数理解析研究所講究録
第 855 巻 1993 年 153-165

153

Towards the Kazhdan-Lusztig multiplicity formulafor generalized Kac-Moody algebras
BY SATOSHI NAITO (内藤聡)
Department of Mathematics, Faculty of Science, Shizuoka University,
836 Ohya, Shizuoka 422, Japan

1. GENERAIIZED KAC-MOODY AIGEBRAS
1.1. Let $A=(a_{ij})_{i,j\in I}$ with $I=\{1,2, \ldots n\}$ be a real
following conditions:(C1) $a_{ii}=2$ , or $a_{ii}\leq 0(i\in I)$ ;(C2) $a_{ij}\leq 0(i\neq j)$ , andif $a_{ii}=2$ ;(C3).

$n\cross n$matrix satisfying the$a_{ij}\in Z$$a_{ij}=0\Leftrightarrow a_{ji}=0$

We call such a matrix a GGCM ( $=generalized$ , generalized Cartan matrix).
For any GGCM $A=(a_{ij})_{i,j\in I}$ , we have a triple
satisfying the following (see [6, Chap.1]):$(\mathfrak{h}, \Pi=\{\alpha_{i}\}_{i\in I}, \Pi^{V}=\{\alpha_{i}^{\vee}\}_{i\in I})$

(R1) is a finite-dimensional (complex) vector space such thatrank ;(R2)
is linearly independent, andlinearly independent, where;$\mathfrak{h}$$\dim_{\mathbb{C}}\mathfrak{h}=2n-$$A$$\Pi=\{\alpha_{i}\}_{i\in I}\subset \mathfrak{h}^{*}$$\Pi^{\vee}=\{\alpha_{i}^{\vee}\}_{i\in I}\subset \mathfrak{h}$is$\mathfrak{h}^{*}:=Hom_{\mathbb{C}}(\mathfrak{h}, C)$

(R3)

and

$\mathfrak{h}^{*}$$\langle\alpha_{j}, \alpha_{i}^{v}\rangle=a_{ij}(i,j\in I)$, where$\langle\cdot, \cdot\rangle$

denotes a duality pairing between$\mathfrak{h}$.

The above triple is called a realization of .
From now on throughout this paper, we assume that the GGCM
is symmetrizable, i.e., that there exists a diagonal matrix such that $\det D\neq 0$ and $DA$
is symmetric.
A generalized Kac-Moody algebra ( $=GKMal$ gebra) associated to a symmetrizable GGCM
is a Lie algebra $g(A)$ (over the complex number field C)
$A$

$A$

$D$

$A=(a_{ij})_{i_{\dot{\theta}}\in I}$

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generated by the above vector space and the elements
following relations (see [1], or [6, Chap.11]):$\mathfrak{h}$(F1)$e_{i},$$f_{i}(i\in I)$

satisfying the$\{\begin{array}{l}[h,h’]=0(h,h’\in \mathfrak{h})[h,e_{i}]=\langle\alpha_{i},h)e_{i},[h,f_{i}]=-\langle\alpha_{i},h\rangle f_{i}(h\in \mathfrak{h},i\in I)[e_{i},f_{j}]=\delta_{ij}\alpha_{i}^{\vee}(i,j\in I)\end{array}$(F2) $(ade_{i})^{1-a_{ij}}e_{j}=0,$(F3) $[e_{i}, e_{j}]=0,$$(a_{ii}=2, j\neq i)$$\langle adf_{i})^{1-a:j}f_{i}=0$$[f_{i}, f_{j}]=0$$(a_{ii}, a_{jj}\leq 0, a_{ij}=0)$,.

Then, we have the root space decomposition of $g(A)$ with respect to the Cartan, whereissubalgebrais thethe set of positive roots, $\Delta_{-}(=-\Delta_{+})$ is the set of negative roots, and. Note that multroot space attached to a root$\Delta_{+}(\subset\Sigma_{i\in I}Z_{\geq 0}\alpha_{i})$$\mathfrak{h}:\mathfrak{g}(A)=\mathfrak{h}\oplus\Sigma_{\alpha\in\Delta_{+}}^{\oplus}\mathfrak{g}_{\alpha}\oplus\Sigma_{\alpha\in\Delta_{-}}^{\oplus}\mathfrak{g}_{\alpha}$$\mathfrak{g}_{\alpha}$$(\alpha):=$$\alpha\in\Delta=\Delta_{+}\cup\Delta_{-}\subset \mathfrak{h}^{*}$$\dim_{C}\mathfrak{g}_{\alpha}=\dim_{C}\mathfrak{g}_{-\alpha}<+\infty(\alpha\in\Delta_{+})$

Put$\mathfrak{n}_{+}:=\Sigma_{\alpha\in\Delta_{+}}^{\oplus}g_{\alpha},$.

$\mathfrak{n}_{-}:=\Sigma_{\alpha\in\Delta_{+}}^{\oplus}\mathfrak{g}_{-\alpha}$, and$b:=\mathfrak{h}\oplus n_{+}$.

$;=$$:=\{i\in I|a_{ii}\leq 0\}$ . Let
and$;=\{\alpha_{i}\in\Pi|i\in I^{im}\}$$\{\alpha_{i}\in\Pi|i\in I^{re}\}$ be the set of real simple roots, andthe set of imaginary simple roots.
if $a_{ij}=0$ . (Remark, we say thatFor
is perpendicular tothat an imaginary simple rootis perpendicular to itself if $a_{ii}=0.$ ) For.
, we say thatandis perpendicular to if, andNow, fix an element
if $a_{ii}=2$ }. Then, we define a subset
of to be the set of all sums of (notnecessarily distinct,) pairwise perpendicular, imaginary simple roots perpendicular
to . Note that
contains the setI, wewith $a_{jj}=0$ } by definition. For an elementput.

1.2. We put$I^{re}$$:=\{i\in I|a_{ii}=2\}$ ,$I^{im}$$\Pi^{re}$$\Pi^{im}$$\alpha_{i},$$\alpha_{j}\in\Pi^{im}$$\alpha_{i}$$\alpha_{j}$$\alpha_{i}\in\Pi^{im}$$\alpha_{i}\in\Pi^{im}$$\lambda\in \mathfrak{h}^{*}$$\lambda$$\alpha_{i}$$\langle\lambda, \alpha_{i}^{V}\rangle=0$$\Lambda\in P_{+}:=\{\lambda\in \mathfrak{h}^{*}|\langle\lambda, \alpha_{i}^{v}\rangle\geq 0(i\in I)$$\mathcal{A}(\Lambda)$$Z_{\geq 0}$$\Lambda$$\mathcal{A}:=\mathcal{A}(0)$$\langle\lambda, \alpha_{i}^{\vee}\rangle\in$$\mathfrak{h}^{*}$$\{0\}\cup\Pi^{im}\cup\{m\alpha_{j}$$\Pi^{im}$$m\in Z_{\geq 2},$ $\alpha_{j}\in$$\beta=\Sigma_{i\in I^{im}}k_{i}\alpha_{i}(k_{i}\in Z_{\geq 0})$$ht(\beta)=\Sigma_{i\in I^{m}}.k_{i}$1.3. For $i\in I^{re}$ , let
be the simple reflection of
given by:(A). The Weyl group $W$ of
genis the subgroup oferated by the $r_{i}’ s(i\in I^{re})$ . For an element $w\in W,$
denotes the length of.
Let
be the set of real roots,
the set of imaginary
roots. For a real root $\alpha=w(\alpha_{i})(w\in W, \alpha_{i}\in\Pi^{re})$ , we define the reflectionof$\mathfrak{h}^{*}$$r_{i}$$\langle\lambda, \alpha_{i}^{v}\rangle\alpha_{i}$$\in \mathfrak{h}^{*}$$r_{i}(\lambda)=\lambda-$$GL(\mathfrak{h}^{*})$$\mathfrak{g}(A)$$\ell(w)$$w$$\Delta^{re}:=W\cdot\Pi^{re}$$\Delta^{im}:=\Delta\backslash \Delta^{re}$$r_{\alpha}$

155, where)with respect to by:is the dual real root of . Note that $r_{\alpha}=wr_{i}w^{-1}\in W$ .
$\mathfrak{h}^{*}$$r_{\alpha}(\lambda)=\lambda-\langle\lambda,$$\alpha$$\alpha^{\vee}$$\alpha(\lambda\in \mathfrak{h}^{*})$$\alpha^{V}:=w(\alpha_{i}^{\vee})\in \mathfrak{h}$$\alpha$, we denote by
1.4. Forwith
the Verma moduleis the one-dimensional
highest weight over the GKM algebra $g(A)$ . Here,-module with weight , on whichacts triviaUy. As is well-known, the Vermamoduleis the universal highest weight $g(A)$ -module with highest weight ,. Then, we define
and has a unique maximal proper g(A)-submodule, so thatto be the quotient $g(A)$ -module ofis the irreduciblebyhighegt weight $g(A)$ -module with highest weight .
$V(\lambda)$$\lambda\in \mathfrak{h}^{*}$$U(\mathfrak{g}(A))\otimes_{U(b)}C(\lambda)$$C(\lambda)$$\lambda$$\lambda$$\mathfrak{h}$$\mathfrak{n}_{+}$$V(\lambda)$$\lambda$$L(\lambda)$$V^{/}(\lambda)$$V(\lambda)$$L(\lambda)$$V^{/}(\lambda)$$\lambda$

2. BRUHAT ORDERING AND KAZIIDAN-LUSZTIG POLYNOMIALS
2.1. Here, we extend the notion of the Bruhat ordering on the Weyl group $W$ tothat on the direct product setof $W$ and $A=A(O)$ as follows.
$W\cross A$$w_{2}\in W$ . We write
if there
Definition 2.1 (Bruhat ordering). Let
and $\ell(w_{1})=l(w_{2})+1$ . Moreover,exists somesuch that$w_{r}\in W$ suchfor $w,$ $w^{/}\in W$ , we write
if $w=w^{/}$ or if there exist
that$w_{1}arrow w_{2}$$w_{1},$$\gamma\in\Delta^{re}\cap\Delta_{+}$$w_{1}=r_{\gamma}w_{2}$$w\geq w^{/}$$w_{1},$ $\ldots$$warrow w_{1}arrow\cdotsarrow w_{r}arrow w’$

Definition 2.2 (cf. [11, Definition 2.2]). Let
if there exists somesuch that, we write$\alpha_{j}\in\Pi^{im}$$\Sigma_{k\in I^{im}}m_{k}\alpha_{k},$$\beta_{1},$.

$\beta_{2}\in \mathcal{A}$$\beta_{1}=\beta_{2}+\alpha_{j}$$\beta’=\Sigma_{k\in I^{im}}m_{k}’\alpha_{k}\in \mathcal{A}$Definition 2.3 (cf. [11, Definition 2.3]). For$\beta\geq\beta’$if. We write$\beta_{1}arrow\beta_{2}$. Moreover, for$m_{k}\geq m_{k}’$$\beta=$for all $k\in I^{im}$ .

$(w_{1}, \beta_{1}),$ $(w_{2}, \beta_{2})\in W\cross \mathcal{A}$, we write$(w_{1}, \beta_{1})arrow(w_{2}, \beta_{2})$if$w_{1}arrow w_{2}$Moreover, for $(w, \beta),$$\beta\geq\beta’$and$\beta_{1}=\beta_{2}$,$(w^{/}, \beta^{/})\in W\cross \mathcal{A}$orif, we write$w_{1}=w_{2}$and$\beta_{1}arrow\beta_{2}$$(w, \beta)\geq(w^{/}, \beta^{/})$.

if $w\geq w^{/}$ and.

2.2. Here, we review the definitions of the Kazhdan-Lusztig polynomials and theinverse Kazhdan-Lusztig polynomials, and then give their certain extensions. We
first note that the Weyl group $W$ of the GKM algebra $g(A)$ is a Coxeter group156
with canonical generator system $\{r_{i}|i\in I^{re}\}$ . The Hecke algebm $H(W)$ of$W$ is the associative algebra over the Laurent polynomial ring(in theindeterminate ) which has a freewith the following-basisrelations:(H1) $T_{w}T_{w’}=T_{ww’}$ if $\ell(ww’)=\ell(w)+\ell(w’)$ $(w, w’\in W)$ ;(H2) $(T..+1)(T_{r_{i}}-q)=0$ $(i\in I^{re})$ .
$Z[q^{\frac{1}{2}}, q^{-\frac{1}{2}}]$$Z[q^{\frac{1}{2}}, q^{-\frac{1}{2}}]$$q^{\frac{1}{2}}$$\{T_{w}\}_{w\in W}$

be the involutive automorphism of the ring $H(W)$ defined by:$\iota(T_{w})=(T_{w^{-1}})^{-1}(w\in W)$ . Then, we know the following proposition due
to Kazhdan and Lusztig [9].
Let$\iota(q^{\frac{1}{2}})=$$\iota$$q^{-\frac{1}{2}},$PROPOSITION 2.4 ([9]). For each $w\in W$ , there exists a unique element $C_{w}\in$aving the following properties:(1) $\iota(C_{w})=C_{wi}$(2),$\mathcal{H}(W)\Lambda$$C_{w}=(-1)^{l(w)}q^{z\perp_{2}w\Delta} \sum_{y\leq w}(-1)^{l(y)}q^{-l(y)}\iota(P_{y,w}(q))T_{y}$where $P_{w,w}=1$ , and $P_{y,w}(q)$ is a polynomial with integer coefhcients in the indeterminate of degree $\leq(1/2)\cdot(\ell(w)-\ell(y)-1)$ for $y\leq w$ .
Moreover, the elements $C_{w}(w\in W)$ form aasis of $?t(W)$ .
$q$$keeZ[q^{\frac{1}{2}}, q^{-\frac{1}{2}}]- b$

The above polynomials $P_{y,w}(q)\in Z[q](y\leq w)$ are called the Kazhdan-Lusztig
polynomials. We set $P_{y,w}(q):=0$ unless $y\leq w$ .
Now, for, we define a polynomial
in by$\beta,$$\beta’\in \mathcal{A}=A(0)$$P_{\beta,\beta^{l}}(q)$$q$$P_{\beta)}\rho’(q):=\{\begin{array}{l}1if\beta’\geq\beta 0otherwise\end{array}$Moreover, for$(w, \beta),$$(w’, \beta’)\in W\cross A$, we put$P_{(w,\beta),(w’,\beta’)}(q):=P_{w,w’}(q)\cdot P_{\beta,\beta’}(q)$,and call this polynomial the extended Kazhdan-Lusztig polynomial.
It is also known (see [10], and also [8, \S 5]) that there exist the inverse KazhdanLusztig polynomials $Q_{w,y}(q)\in Z[q](w\leq y\in W)$ for the Coxeter group $W$ suchthat$\sum_{w\leq y\leq w’}(-1)^{\ell(y)-l(w)}Q_{w,y}(q)P_{y,w^{r}}(q)=\delta_{w,w^{l}}$We set $Q_{w,y}(q):=0$ unless $w\leq y$ .

$(w\leq w’)$.

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For $(w, \beta),$$(w’, \beta’)\in W\cross \mathcal{A}$, we put$Q_{(w,\beta)(w’,\beta’)})(q)))$, and call this polynomial the extended inverse Kazhdanwhere
Lusztig polynomial.
Then, it is easy to see the following.
$Q_{\beta,\beta’}(q):=P_{\beta,\beta’}(q)$For $(w, \beta),$$\sum$$(w^{/}, \beta^{/})\in W\cross \mathcal{A}$, we have.

$(-1)^{(l(y)+ht(\gamma))-(l(w)+ht(\beta))}Q_{(w,\beta),(y,\gamma)}(q)P_{(y,\gamma),(w’,\beta’)}(q)=\delta_{(w,\beta),(w’,\beta’)}$$(y,\gamma)\in W\cross A$

3. IRREDUCIBI, SUBQUOTIENTS AND EMBEDDINGS OF VERMA MODULES$E$

is said to be -diagonalizable if
A $g(A)$ -module
admits a weight space, wheredecomposition:is the weight space of weight. We
denote by $P(V)$ the set of all weights of . We call an -diagonalizable module
a weight module if
for all.
, following [13, \S 2], we define the category
Now, forto be the fullsubcategory of the category of all $g(A)$ -modules whose objects are weight modules, we write. For
such that
if $\lambda-\mu\in$
$V$

$V$

$\mathfrak{h}$$\tau\in \mathfrak{h}^{*}$$V_{\tau}$$V=\Sigma_{\tau\in \mathfrak{h}^{*}}^{\oplus}V_{\tau}$$V$$\mathfrak{h}$$\dim_{\mathbb{C}}V_{T}<+\infty$$V=\Sigma_{\tau\in P(V)}^{\oplus}V_{r}$$C(\lambda)$$\lambda\in \mathfrak{h}^{*}$$V$$\tau\in \mathfrak{h}^{*}$$P(V)\subset\lambda-\Sigma_{i\in I}Z_{\geq 0}\alpha_{i}$$\lambda,$$\mu\in \mathfrak{h}^{*}$$\mu\leq\lambda$.
Here, we recall from [3, Definition 3.5] the definition of the multiplicity [V :$L(\mu)]$ of $L(\mu)$ in(in [3], the multiplicity [V : $L(\mu)$ ] isfor a module
defined for in a wider category ).
$\Sigma_{i\in I}Z_{\geq 0}\alpha;$$V$$V\in C(\lambda)$$V$$\mathcal{O}$PROPOSITION3.1([3, Proposition 3.2]). Let
there exists a finite in creasin ffl tration$\lambda,$$\mu\in \mathfrak{h}^{*}$, and $V\in C(\lambda)$ . Then,$g$$0=V_{0}\subset V_{1}\subset\cdots\subset V_{t}=V$of $g(A)- s$ ubmodules ofsuch th at for each $j(1\leq j\leq t)$ the quotien mod$ule$, or has no weights witheither is isomorphic to some$V$

$V_{j}/V_{j-1}$$\tau\geq\mu$.

$t$$L(\mu;)(\mu_{j}\in \mathfrak{h}^{*})$$\tau$

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We call the above filtration a local composition series of at . We know thatthe cardinality of the set { $1\leq j\leq t$ I $V_{j}/V_{j-1}\cong L(\mu)$ } is independent of the
choice of the local composition series of at . So, we call it the multiplicity of$L(\mu)$ in, and denote it by [V : $L(\mu)$ ].
Now, we choose and fix an element
such that.
From now on, we shall use the notation
$V$

$V$

$\mu$

$\mu$

$V$

$\langle\rho, \alpha_{i}^{V}\rangle=(1/2)\cdot a_{ii}(i\in I)$$\rho\in \mathfrak{h}^{*}$$(w, \beta)0\Lambda$

for$:=w(\Lambda+\rho-\beta)-\rho$

and.
We recall the following two theorems, which are essentially proved in [11].
$(w, \beta)\in W\cross \mathcal{A}$$\Lambda\in P_{+}$

THEOREM 3.2 (cf. [11, Proposition 2.11]). Fix. Then, we have$\Lambda\in P_{+}$. Let$(w_{1}, \beta_{1}),$$(w_{2}, \beta_{2})\in$$W\cross \mathcal{A}(\Lambda)$$\dim_{C}Hom_{g(A)}(V((w_{1}, \beta_{1})0\Lambda),$ $V((w_{2}, \beta_{2})0\Lambda))\leq 1$.

Note that any nonzero $g(A)$ -module homomorphism between two Verma modules is injective. So, we may write$V((w_{1}, \beta_{1})\circ\Lambda)\subset V((w_{2}, \beta_{2})\circ\Lambda)$

when the equality holds in the above theorem.
THEOREM 3.3 (cf. [11, Proposition 2.12]). Let. Then,$\Lambda\in P_{+},$$(w_{1}, \beta_{1}),$ $(w_{2}, \beta_{2})\in$$W\cross \mathcal{A}(\Lambda)$$V((w_{1}, \beta_{1})0\Lambda)\subset V((w_{2}, \beta_{2})0\Lambda)$$\Leftrightarrow$$(w_{1},\beta_{1})\geq(w_{2}, \beta_{2})$$\Leftrightarrow$$[V((w_{2}, \beta_{2})\circ\Lambda) :L((w_{1}, \beta_{1})\circ\Lambda)]\neq 0$.

4. TRANSLATION FUNCTORS, we define the translation functor
Here, forfrom the category $C(\mu)$, which is a generalization to GKM algebras of the one definedto the categoryin [12] by W. Neidhardt. This functor enables us to show that the multiplicity$\lambda,$$\mu\in \mathfrak{h}^{*}$$C(\lambda)$$T_{\mu}^{\lambda}$

159[V $((w,$: $L((w’,$] $((w, \beta),$does not depend on
the choice of.
Since we assume that the GGCM is symmetrizable, there exists a nondegenerate, symmetric, invariant bilinear form
on $g(A)$ . Recall that the restrictionof this bilinear form
to is also nondegenerate, so that it induces on. Then,nondegenerate, W-invariant bilinear form, which we again denote by
we can define the so-called (generalized) Casimir operator on the modules in
the category, or more generally in the category(see [6, Chaps.2
and 9]). Further, under the action of , a module $V\in C(\lambda)$ decomposes into thedirect sum$(w’, \beta’)\in W\cross \mathcal{A}(\Lambda))$$\beta’)0\Lambda)$$\beta)0\Lambda)$$\Lambda\in P_{+}$$A$$(\cdot|\cdot)$$(\cdot|\cdot)$$\mathfrak{h}^{*}a$$\mathfrak{h}$$(\cdot|\cdot)$$V$$\Omega$$C(\lambda)(\lambda\in \mathfrak{h}^{*})$$\mathcal{O}$$\Omega$$V= \sum_{k\in \mathbb{C}}^{\oplus}V^{(k)}$of generalized eigenspaces
for the eigenvalue $k\in C$ ofwith highest weighthighest weight $g(A)$ -module, where I
operator
denotesfor$V^{(k)}$$V$$(|\lambda+\rho|^{2}-|\rho|^{2})I_{V}$Definition 4.1. For
the categoryby$\lambda,$$\mu\in \mathfrak{h}^{*}$$\lambda\in \mathfrak{h}^{*},$$(\mu|\mu)$$\mu|^{2}$, define the functor$T_{\mu}^{\lambda}$$\Omega$. Note that on a$\Omega$$\mu\in \mathfrak{h}^{*}$

acts as the scalar.

from the category $C(\mu)$ to$C(\lambda)$$(V\in C(\mu))$$T_{\mu}^{\lambda}(V):=(V\otimes_{\mathbb{C}}L(\lambda-\mu))^{(|\lambda+\rho|^{2}-|\rho|^{2})}$

which we call the translation functor from

$\mu$

to

$\lambda$,.

Remark. By [4, Proposition 4.6], we see that the functor$T_{\mu}^{\lambda}$

is an exact functor.

Now, following the general line of [12], we can prove a series of propositions
below.
PROPOSITION 4.2. Let$\Lambda\in P_{+},$ $(w, \beta)\in W\cross \mathcal{A}$. Then, we have$T_{0}^{\Lambda}(V((w,\beta)00))\cong\{\begin{array}{l}V((w,\beta)o\Lambda)if\beta\in \mathcal{A}(\Lambda)0if\beta\not\in \mathcal{A}(\Lambda)\end{array}$

PROPOSITION 4.3. Let$\Lambda\in P_{+},$ $(w, \beta)\in W\cross \mathcal{A}$. Then, we ave$h$$T_{0}^{\Lambda}(L((w, \beta)\circ 0))\cong L((w, \beta)\circ\Lambda)$

Remark. $T_{0}^{\Lambda}(L((w, \beta)00))=0$ unless$\beta\in \mathcal{A}(\Lambda)$

.

or

$0$

.

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PROPOSITION 4.4. Let$\Lambda\in P_{+},$ $\mu\in \mathfrak{h}^{*},$ $(w, \beta)\in W\cross \mathcal{A}(\Lambda)$. If$0=V_{0}\subset V_{1}\subset\cdots\subset V_{t}=V((w, \beta)\circ 0)$

is a local composition series of $V((w, \beta)00)$ at$\mu-\Lambda$, then$0=T_{0}^{A}(V_{0})\subset T_{0}^{\Lambda}(V_{1})\subset\cdots\subset T_{0}^{\Lambda}(V_{t})\cong V((w, \beta)0\Lambda)$

is a $locaI$ composition series of $V((w, \beta)0\Lambda)$ at .
$\mu$

Using Propositions4.3and4.4, we can show the fo11owing, which is one of ourmain results.
THEOREM 4.5. Let[V $((w,$$\Lambda\in P_{+}$$\beta)0\Lambda)$. Then, for any: $L((w^{/},$$\beta’)0\Lambda)$$(w, \beta),$ $(w^{/}, \beta^{/})\in W\cross A(\Lambda)$, we have] $=[V((w, \beta)00) : L((w^{/}, \beta’)00)]$ .

As a corollary of the proof of Theorem 4.5, we obtain

COROLLARY 4.6. Let$\Lambda\in P_{+},$ $(w, \beta)\in W\cross \mathcal{A}(\Lambda)$. Then, we $have$$T_{0}^{A}(L((w, \beta)00))\cong L((w, \beta)0\Lambda)$.

5. GENERALIZATION OF TIIE KAZIIDAN-LUSZTIG CONJECTURE
5.1. Here, let $A_{J}=(a_{ij})_{i,j\in J}$ be a symunetrizable GCM ( generalized Cartan
matrix) indexed by a finite set , and let
be a Kac-Moody algebra overassociated towith the Cartan subalgebra , simple roots, simple coroots, and the Weyl group.
In addition, let $P_{w,w’}(q)(w, w^{/}\in W_{J})$ be the Kazhdan-Lusztig polynomials for the(see \S 2.2).
Coxeter group, we denote byFor
the Verma module with highest, we
its unique irreducible quotient. For
weight over , and bydenote by(see \S 3).
the multiplicity ofinFirst, we recall the following celebrated result due to Kashiwara $[7, 8]$ , or Casian
[2].
$=$

$J$

$C$

$g_{J}:=\mathfrak{g}(A_{J})$$A_{J}$$\Pi_{J}=\{\alpha_{i}\}_{i\in J}(\subset$$\mathfrak{h}_{J}$$\Pi_{J}^{\vee}=\{\alpha^{\vee}:\}_{i\in J}(\subset \mathfrak{h}_{J})$$\mathfrak{h}_{J}^{*})$$W_{J}(\subset GL(\mathfrak{h}_{J}^{*}))$$W_{J}$$\lambda\in \mathfrak{h}_{J}^{*}=Hom_{\mathbb{C}}(\mathfrak{h}_{J}, C)$$\lambda$$L_{J}(\lambda)$$g_{J}$$[V_{J}(\lambda) :$V_{J}(\lambda)$

L_{J}(\mu)]$$\lambda,$$L_{J}(\mu)$$V_{J}(\lambda)$$\mu\in \mathfrak{h}_{J}^{*}$

161
be a Kac-Moody algebra associatedTHEOREM 5.1 ([2], [7, 8]). Let. Assume that is an element of
to a symmetrizable GCM
such thatfor $aili\in J$ . Then, for any $w,$ $w’\in W_{J}$ , we have$g_{J}=\mathfrak{g}(A_{J})$$A_{J}$$\Lambda$$\mathfrak{h}_{J}^{*}$$\langle\Lambda, \alpha_{i}^{v}\rangle\in Z_{\geq 0}$$[V_{J}(w(\Lambda+\rho_{J})-\rho_{J}) :Here,$\rho_{J}$is a fixed element of$\mathfrak{h}_{J}^{*}$.

L_{J}(w’(\Lambda+\rho_{J})-\rho_{J})]=P_{w,w’}(1)$such that \langle$\rho_{J},$$\alpha_{i}^{v}$} $=1$ for a1J $i\in J$ .

5.2. We now return to the setting of \S 1-\S 4. Note that we assume that the GGCM
is symmetrizable. In [11], we have essentiaily proved the following theorem.

$A$. Then,THEOREM 5.2 (cf. [11, Proposition 2.9]). Let
any irred ucible subq uotient of $V((w, \beta)0\Lambda)$ is isomorphic tofor. Moreover, the converse statement
some
with
also holds.
$\Lambda\in P_{+},$$(w, \beta)\in W\cross \mathcal{A}(\Lambda)$$L((w^{/}, \beta’)0\Lambda)$$(w^{/}, \beta’)\in W\cross \mathcal{A}(\Lambda)$$(w^{/}, \beta’)\geq(w, \beta)$

Therefore, the multiplicities $[V((w, \beta)0\Lambda) : L((w’, \beta^{/})0\Lambda)]((w, \beta),$ $(w’, \beta’)\in$
are of great interest. Here, we shall derive some partial information
about the above multiplicities from Theorem 5.1, which is for the case of KacMoody algebr as.
Remark that the submatrix
of a symmetrizable GGCM
$A=(a_{ij})_{i,j\in I}$ is a symmetrizable GCM. Let
be the Lie subalgebra of $g(A)$, whereis a certain good subspace of ,generated by.
such that the triple
is a realization of the GCM) Then,is canonically(Here,
denotes the restriction oftoisomorphic to a Kac-Moody algebra
over as sociated to the GCM. In fact, we havewith the Cartan subalgebra$W\cross \mathcal{A}(\Lambda))$$A_{I^{re}}$$:=(a_{ij})_{i,j\in I^{re}}$$g_{I^{rc}}$$\mathfrak{h}_{I^{re}}U\{e_{i}, f_{i}|i\in I^{re}\}$$\mathfrak{h}$$\mathfrak{h}_{I^{re}}$$A_{I^{r*}}$$(\mathfrak{h}_{I^{re}}, \{\alpha_{i}|_{\mathfrak{h}_{t^{re}}}\}_{i\in I^{re}}, \{\alpha_{i}^{V}\}_{i\in I^{re}})$$\alpha_{i}|_{\mathfrak{h}_{I^{re}}}$$\alpha_{i}$$g(A_{I^{re}})$$\mathfrak{h}_{I^{re}}.$$g_{I^{re}}$$C$$A_{I^{re}}$$\mathfrak{h}_{I^{re}}$$\mathfrak{g}_{I^{re}}=\mathfrak{h}_{I^{re}}\bigoplus_{\alpha\in}\sum_{\Delta_{I^{r*}}}\oplus g_{\alpha}$,, or rather its restriction to, can be rewhere. From now on, we canonically identify thegarded as the root system of.
of $g(A)$ with
subalgebra
Then, we have the following by exactly the same argument as the one for [15,Theorem 2.3], abusing the notation of \S 5.1 under the above identification.
$\Delta_{I^{re}}$$:=\Delta\cap(\Sigma_{i\in I^{re}}Z_{\geq 0}\alpha_{i})$$\mathfrak{g}_{I^{r}}$.

$(g_{I^{re}}, \mathfrak{h}_{I^{re}})$$\mathfrak{g}(A_{I^{r*}})$$\mathfrak{h}_{I^{re}}$162
PROPOSITION 5.3. Let
have$\lambda,$$\mu\in \mathfrak{h}^{*}$. Assume that$\lambda-\mu\in\Sigma_{i\in I^{rc}}Z\alpha_{i}$$[V(\lambda):L(\mu)]=[V_{I^{re}}(\lambda|_{\mathfrak{h}_{I^{re}}}):L_{I^{re}}(\mu|_{\mathfrak{h}_{I^{re}}})]$

Here, forand

$L_{I^{rc}}(\tau|_{\mathfrak{h}_{t^{re}}})$.

is the Verma module over the Kac-Moody algebra,, whose highest weightis the restriction of tois its unique irreducible quotien .

$\tau\in \mathfrak{h}^{*},$$\mathfrak{g}_{I^{re}}(\cong g(A_{I^{re}}))$. Then, $we$

$V_{I^{re}}(\tau|_{\mathfrak{h}_{I^{re}}})$$\tau|\mathfrak{y}_{I^{re}}\in \mathfrak{h}_{I^{re}}^{*}$$\tau$$\mathfrak{h}_{I^{re}}$$t$

As a direct consequence of Theorem 5.1 and Proposition 5.3, using Theorem
3.3, we obtain the following theorem.

THEOREM 5.4. Let$(w, \beta),$$\Lambda\in P+,$

[V $((w,$$\beta)0\Lambda)$$(w^{/}, \beta^{/})\in W\cross \mathcal{A}(\Lambda)$: $L((w^{/},$$\beta^{/})0\Lambda)$]. Then, we$\geq P_{(w,\beta),(w’,\beta’)}(1)$$\Lambda ave$,is the extended Kazhdan-Lusztig polynomial (introduced in, or if $w=w^{/}=1$ .
\S 2.2). Moreover, the $eq$ uality holds ifwhere$P_{(w,\beta),(w’)\beta)}(q)$$\beta=\beta’$Now, recall that the Weyl group $W$ of the GKM algebra $g(A)$ is by definition
the subgroup ofgenerated by the simple reflections $r_{i}(i\in I^{re})$ . However,$W$ by itself seems to be too small for the description of the representation theoryof $g(A)$ . Actually, from Theorems 3.3 and 5.2, we have an impression that theof $W$ and behaves as if it were the true (Weyl group” ofdirect productthe GKM algebra $g(A)$ .
On the other hand, in the case where $a_{ii}\neq 0(i\in I)$ , the set $\mathcal{A}=A(0)$ consistsof all sums of distinct, pairwise perpendicular, imaginary simple roots. So, in
this case,can be embedded into the Coxeter group $(Z/2Z)^{m}$ with $m$, via the identification of an imaginary simple root
the cardinality of the set. Hence,can be embedded into thewith a generator
direct product $W\cross(Z/2Z)^{m}$ of Coxeter groups, together with the Bruhat ordering(see Definitions 2.1-2.3).
Under this embedding, the Kazhdan-Lusztig polynomial associated to the elements $(w, \beta),$should be just the extended Kazhdan-Lusztig
polynomial
defined in \S 2.2. (Here, note that the Kazhdan-Lusztig
polynomial
for the Coxeter group $Z/2Z$ with generator 1 is
identically equal to 1.) Therefore, it seems natural to us to suggest the following
conjecture.
$GL(\mathfrak{h}^{*})$$W\cross \mathcal{A}$$\mathcal{A}$$\mathcal{A}=\mathcal{A}(0)$$I^{im}$$\alpha_{j}\in\Pi^{im}$$\overline{1}\in Z/2Z$$(w’, \beta’)\in W\cross \mathcal{A}$$P_{(w,\beta),(w’,\beta’)}(q)$$P_{\overline{0},\overline{1}}(q)(\overline{0}, \overline{1}\in Z/2Z)$$W\cross A$

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CONJECTURE. Assume that the GGCM $A=(a_{ij})_{i,j\in I}$ satisfies the condition that$a_{ii}\neq 0(i\in I)$ . Let. Then, we have$\Lambda\in P_{+},$[V $((w,$$(w, \beta),$$\beta)0\Lambda)$:$(w^{/}, \beta’)\in W\cross \mathcal{A}(\Lambda)$$L((w^{/},$ $\beta’)\circ\Lambda)$] $=P_{(w,\beta),(w’,\beta’)}(1)$ .

5.3. Since this paper was prepared, we have succeeded in proving that the above
conjecture is true. We now sketch briefly the idea of the proof. From now on, we
assume that the GGCM $A=(a_{ij})_{i,j\in I}$ is symmetrizable, and satisfies the $co$ ndition
that $a_{ii}\neq 0(i\in I)$ . Note that in this case the set
consists of all sums of
distinct, pairwise perpendicular, imaginary simple roots perpendicular to.
Then, we can prove the following generalization to GKM algebras of Jantzen’s
character sum formula corresponding to a quotient of two Verma modules (cf. [5]and [14]).
$A(\Lambda)$$\Lambda^{rightarrow}\in P_{+}$

THEOREM 5.5. Let $g(A)$ be a $GKM$ algebra associ ated to a symmetrizable GGCM
$A=(a_{ij})_{i,j\in I}$ sa tisfying the condition that $a_{ii}\neq 0(i\in I)$ .
Fix
Let$\alpha=w(\alpha_{j})\in\Delta_{+}$ , where $w\in W$ andwith. We set$\lambda:=w(\Lambda+\rho)-\rho=(w, 0)0\Lambda,$ $\mu:=\lambda-\alpha=w(\Lambda+\rho-\alpha_{j})-\rho=(w, \alpha_{j})0\Lambda$ , and$N(\lambda):=V(\lambda)/V(\mu)$ (see Theorem 3.3). Then,
has a $g(A)$ -module fiItration$\Lambda\in P+\cdot$$\alpha_{j}\in\Pi^{im}$$\langle\Lambda, \alpha_{j^{\vee}}\rangle=0$$N(\lambda)$$N(\lambda)=N(\lambda)_{0}\supset N(\lambda)_{1}\supset N(\lambda)_{2}\supset\cdots$such that:
(1)
(2)
ch

$N(\lambda)/N(\lambda)_{1}\cong L(\lambda)$

as a $g(A)$ -mod $ule$;$N(\lambda)_{i}$$\sum_{i\geq 1}$$= \sum_{\beta\in\Delta+}$$\sum_{j>1}$ch$V( \lambda-j\beta)-\sum_{\gamma\in\Delta+}$$2(\lambda+\rho|\beta\overline{)}=j(\beta|\beta)$-ch$V(\lambda-\alpha)$$\sum_{m\geq 1}$

ch $V(\lambda-\alpha-m\gamma)$$2(\lambda-\alpha+\rho|\gamma)=m(\gamma|\gamma)$.

Here, ch denotes the formal character.
By double induction on $\ell(w’)-l(w)$ and $ht(\beta’)-ht(\beta)$ , using Theorem 5.4
as the starting point of the induction and Theorem 5.5 for the induction step,
we can prove that the above conjecture holds under the condition on the GGCM
$A=(a_{ij})_{i,j\in I}$ that $a_{ii}\neq 0(i\in I)$ .

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As a consequence, we obtain the following theorem.

THEOREM 5.6. Let $g(A)$ be a $GKM$ algebra. Assume that the symmetrizable
GGCM $A=(a_{ij})_{ij\in I}$ satisfies the condition that $a_{ii}\neq 0(i\in I)$ . Let.
, we haveThen, for$\Lambda\in P_{+}$$(w, \beta)\in W\cross \mathcal{A}(\Lambda)$

ch $V((w, \beta)0\Lambda)=\sum_{(w}))$
EquivaIently, for$(w, \beta)\in W\cross \mathcal{A}(\Lambda)$

ch $L((w’, \beta’)0\Lambda)$ .

, we $have$

ch $L((w, \beta)0\Lambda)$$= \sum_{(w^{l},\beta’)\in W\cross A(A)}(-1)^{(l(w’)+ht(\beta’))-(l(w)+ht(\beta))}Q_{(w,\beta),(w’,\beta’)}(1)$ch$V((w^{/}, \beta’)0\Lambda)$,

where $Q_{(w,\beta),(w’,\beta’)}(q)((w’, \beta’)\in W\cross A(\Lambda))$ are the $ex$ tended Kazhdan-Lusztig
polyn onials.

Remark. It is well-known that ch$V( \lambda)=e(\lambda)\cdot\prod_{\alpha\in\Delta+}(1-e(-\alpha))^{-mult(\alpha)}$$(\lambda\in \mathfrak{h}^{*})$

,

where
is a formal exponential for(see [6, Chap.10]). Moreover, weknow that $Q_{1,w’}(1)=1(w^{/}\in W)$ . Therefore, in view of the Weyl-Kac-Borcherds
character formula for(see [1], or [6, Chap. 11]), the condition on the
GGCM $A=(a_{ij})_{i,j\in I}$ that $a_{ii}\neq 0(i\in I)$ of Theorem 5.6 seems to be necessary.
$e(\tau)$$\tau\in \mathfrak{h}^{*}$$L(\Lambda)(\Lambda\in P_{+})$

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