvious that for p £ Q+\ {0} ,

Pß = Ker(p\-) + (l/2)p-p,

where Ker(p\-) := {A £ b* \ (p\X) = 0} .Suppose that Pß = PJa for some 0 ^ p £ Q+\Z>X -A+ and j £ Z>i, a £ A+ .

Then we have that Ker(p\-) = Ker(v'a|-) = Ker(a|-) and that p-ja £ Ker(a|-).

Because (-]•) is nondegenerate on b*, Ker(//|-) = Ker(a|-) implies p = qa for

some positive rational number q £ Q.

In the case where (a\a) ^ 0, we have

0 = (a\p - ja) = (q - j)(a\a),

and hence q = j. This is a contradiction, since p <fc Z>x • A+ .In the case where (a\a) = 0, we have

a£A+f]Aim = II w(K),

w€W

where K is as in §1.4. So we have a = w(x) for some w £ W, x £ K. Here

note that p = qa = w(qx) £ Q+\{0} , and that q is a positive rational number.

Then it is obvious that qx £ Q+ \ {0} , (qx, aV) > 0 (/ € Ire), and supper)

is connected. In addition, we have qx £ [jj>2 J ' H"" > since we have assumed

that an ^0 (i £ I). Thus, we get qx £ K, so that p itself is a positive

imaginary root. This is a contradiction, since p £ Z>x • A+ .

From the above remark, we see immediately that, for 0 ^ p £ Q+ \ Z> i • A+

and y £ Z>x • A+ , Pß n Py is a closed, nowhere-dense subset of Pß . Hence

PßC\3 is dense in Pß for 0 ^ p £ Q+ \ Z> t • A+ , because

3 = {X £ b* | V(X) is irreducible} = h* \ |J Py.y€Z>x-A+

Now the rest of the argument in the proof of [GW, Lemma 4.4] carries over

unchanged, by using Lemma 4.3.6 instead of [GW, Corollary 4.3]. D

We need a few more lemmas for the main result of this subsection. For

a £ W-nim , let 0Q> i £ f/(n_ @b)-a be the element in Theorem 4.3.4. Looking

upon £/(n_ © b) as canonicafiy equal to Í7(n_) ®c U(b) , we define for X £b*,

0Q,i(A):=(id®A)(r7a,,)ec7(n_)_a.

It is clear that

ea,iVX = 6aA(X)vx, and x(0a,i(X)) = l (X£b*)-

We put, for p £ Q+ ,

qß(X) := x(S-ß(X)) (X e h*).

Then, as an obvious consequence of Lemma 4.3.5, we have the following.



Lemma 4.3.8 (cf. [RW, Lemma 3.8]). Let p£Q+\{0}.

(1) Ify£ C/(n_)_„, then (S-ß(X),y)x = x(y)Rß(X) for leí,*.(2) (S-ß(X), S^ß(X))x = qß(X)Rß(X) for X £ fr* .

Now, for a £ W • 77,m , we set

VaA:={X£ b*aJ | Ljfi(X) ¿ 0 for ß £ A+ , ; 6 Z>, such that jß<a, jß ¿ a),

and

(h; ,)c := {¿ e h*,, I Ly(X) ¿ 0 for y 6 Q+ \ {0} such that y * a}.

Remark 4.3.9. From the assumption that a,, ^0 (i £ I), it immediately

follows that VaX is Zariski-dense in b*a , and that (h* x)c is not empty.

Lemma 4.3.10 (cf. [RW, Lemma 3.9]). Let a £ W • 77"" .

(1) IfX£b*a x,then S-a(X) = qa(X)6aA(X).

(2) IfX£VaA,then qa(X)¿0.

Sketch of proof. For the assertion (1), we first note that if A £ Va,x , then by

[KK, Proposition 4.1(a)],

dimcHomflM)(F(A-a), V(X)) = 1.

(Recall that dimc Qa = 1 for a £ W • 77 .) Then the rest of the argument used

to prove [RW, Lemma 3.9 (1)] applies to our case.

For the assertion (2), we take an element Ç £b* such that (Qn) ^ 0 for any

*! £ Q+\ {0} , instead of p £ b* in the proof of [RW, Lemma 3.9]. Then theproof is exactly the same as that of [RW, Lemma 3.9 (2)]. D

Finally, we come to the main part of this subsection. Let a = Y^iei m¡ai e

W • 77"" , A £ b*a , , and let 0Q,i € £/(n_ © h)_„ be the element in Theorem

4.3.4. Then it easily follows that 6a t x (X)vx = 6a, i vx £ V(X) is a nonzero highest

weight vector of weight A - a, where vx is the canonical generator of V(X).

Hence we get an embedding

T(X):V(X-a)^V(X),

where T(X)vx_a = 6atX(X)vx with vx_a the canonical generator of V(X - a).

Now we set

N(X):= V(X)/T(X)V(X-a).

We fix root vectors fn+x, fn+2, ... £ n_ = £®e¿)+ fl_a so that fx,f2,... is

a basis of n_ by root vectors. We set da,x := fxmi ■■■ f/f". Then, for each

p £ Q+ , we can construct a complementary subspace V-ß to f/(n_)_M+a • 8a¡ x

in U(ri-)-ß , such that for all £ 6 b*,

V_ß © (U(n-)-ß+a - 0Q,i(í)) = Uin-j-jt

as a vector space (see [S, §3]). Now we set V := Y^%q+ V-ß ■

Then, for ¿; 6 b*a , , V is isomorphic to N(£) as a vector space via the map

n given byy/ç(Y) = Yvi + T(()V(Z-a) (Y £ V).


3908 SATOSHINAITO

Therefore, via the above map ipt, we may look upon V as a g(^)-module

with an action n( of g(A). (Here we remark that, for Y £ V-ß (p £ Q+) and

h £ b, we have n^(h)Y = (Ç - p, h)Y.)On the other hand, the Shapovalov form (• , •),* induces a contravariant

bilinear form (again denoted by (• , •){ ) on N(Ç) = V(Ç)/T(Ç)V(Ç - a), since

T(£,)V(S, - a) is obviously contained in the radical of the Shapovalov form

on V(Ç). So, pulling back this form (• , •),» on 7V(<^) by ^ , we can define

a symmetric, 7t¿-contravariant bilinear form (•!•){ on V (depending on £, £

K,i)-Later, we shall need the following lemma.

Lemma 4.3.11. Let cj e (h* i)c- Then the form (-|-)í on V is nondegenerate.

Proof. We show that (• , •){ on N(£,) is nondegenerate. This is equivalent to

the irreducibility of N(Ç), since 7V(¿) is a highest weight a(^)-module with

highest weight £. Hence it suffices to show that if, for some n £ Q+, there

exists a nonzero highest weight vector v £ N(Ç)ç_n , then n = 0.Suppose that n ^ 0. Then it readily follows from the definition of the

Casimir operator Q and the fact that Q acts on N(Ç) by the scalar (Ç+p\Ç+p)

- (P\P) that

Cv = ((Í + p\S + p) - (p\p))v

= ((i - n + p\Ç - n + p) - (p\p))v.

So we get 2(£ + p\n) = (n\r¡). Therefore, from the assumption of the lemma, it

follows that n = a.Thus we obtain a nonzero weight vector v £ V(£,)^_a of weight ¿; - a such

that v i T(Ç)V(Z - a) and n+v £ T(Ç)V(£ - a). Now it is obvious thatn+v = 0. So we have got two linearly independent embeddings of V(Ç-a) into

V(£). This is a contradiction, since we have dimcHomB(^)(F(¿;-a), V(£)) = 1

by [KK, Proposition 4.1(a)]. This proves the lemma. D

We fix an element a = w(a.j) (w £ W ,a¡ £ 77"") of W • 77"" , and

an element A e h* , . Let co £ b* be an element such that (œ\a) = 0 and

(u)\ß)*0 (ß£A;\{a}).Now we give a construction of the desired filtration of N(X) = V(X)/

T(X)V(X - a) by g(¿1)-submodules. We look upon C[t] ®c V = ¿®€ß+ C[í] <8>c

V-ß as the direct sum of the spaces of all polynomials in one variable t with

values in V-ß (p £ Q+). Set, for each i £ Z>o ,

N(X)t := {^(/(0)) £ N(X) | / £ C[i] ®c V, (f(t)\v)x+lw £ t'C[t] for all v £ V}.

Then we get a filtration (called Jantzen's filtration)

N(X) = N(X)0dN(X)xdN(X)2?---

of N(X) by g(v4)-submodules. Here N(X)/N(X)X is isomorphic to L(X) asa g(^)-module. Furthermore, the following theorem (= the first assertion of

Theorem II) holds.

Theorem 4.3.12 (cf. [RW, Theorem 3.10]). Let q(A) be a GKM algebra asso-ciated to a symmetrizable GGCM A = (ay/ï./g/ satisfying the condition that



Oft ¿0 (i £ I). Let a = w(af £ W • 77"" with w £ W and a¡ £ 77"", andlet X £ b*a x. Then, for the filtration

N(X) = N(X)oDN(X)xDN(X)2D---

of N(X) given above, the following equality holds in the algebra % :

(4.3.1)£ chN(X)i= J2 E ^hV(X-lß);>l ß<EA+ /€Z>,

Iß

- y y^ ch V(X -a- my) - a(X) ch V(X - a),y£A+ meZ>i

Lmy{X-a)=Q

where a(X) £ Z, the roots ß £ A+ and y £ A+ are taken with their multiplicities.

Sketch of proof. Let p £ Q+ . Fix a basis {vx, ... , vd } of V-ß (c U(ri-)-ß).We set, for cj e b* , ,

' * 'a,1 '

P>i-ß(i) ■= det(<w,|^)5)i<i;7<^ = det((tA/, Uy)i)i<i,/<«/,-

It is clear that X + tea £ b*a x for all t £ C. We note that if A + t0co

e (K \Y (^ 0) f°T some ¿o £ C, then the form (-|-)/i-r,ocü is nondegenerate

on F = £®eQ+F_„ by Lemma 4-3-U' and so D'iM^-p(l + (o«) =

det((v,|t;J)/i+íou))1<,i7<^ 9¿ 0 for all /a € Q+. Since such t0 £ C clearly ex-

ists, the polynomial map 7)^+to_/J(A + tco) in í is not identically equal to

zero. Hence, for each p £ Q+ , there exists some e = e(p) > 0 such that

D'k+tco-p(x + tco)^0 for all teC with 0 < |r| < e .

Then, using [J, Lemma 5.1] applied to C[r] ®c V-M, we can easily deduce

that

ordo(D'l+ta_ll(X + tco)) = £ dimc(N(X)i n N(X)x_ß),'-'

where ordo(/(i)) (f(t) /0e C[a]) denotes the largest power of t dividing

/(/). Thus, we obtain

(4.3.2) j; chN(X)i = e(X) • E ord0(Z>l+icu_„(A + tco))e(-p).¡>\ peQ+

Hence we have only to compute ordo(D'l+ta_ (X + tco)) for p £ Q+ .

By the same argument as in the proof of [RW, Theorem 3.10], using Theorem

4.1.1, we obtain

OTd0(D'x+ta_li(X + tœ))

M,« =EE^-//?)-E E K(p-a-my)(4.J.JJ ßcJ+ l^z>¡ y€¿+ meZ>i

Llß(k)=0 Lmy{X-a)=0

+ K(p - a) • ordo(<?a(A + tea)) - K(p - a) • ordo(Ra(X + tco)),

where the roots ß £ A+ and y £ A+ are taken with their multiplicities, and

K(h) (n £ Q+) is the generalized Kostant partition function. (We have also

used the assumption that a„ ^ 0 (a G /) in this computation.)


3910 SATOSHI NAITO

Here we should mention the following misprints in the proof of [RW, The-

orem 3.10]: the determinant Dí^ka_fl(^ - ka) in the equation (3.12) should

be replaced by D^_ka)-^M^ka-¡(^ - ka), and similarly, Dv_ka_fl(u - ka) in the

equation (3.13) should be replaced by D(V_ka)_^_ka)(v - ka) (in our case,

AC = 1) .

Now we recall that for x £ b*,

(4.3.4) ch V(x) = e(x) ■ £ K(n)e(-n).

Then, from the identities (4.3.2), (4.3.3), and (4.3.4), we can easily derive the

desired formula (4.3.1) in the algebra S'. This completes the proof of thetheorem. G

The formula (4.3.1) in Theorem 4.3.12 is called Jantzen's character sum for-

mula.

5. Some further information about the constant a(X)

Let A £ P+, w £ W, and a¡ £ Wm with (A\af) = 0. We put A :=w(A + p) - p £ b*, a:= w(af) £ W ■ //"" c A+ . Then we have

2(A + p\a) - (a\a) = 2(w(A + p)\w(aj)) - (w(aj)\w(aj))

= 2(A + ola,) - (a/la,)

= 20|a,-) - (a}\af)= 0.

Thus we have A e h* x, and hence we can apply Theorem 4.3.12 to this case.

In this section, we first apply Theorem 4.2.1 to obtain that [V((w ,0)oA):

L((w , af) o A)] = 1. Then we apply Theorem 4.3.12 to show that the constant

a(X) in Theorem 4.3.12 is equal to 1 in the above case (which is the second

assertion of Theorem II). The latter fact plays an important role in the proof

of the Kazhdan-Lusztig conjecture in §6.

5.1. Application of the character sum formula for a Verma module. Now we

shall make use of Jantzen's character sum formula for a single Verma module

to compute the multiplicity [V((w, 0) o A) : L((w , af o A)]. Let q(A) be a

GKM algebra associated to a symmetrizable GGCM A = (a(J), ;6/ with a,, ^ 0(a £ I). Note that, in this case, the set sé (A) consists of all sums of distinct,

pairwise perpendicular, imaginary simple roots perpendicular to A £ P+ . In

particular, sé (A) is a finite set.

Theorem 5.1.1. Let A £ P+, w £ W, and ß, ß' £ sé (A). If ß' ^ ß andht()S') = ht(/3) + 1, then we have [V((w , ß) o A) : L((w , ß') o A)] = 1.

Proof. First we note that by the assumption of the theorem, there exists af £

nim with (A\aj) = 0, (ß\atj) = 0 such that ß' = ß + aj . So we have

w(A + p-ß')-p = w((A - ß) + p-af- p,

where A - ß £ P+ , (A - ß\aj) = 0. Hence we may assume that ß = 0, and

that ht(/?')= 1.



Put A := w(A + p) - p £b*, p := w(A + p - af - p = (w , af o A . Then,by Theorem 4.2.1, the Verma module V(X) has a filtration

V(X)dV(X)xdV(X)2d-..

by a(^l)-submodules such that V(X)/V(X)X =■ L(X), and such that

(5.1.1) e ch v% = E E ch m - iß)-i>\ ßeA+ /€Z>,

2(X+p\ß)=l(ß\ß)

Since we have the exact sequence

0 —» V(X)X — V(X) —* L(X) -^ 0,

we get

[V(X) : L(p)] = [V(X)X : L(p)] + [L(X) : L(p)\.

Here we recall that, for A £ P+ and (wx, ßx), (w2, ß2) £ Wxsé(A), (wx, ßx)o

A = (w2, ß2) o A if and only if (wx, ßx) = (w2, ß2), since (A + p- /?,, a^) £Z>i (i = 1, 2) for Ac £ Ve (cf. the proof of [K, Proposition 3.12]). So we

obtain \L(X) : L(p)\ = 0. Thus, we have to show that [V(X)X : L(p)] = 1.Note that, in the formula (5.1.1), all ch V(X)¡ (i > 1) on the left-hand side,

and all chV(X - Iß) (ß £ A+, I £ 1>x) on the right-hand side are clearly(possibly infinite) sums of some scalar multiple of chL(X - h) (n £ Q+) by

Proposition 3.2.1. Furthermore, chL(X-n) (n £ Q+) are linearly independent

over C (i.e., J^rj€Q+a^ ch L(X - n) = 0 with an £ C implies an = 0 for all

n £ Q+). Therefore, we have that

[V(X)x:L(p)]>[V(X)2:L(p)]>.-. ,

and that

E W(X)i : L(p)] =J2 E \-V(k - lß) ■■ ¿Ml-'>i

Let us consider the equation

or equivalently,

1>1 ß€A+ /ez>,

2(X+p\ß)=l(ß\ß)

2(X + p\ß) = l(ß\ß),

2(w(A + p)\ß) = l(ß\ß) (ß£A+,l£Z>x).

By Lemma 4.3.1, the above equation is equivalent to either of the following:

(a) ß £ A+ n Are and rßw ^ w ;

(b) /= l,and w~x(ß)£nim with (A\w~x(ß)) = 0.

(Note that in both cases, we have dimc Qß = 1.)In case (a), we have / = (w(A + p), ßv), so that

X-lß = w(A + p)-p- (w(A + p), ßy)ß

= rß(w(A + p))-p

= (rßw , 0) o A.

In this case, we have

[V(X - lß) : L(p)] = [V((rßw, 0) o A) : L((w, aj) oA)] = 0,


■■{

3912 SATOSHI NAITO

since, by Theorem 3.3.3, [V((rßW, 0) o A) : L((w, af o A)] ^ 0 implies(w , af ^ (rßW , 0). This contradicts the assumption that rßW ^ w .

In case (b), we have, for some ak := w~x(ß) £ 77"" with (A\ak) = 0,

X-lß = X-ß

= w(A + p)- p-w(ak)

= w(A + p - ak) - p

= (w, ak)oA.

In this case, we have, by Theorem 3.3.3,

[V(X - Iß) : L(p)\ = [V((w , ak) o A) : L((w , af o A)]

1 if ak = af ,

0 otherwise.

To sum up, we have obtained

V^ rr//i\ T / \i iYj[V(X)l:L(p)]=l.«>i

Therefore, we get [V(X)X : L(p)] = 1, [V(X)¡ : L(p)] = 0 (/ > 2). Thus, wehave proved the theorem. D

5.2. The constant a(X). To show the second assertion of Theorem II, again we

compute the multiplicity [V((w, Q)oA) : L((w, a¡)oA)\ = [V(X) : L(p)] above,

this time by using Theorem 4.3.12. Let A £ P+ , and let a = w(af £ W • 77""with (A\af = 0. Then A = w(A + p) - p £ b*a , , as shown before. So, since

X- a = w(A + p) - p - w(af

= w(A + p-atj) - p

= (w,afoA

= P,

we have an embedding

T(X) : V(p) = V((w , af oA)^ V((w ,0)oA) = V(X).

Therefore, we obtain

l = [V(X) :L(p)]

= [V((w , afo A) : L((w , aj)o A)] + [V(X)/T(X)V(X- a) : L((w , afo A)],

and hence[V(X)/T(X)V(X - a) : L((w, aj) o A)] = 0.

Using this fact, we get the following proposition (= the second assertion of

Theorem II).

Proposition 5.2.1. Let q(A) be a GKM algebra associated to a symmetrizable

GGCM A = (aif)iJe¡ with a„ #0 (i £ I). Let A £ P+, w £ W, and aj £n,m with (A\af) = 0. Put X:= w(A + p)-p £b*, and a:=w(aj)£ W-W" .Then X £ h* ,. Moreover, in this case, the constant a(X) in the formula (4.3.1)

in Theorem 4.3.12 is equal to 1.

Proof. Put 7V(A) := V(X)/T(X)V(X - a) and p := (w , af o A . Then we have[N(X) : L(p)] = 0, as seen above. So we have [N(X)¡ : L(p)] = 0 (a > 1).



Note that, in formula (4.3.1) in Theorem 4.3.12, all chN(X)¡ (i > 1) onthe left-hand side, and all ch V(X -Iß) (ß £A+, I £ Z>0 , ch V(X - a - my)(y £ A+, m £ Z>i) on the right-hand side are clearly (possibly infinite) sums ofsome scalar multiple of chF(A - n) (n £ Q+) by Proposition 3.2.1. Further-

more, ch L(X - n) (n £ Q+) are linearly independent over C (see the proof ofTheorem 5.1.1).

Therefore, we have

o = E ww¡: L^/>i

= E E [V(X-lß):L(p)]-Y, E [V(X-a-my):L(p)\ß€A+ A€Z>! y€á+ m€ZZ\

Llß(X)=0 Lmy(X-a)=0

-a(X)[V(X-a):L(p)].

It has already been seen in the proof of Theorem 5.1.1 that

E E [V(X-lß):L(p)]=l.ßeA+ it»

Llß(X)=0

Now consider the equation

Lmy(X-a) = 0,

or equivalently,

2(w(A + p- aj)\y) = m(y\y) (y £ A+, m £ Z>i).

By Lemma 4.3.1, the above equation is equivalent to either of the following:

(a) y £ A+r\ Are and ryw > w ;

(b) m = 1, and w~x(y) £ 77'm with (A - af\w~x(y)) = 0.

(Note that in both cases, we have dime 9y = 1 ■)

In case (a), we have m — (w(A + p - af, yv), so that

X- a - my = w(A + p) - p - w(af - my

= w(A + p-af)-my - p

= ry(w(A + p-af))- p

= (A-j.it;, af) o A.

In this case, we have, by Theorem 3.3.3,

[F(A -a-my): L(p)] = [V((ryw , a¡) o A) : L((w , oy) o A)] = 0,

since ryw > w .

In case (b), we have, for some ak := w~x(y) £ 77"" with (A - afak) = 0,

X- a- my = w(A + p) - p - w(a¡) - w(ak)

= w(A + p - (af + ak)) - p

= (w , aj + ak) o A.

Here note that a7 +ak £sé(A). In this case, we have, by Theorem 3.3.3,

[V(X -a-my): L(p)\ = [V((w , a¡ + ak) o A) : L((w , aj) o A)] = 0.


3914 SATOSHI NAITO

Summarizing, we obtain

0=1- a(X)[V(X - a) : L(p)] = 1 - a(X)[V((w , af o A) : L((w , af o A)]

= 1 -a(X),

and hence a(X) = 1. Thus the proposition has been proved. D

6. Proof of the conjecture

Fix A £ P+. In this section, we shall prove the Kazhdan-Lusztig con-

jecture for GKM algebras q(A) associated to a symmetrizable GGCM A =

(a¡j)i j€i with an #0 (i £ I), that is, that the multiplicity [V((w, ß) o A) :

L((w', ß')oA)] for (w,ß),(w',ß')£Wxse(A) is equal to Plw,fi),{m.,fi.)(l),where P(W ß) (W- ß-)(q) is the extended Kazhdan-Lusztig polynomial introduced

in § 2.Furthermore, we shall describe the (formal) character of the irreducible high-

est weight a(^4)-module L((w, ß) o A) with highest weight (w , ß) o A for

(w, ß) £ W x sé (A) in terms of the extended inverse Kazhdan-Lusztig poly-

nomials Q{w,ß)Aw',ß')(q) ((w1, ß') £ W x sé (A)).

6.1. Proof of the multiplicity formula. In [N], we proved the following theorem

(without assuming that a„ ^ 0 (i £ I)).

Theorem 6.1.1 ([N, Theorem 5.3]). Let g(A) be a GKM algebra associated to

a symmetrizable GGCM A = (a¡j)ij€i. Let A £ P+, and (w, ßx), (z, ßf) £W xsé(A). Then we have

[V((w , ßx) o A) : L((z, ß2) o A)]> P{w,ßlh{z,ß2)(l),

where P(W,ßx),(Z,ß1)(q) is the extended Kazhdan-Lusztig polynomial. Moreover,

the equality holds if ßx = ß2 or if w = z = 1.

Furthermore, we conjectured that, under the assumption that a,, ^ 0 (i £ I),

the equality holds for any (w , ßx), (z, ßf) £ W xsé(A) in the above theorem.

Now we are ready to prove this conjecture.

Theorem 6.1.2 (multiplicity formula). Let q(A) be a GKM algebra associated

to a symmetrizable GGCM A = (a¡j)ije¡ satisfying the condition that a,, ^ 0

(i £l). Let A£P+, and (w, ßx), (z, ßf) £ W x sé (A). Then we have

[V((w,ßx)oA):L((z,ß2)oA)] = PiwJlh{z,ß2)(l),

where P(W, ̂, ), (Z, ft)(i) is the extended Kazhdan-Lusztig polynomial.

Proof. By Theorem 3.3.3 and the definition of the extended Kazhdan-Lusztig

polynomials, it suffices to show that, for (z, ßf) ^ (w, ßx) £ W xsé(A),

[V((w,ßx)oA):L((z,ß2)oA)] = Pw,2(l),

where Pw>z(q) is the usual Kazhdan-Lusztig polynomial (associated to w , z £

W ) for the Weyl group W. We give a proof of this by double induction on

ht(/52) - ht(/?i) and £(z) - £(w). When ht(ßf) - ht(^i) = 0, the assertion istrue by Theorem 6.1.1. When l(z) - i(w) = 0, the assertion can be proved by

exactly the same argument as the one below. So we omit its proof.



Now assume that ht(/?2) - ht(^.) = t > 1, l(z) - l(w) = s > 1. We firstnote that

(z,ßf)oA = z(A + p-ßf)-p

= z((A-ßx) + p-(ß2-ßx))-p

= (z,ß2-ßx)o(A-ßx),

where A - ßx £ P+ , ß2 - ßx £ sé (A - ßx), and that

(w, ßx)oA = w(A + p- ßx) - p

= (w,0)o(A-ßx).

Therefore, we may assume that ßx = 0.

Since ht(/?2) = t > 1, we can write ß2 = ß' + a¡ for some ß' £ sé (A),

a¡ £ 77"" such that (A\af) = (ß'\a.f) = 0. Put A := (w , 0) o A = w(A + p) - pand a := w(otj) £ W • 77"" . Then, as before, we have an embedding

T(X) : V(X -a) = V((w , a7) o A) ¿> V((w , 0) o A) = V(X).

Put p := (z, ß2) o A and N(X) := V(X)/T(X)V(X - a). Then we have

[V(X) : L(p)] = [N(X) : L(p)] + [V(X - a) : L(p)].

It follows from the induction hypothesis that

[V(X-a):L(p)] = [V((w,af)oA):L((z,ß2)oA)] = Pw,z(l),

since ht(/?2) - ht(a;) = t - 1 . Thus we must show that

[V(X)/T(X)V(X - a) : L(p)] = [N(X) : L(p)] = 0.

On the other hand, by Theorem 4.3.12, the module N(X) has Jantzen's fil-tration

N(X)dN(X)x dN(X)2d--- .

Here it is obvious that

[7V(A) : L(p)} = [N(X)/N(X)X : L(p)] + [N(X)X : L(p)]

= [N(X)x:L(p)],

since N(X)/N(X)X = L(X), and X ^ p. Hence what we have to show is that

[N(X)x:L(p)] = 0.We will prove that £w>i [N(X)¡ : L(p)] = 0. By Jantzen's character sum

formula (4.3.1) in Theorem 4.3.12, together with Proposition 5.2.1, we have (as

in the proof of Proposition 5.2.1)

E imm ■■ un)] = E E rF(A - lß): Lw\i>\ ßeA+ /ez>,

Llf(X)=0

"E E [V(X-a-my):L(p)]-[V(X-a):L(p)].y€A+ meZ>,

Lmy(X-a)=0

Now let ß £ A+, I £ Z>i be such that L¡P(X) = 0, or equivalently, suchthat 2(w(A + p)\ß) = l(ß\ß). Then, as in the proof of Theorem 5.1.1, we have

either of the following.


3916 SATOSHI NAITO

Case (a): ß £ A+C\Are with rßW ^ w . In this case, it follows from Theorem

3.3.3, together with the induction hypothesis, that

[V(X - Iß) : L(p)] = [V((rßw , 0) o A) : L((z, ßf) o A)]

f Prßw,z(l) if Z^rßW,

\ 0 otherwise,

since z ^ rßW ^ w implies l(z) - l(rßw) % i(z) - ¿(w) = s (note that

rßw ¿ w).

Case (b): / = 1, ß = w(ak) for some ak £ 77"" with (A\ak) = 0. Inthis case, it follows again from Theorem 3.3.3, together with the induction

hypothesis, that

[V(X - Iß) : L(p)] = [V((w , ak) o A) : L((z, ßf) o A)]

(Pw,z(l) ifß2^ak,

\ 0 otherwise,

since ht(/?2) - ht(ak) = t - 1.Therefore, we deduce that

E E [V(X-lß):L(p)]= E PrßW,z(l) + ht(ß2)Pw,z(l).ß€J+ /ez>, ß€A+nAre

Llfi(X)=0 z^rßw^w

Here we remark that for fixed z, w £ W, the cardinality of the set {ß £A+ n Are I z ^ rßW ^ w} is finite.

Now let y £ A+, m £ Z>x be such that Lmy(X - a) = 0, or equivalently,

such that 2(w(A + p - af\y) = m(y\y). Then, as in the proof of Proposition

5.2.1, we have either of the following.Case (c): y £ A+C\Are with ryw ^ tu . In this case, it follows from Theorem

3.3.3 and the induction hypothesis that

[V(X- a-my): L(p)] -- ■■ [V((ryw , aj) o A) : L((z, ß2) o A)]

Pryw,z(l) if Z^ryW,

0 otherwise,= {

since i(z) -l(ryw) % i(z) - i(w) = s and ht(/?2) - ht(af = t - 1.

Case (d): m = 1, y = w(ak) for some ak £ 77"" with (A - afak) = 0.In this case, it follows again from Theorem 3.3.3 and the induction hypothesis

that

[K(A -a-my): L(p)] = [V((w , aj + ak) o A) : L((z, ßf) o A)]

^(Pw,z(l) ifß'Zak,

\ 0 otherwise,

since ht(/?2) - ht(a; + ak) = t - 2 .

Therefore, we deduce that

E E [V(X-a-my):L(p)]= E Pyw,z(l)+ ht(ß')Pw,z(l).yeA+ m€Z>, y€A+nA"

Lmy{X-a)=0 z^ryw^w



Summarizing the above argument, we obtain that

Y^[N(X)i:L(p)]= E Prßw,z(l) + tPw,z(l)1>1 ß€A+nA'e

z^rßw^w

- E PryW,z(l)-(t-l)PW,z(l)-Pw,z(l)y€A+r\Are

z^ryw^w

= 0.

Thus, we have completely proved the theorem. D

6.2. Proof of the character formula. Here we shall derive the character formula

for the irreducible highest weight g(^4)-module L((w, ß) o A) with A £ P+ ,(w, ß) £ W x sé (A) from the multiplicity formula above.

Before proceeding to the proof of the character formula, we prepare the

following lemma.

Lemma 6.2.1. We use the notation of §2.2. For ß, ß' £ sé = sé(0), we have

Y,(-l)bt{y)-ht{ß)Qß,y(q)Py,ß'(q) = Sß,ß'-

yes<?

Furthermore, for (w, ß), (w', ß') £ W x sé , we have

E (-^^^^»-^^^»ô^,,,,^,,)^)^,,),^,,^«){y ,y)ewxs/

= 0~(w,ß),(w< ,ß')-

Proof. The second assertion immediately follows from the first one and the def-

inition of the extended inverse Kazhdan-Lusztig polynomials. Hence it suffices

to show the first assertion. Furthermore, we may assume that ß < ß'. Then

we have

Y,(-l)htM-hmQß,y(q)Py,ßfq)= E (-DhM)y€si? y€stC

ß<y<ß'■

= E (-Dhtwyes/

o^y^ß'-ß

= (1 - ljW-^)

= °ß,ß' >

which proves the lemma. O

Finally, we come to the main result of this paper.

Theorem 6.2.2 (character formula). Let q(A) be a GKM algebra associated to a

symmetrizable GGCM A = (a¡f)ij€j with Qa ̂ 0 (i £ I). Let A £ P+ . Then,for (w, ß) £ W xsé(A), we have (equality in the algebra f)

chV((w,ß)oA)= E P(w,ß)AW',ß^)chL((W',ß')oA).

{w',ß')eWxjf(A)


3918 SATOSHI NAITO

Equivalently, for (w, ß) £ W x sé (A), we have (again equality in the algebra

9)

ch L((w , ß) o A) = E (- if&'r^'V-VW+W»{w',ß')€lVxjf(A)

■Q(w,ß),{w,ß')(l)chV((w',ß')oA).

Proof. The first assertion follows directly from Theorem 6.1.2, by using Propo-

sition 3.2.1, Definition 3.2.2, and Theorem 3.3.1. The second assertion imme-

diately follows from the first one by using Lemma 6.2.1. D

It is a well-known fact that ch V(x) = e(x) • R~x for x £ b*, where R :=

Yla€A+(l -e(-a))mult(a) (see [K, Chapter 10]). Hence, taking (w, ß) = (1, 0) e

W xsé(A) in the above theorem, we have

e(p)-R-chL(A)= E (-l)e{w')+mß,)Qx,w>(l)e(w'(A + p-ß')).

(w' ,ß')€Wxsf(A)

Here it is known that Qx ,wfl) = 1 for all w' £ W .Now recall that the set sé (A) coincides with the set S^(A) under the con-

dition on the GGCM A = (fl/y)f,;e; that a,, ^ 0 (i £ I). Thus we have re-covered the Weyl-Kac-Borcherds character formula for the irreducible highest

weight 0(^4)-module L(A) with highest weight A £ P+ , although the characterformula for L(A) itself is proved without the condition above on the GGCM

A (see [Bl] or [K, Chapter 11]).

Remark 6.2.3. From the above argument, we see that the restriction on the

GGCM A = (a¡f)ije¡ in Theorems 6.1.2 and 6.2.2, that a,-,- ̂ 0 (i £ I), isessential. We do not know what the multiplicity [V((w , ß)oA) : L((w', ß')oA)]

is, for (w , ß), (w', /?') £ W x sé (A), when this restriction is removed.

References

[Bl] R. E. Borcherds, Generalized Kac-Moody algebras, J. Algebra 115 (1988), 501-512.

[B2] _, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992),

405-444.

[C] L. Casian, Kazhdan-Lusztig multiplicity formulas for Kac-Moody algebras, C. R. Acad. Sei.

Paris 310 (1990), 333-337.

[DGK] V. V. Deodhar, O. Gabber, and V. G. Kac, Structure of some categories of representations

of infinite-dimensional Lie algebras, Adv. Math. 45 (1982), 92-116.

[GW] R. Goodman and N. R. Wallach, Whittaker vectors and conical vectors, J. Funct. Anal. 39

(1980), 199-279.

[J] J. C. Jantzen, Moduln mit einem Höchsten Gewicht, Lecture Notes in Math., vol. 750,

Springer, Berlin and New York, 1979.

[K] V. G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, Cambridge,

1990.

[KK] V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infi-

nite-dimensional Lie algebras, Adv. Math. 34 (1979), 97-108.

[Ka] M. Kashiwara, Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra, The

Grothendieck Festschrift, II (P. Cartier et al., eds.), Progress in Math., vol. 87, Birkhäuser,

Boston, 1990, pp. 407-433.



[KT] M. Kashiwara and T. Tanisaki, Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody

Lie algebra. II: Intersection cohomologies of Schubert varieties, Operator Algebras, Uni-

tary Representations, Enveloping Algebras, and Invariant Theory (A. Connes et al., eds.),

Progress in Math., vol. 92, Birkhäuser, Boston, 1990, pp. 159-195.

[KL1] D. A. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras,

Invent. Math. 53 (1979), 165-184.

[KL2] _, Schubert varieties and Poincaré duality, Proc. Sympos. Pure Math., vol. 36, Amer.

Math. Soc, Providence, RI, 1980, pp. 185-203.

[N] S. Naito, Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras. I: Towards the

conjecture, Comm. Algebra 23 (1995), 703-736.

[RW] A. Rocha-Caridi and N. R. Wallach, Highest weight modules over graded Lie algebras : Res-

olutions, filiations and character formulas, Trans. Amer. Math. Soc. 277 (1983), 133-162.

[S] N. N. Shapovalov, On a bilinear form on the universal enveloping algebra of a complex

semisimple Lie algebra, Functional Anal. Appl. 6 (1972), 307-312.

[W] N. R. Wallach, A class of non-standard modules for affine Lie algebras, Math. Z. 196 (1987),

303-313.

Department of Mathematics, Shizuoka University, 836 Ohya, Shizuoka 422, Japan

E-mail address: smsnait9sci.shizuoka.ac.jp


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