数理解析研究所講究録
1017 巻 1997 年 54-69

54

GENERALIZED KAC-MOODY ALGEBRA に対する
HARISH-CHANDRA HOMOMORPHISM について内藤

聡

(SATOSHI NAITO)

筑波大学

数学系

Dedicated to Professor Yasuo Yamasaki

INTRODUCTION

Let$\mathrm{g}(A)$be the complex contragredient Lie algebra associaed to a symmetrizable

real square matrix $A=(a_{ij})_{i,j\in I}$ indexed by a finite set$I$(see [K1] and [KK] fordetails). In [K2], Kac introduced a complex associative algebra$U_{F}(\mathrm{g}(A))$, which canbe thought of as a certain completion of the universal enveloping algebra
the contragredient Lie algebra$H$$\mathrm{g}(A)$and the algebra$\mathcal{F}$of complex-valued fuctions on the set$Z_{F}$of the algebra$\mathfrak{h}^{*}\backslash L$, wherethe union of certain infinitely many affine hyperplanes in the algebraic dual

Cartan subalgebra$\mathfrak{h}$of$\mathrm{g}(A)$$\mathfrak{h}^{*}$“vector-valued” functions on the interior $K$ of the complexified Tits cone$\mathrm{g}(A)$

$L$

is

of the.

Moreover, he studied the “holomorphicity” of the elements of the algebra

where

of

. In it he showed that there exists an isomorphism(called the Harish-Chandra homomorphism) between the center$\hat{U}_{F}(\mathrm{g}(A))$$U(\mathrm{g}(A))$$X_{\mathbb{C}}$$\sim$$Z_{F}$

asinthe case..

is the symmetrizable Kac-Moody algebra (i.e., the matrix $A=(a_{ij})_{i,j}\in I$ isa symmetrizable generalized Cartan matrix).

In this paper, we generalize his results in [K2] to the case where$\mathrm{g}(A)$is the sym-

metrizable generalized Kac-Moody algebra (i.e., the complex contragredient Lie algebra
associated to a certain symmetrizable real matrix $A=(a_{ij})_{i,j\in I}$ , called a GGCM).
Typeset by$A_{\mathcal{M}}S-\mathrm{I}\mathrm{E}X$55

1. HARISH-CHANDRA HOMOMORPHISM

In this section we briefly review the setting and some results in [K2], which are valid

for arbitrary symmetrizable contragredient Lie algebras over , hence for symmetrizable$\mathbb{C}$

generalized Kac-Moody algebras over$\mathbb{C}$.

1.1. A completion of the universal enveloping algebra. Let$\mathrm{g}(A)$metrizable generalized Kac-Moody algebra (GKM algebra for short) over
Lie algebra$\mathrm{g}(A)$be the sym$\mathbb{C}$. Then theis nothing but the contragredient Lie algebra associated to a symmetriz-

able real matrix $A=(a_{ij})_{i,j\in I}$ (called a GGCM) indexed by a finite set$I$satisfying thefollowing conditions:(C1) either $a_{ii}=2$ or$a_{ii}\leq 0$(C2) $a_{ij}\leq 0$ if $i\neq j$ , and(C. 3)$a_{ij}.=0\Leftrightarrow a_{ji}=0$

for $i\in I$ ;$a_{ij}\in \mathbb{Z}$for $j\neq i$ if $a_{ii}=2$ ;.

Note that this definition of GKM algebras is due to Kac (see [Kl, Chap. 11]), andslightly different from the original one by Borcherds in [B1] . From now on we follow$)$

the notation of [K1], and freely use results in it (see also our previous papers [N1]-

[N3] .
$)$

Let$\mathfrak{h}$be the Cartan subalgebra of the GKM algebra$\mathrm{g}(A)$. Then, since we have been

assuming that the GGCM $A=(a_{ij})_{i,j\in I}$ is symmetrizable, there exists a nondegenerate
symmetric$\mathbb{C}$-bilinear form$(\cdot|\cdot)$on the dual$\mathfrak{h}^{*}$of , which is invariant under the action$\mathfrak{h}$of the Weyl group W. (Here recall that the Weyl group $W$ of the GKM algebraby definition the subgroup of$GL(\mathfrak{h}^{*})$

generated by the fundamental reflections$\mathrm{g}(A)$$r_{i}$

iswith$a_{ii}=2.)$

Now, for$\alpha\in Q=\sum_{i\in I}\mathbb{Z}\alpha_{i}$, we define the affine linear function$T_{\alpha}(\lambda)=2(.\lambda.+\rho|\alpha)-(\alpha|\alpha)(\lambda\in \mathfrak{h}^{*})$, where$\rho\in \mathfrak{h}^{*}$$T_{\alpha}(\cdot)$is a fixed element of$\mathfrak{h}^{*}$

on$\mathfrak{h}^{*}$

by:such that

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HARISH-CHANDRA HOMOMORPHISM

2

for $i\in I$ . Then we put$(\rho|\alpha_{i})=(\alpha_{i}|\alpha_{i})$$L$.

$\cup\{\lambda\in \mathfrak{h}^{*}|T_{n\beta}(\lambda+\gamma)=0\}$$:=$$n\in \mathbb{Z}^{+}\beta\in\gamma\in Q\Delta\geq 1$Let$\mathcal{F}$be the algebra of -valued functions defined on

dense in$\mathcal{F}$$\mathfrak{h}^{*}\backslash L$$\mathbb{C}$$\mathfrak{h}^{*}$, wherein the usual metric topology, there exists a canonical embedding$S(\mathfrak{h})$the action. Because the set$\pi$

is viewed as the algebra of polynomial functions onof the universal enveloping algebraon the algebra$\mathcal{F}$by:$\pi(e\rho)\varphi(\cdot)=\varphi(\cdot+\beta)$$h(e\rho)=\beta(h)e\rho(\beta\in Q, h\in \mathfrak{h})$By using the action

of

$\pi$

$\varphi(\cdot)\in \mathcal{F}$

and

on

$\mathcal{F}$$\psi(\cdot)\in \mathcal{F}$and$e_{\alpha},$$U(\mathrm{g}(A))\otimes_{\mathbb{C}}\mathcal{F}$$e\rho\in U(\mathrm{g}(A))$

algebra of this associative algebra
the elements$f\otimes\cdot 1-1\otimes\iota(f)$generated by the algebra$\mathcal{F}$

ere

for

and

with

$\alpha,$$\beta\in Q$$U(\mathrm{g}(A))\otimes_{\mathbb{C}}\mathcal{F}$$f\in S(\mathfrak{h})$$U(\mathrm{g}(A))$$\dot{\varphi}(\cdot)\in \mathcal{F}\mathrm{a}\mathrm{n}\mathrm{d}e_{\beta}\sim:’\in\wedge U(\mathrm{g}(A))_{\beta}$. Let$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\dot{\mathrm{t}}\prime \mathrm{x}$of$U_{F}(\mathrm{g}(A))$

be the quotientby the two-sided ideal generated by$U_{F}(\mathrm{g}(A))$

is, and the following relation holds in it:with $\beta\in Q.$$\mathrm{v}\dot{\mathrm{e}}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}$,$\grave{\mathrm{M}}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{o}\dot{}\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}\dot{\mathrm{h}}\mathrm{i}\mathrm{s}\sim$algebra$U_{F}(\mathrm{g}(A))$

spaces as:$U_{F}(9(A))=U(\mathfrak{n}_{-})\otimes_{\mathbb{C}}\mathcal{F}\otimes_{\mathbb{C}}U(\mathfrak{n}_{+})$and canonically contains the algebra,. Then the associative algebra’

decomposes into

, whereby:$\varphi(\cdot)e\rho-e\beta\varphi(\cdot)=e_{\beta(}\varphi(\cdot+\beta)-\varphi(\cdot))$$\mathrm{w}\dot{\mathrm{h}}$$\mathrm{g}(A)$, we can define the structure of an associative$.=$$\varphi(\cdot),$. Here we define$e_{\beta}\in U(\mathrm{g}(A))_{\beta}$$(e_{\alpha}\otimes\varphi(\cdot))(e_{\beta^{\otimes\psi(\cdot))}}:=e_{\alpha\rho\otimes}e(\pi(e_{\beta})\varphi(\cdot))\psi(\cdot)$for$\iota:S(\mathfrak{h})arrow$of the GKM algebra$U(\mathrm{g}(A))$

is.

$U(\mathrm{g}(A))$

algebra on the vector spacefor$\mathfrak{h}^{*}$$\mathfrak{h}^{*}\backslash L$,:.

$U(\mathrm{g}(A))=U(\mathfrak{n}-)\otimes_{\mathbb{C}}S(\mathfrak{h})\otimes_{\mathbb{C}}U(\mathfrak{n}_{+})$.

By putting $\deg(e_{i})=1$ and $\deg(f_{i})=-1$ for $i\in I$ , and $\deg(\mathcal{F})=0$ , we have a$\mathbb{Z}$-gradation of$U_{F}(\mathrm{g}(A))$

as:

‘

,

$\oplus U_{-k}(\mathfrak{n}_{-})\otimes \mathrm{c}\mathcal{F}\otimes \mathbb{C}Um(\mathfrak{n}+)$$U_{F}(\emptyset(A))=\oplus U_{F(}\emptyset(A))_{j}j\in \mathbb{Z}’ U_{f}(\mathfrak{Z}(A))j:=$$m-kjk,m\overline{\overline{\geq}}0$

57

so that we can “complete” it in a canonical way as:,$\hat{U}_{F}(9(A)):=\oplus\hat{U}\tau(\mathrm{g}(A.)j\in \mathbb{Z}\sim.)_{j},\hat{U}_{\mathcal{F}}(\mathrm{g}(A.))_{j}:=m-k-\prod_{k,m\overline{\geq}0^{j}}U-k(\mathfrak{n}_{-})\otimes_{\mathbb{C}}\mathcal{F}\otimes \mathbb{C}U_{m}(\mathfrak{n}_{+})$

where$U_{m}(\mathfrak{n}_{+})$(resp.

is the subspace of$U_{-k}(\mathfrak{n}_{-))}$$U(\mathfrak{n}_{+})$(resp.

$U(\mathfrak{n}$(resp. $-k$ ). Note that the multiplication in $U_{F}(9(A))$ extends to$\hat{U}_{F}(\mathrm{g}(A))$

is an associative algebra containing

Moreover, if$V(\Lambda)$

then the action of$\hat{U}_{F}(\mathrm{g}(A))$

is a highest weight$U(\mathrm{g}(A))$, while the algebraon$\mathcal{F}$$V(\Lambda)$

acts on$\mathrm{g}(A)$$U_{F}(\epsilon(A))$$\varphi(\cdot)\in \mathcal{F}$and$v_{\tau}\in V(\Lambda)_{\tau}$$\hat{U}_{F}(\mathrm{g}(A))$

set of maps

$V(\Lambda)$

For each$k:\triangle\sim+arrow \mathbb{Z}\geq 0$$\beta\in Q_{+}$

such thatby:$\beta$elements of$\triangle\sim+\mathrm{b}\mathrm{e}$,$\tilde{f}_{\alpha}\in 9-\alpha$$\hat{U}_{F}(\mathrm{g}(A))$$\varphi_{k,m}\in \mathcal{F}$and$Z_{F}$.

the center of the asso-$\beta\in Q_{+}=\sum_{i\in I}\mathbb{Z}\geq 0\alpha_{i}$$\{F^{k}\}_{k\in}\mathrm{p}\mathrm{a}\mathrm{r}\beta$)$F^{k}= \prod_{\alpha\in\triangle}\sim\tilde{f}^{k(}+\alpha\alpha$is a root vector for a root

, denote by Par, and put Par:$\mathrm{p}\mathrm{r}o$duct) for$\alpha\in\triangle\sim+\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$

that

the

$= \bigcup_{\beta\in Q}+^{\mathrm{P}\mathrm{a}}\mathrm{r}\beta$of the vector space(finite$\beta$$U(\mathfrak{n}_{-})_{-\beta}$

.

con-

$k=(k(\alpha))_{\alpha\in\overline{\Delta}}+\in$$9-\alpha=\oplus \mathbb{C}\tilde{f}_{\alpha}$. Thenare expresed in the form$\sum_{k,m\in \mathrm{p}\mathrm{a}\mathrm{r}}Fk\varphi k,m\sigma(Fm)$with$\tau\in \mathfrak{h}^{*}$

the multiset in which every positive root$\beta=\sum_{\alpha\in\tilde{\Delta}}+k(\alpha)\alpha$, we can choose a basis

sisting of elements of the form
Par , where,.

with its multiplicity. For.

$\Lambda\in \mathfrak{h}^{*}\backslash L$

can be extended to the action of the algebra

is a weight vector of weight

Now we prepare some notation. Let$\alpha\in\triangle+\mathrm{a}_{\mathrm{P}\mathrm{P}^{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}}}$, so that.

1.2. Harish-Chandra homomorphism. We denote by

ciative algebra$\hat{U}_{\mathcal{F}}(\mathrm{g}(A))$-module with highest weight$\varphi(\cdot)(v_{\tau})=\Psi(\tau)v_{r}$where-)) of degree $m$(infinite sum),.

$|\deg(F^{m})-\deg(F^{k})|<\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$In [K2], Kac proved the following theorem. (Here we also record the full proof by

Kac for later use.)

58

Theorem 1 ([K2, Theorem 1]). Leta$\mathrm{u}n\mathrm{i}q\mathrm{u}e$th atelement$\varphi_{0,0}=\varphi$

be a fun tion on$c$$\varphi\in \mathcal{F}$. Here$\sigma$is the involu tive anti-automorphi

for $i\in I$ , and $\sigma(h)=h$ for$\sigma(e_{i})=f_{i},$ $\sigma(f_{i})=e_{i}$Proof. First we note that an element$h\in \mathfrak{h}$$x\in\hat{U}_{\mathcal{F}}(\mathrm{g}(A))$

a zero operator on each Verma module$\sum_{\beta\in Q}\sum k,m\in \mathrm{p}\mathrm{a}\mathrm{r}\rho\varphi F^{k}k,m\sigma+(Fm)$$\varphi_{0,0}(\Lambda)$

on each Verma module

we will choose$\varphi_{k,m}\in \mathcal{F}$acts as the scalar$z_{\varphi}$with

We denote by$\varphi 0,\mathrm{o}(\Lambda)=\varphi(\Lambda)$

of weight

functions$\varphi_{k,m}$with$\beta,\gamma\in Q_{+}$. Let us fix$k,$ $m\in$$\Lambda\in \mathfrak{h}^{*}\backslash L$

for$\beta\in Q_{+}$

Par7 for. For the matrix

determined by

eig.ht .

$\Lambda$$\in \mathfrak{h}^{*}\backslash$of the form

$z_{\varphi}\in\hat{U}_{F}(\mathrm{g}(A))$$Z_{F}$

if$z_{\varphi}=$. Therefore,in such a way that

$M(\Lambda)_{\Lambda-\beta}$

for each$\lambda\leq\mu\Leftrightarrow\mu-\lambda\in Q_{+}$$v_{\Lambda}\in M(\Lambda)$$\beta\in Q_{+}$$B_{\beta}^{\Lambda}(F^{k}, F^{m})\in \mathbb{C}$on$M(\Lambda)_{\Lambda-\beta}$

is a highest weight vector. Assume that we have already chosen the$\gamma<\beta$, so that we know the matrices$G_{\beta}^{\beta}(\Lambda)$

the condition that$z_{\varphi}$

is determined byacts on$M(\Lambda)_{\Lambda-\beta}$$G_{\gamma}^{\beta}(\Lambda)$$\sigma(F^{k})F^{m}(v\Lambda)=B_{\beta}^{\Lambda}(F^{k}, F^{m})v_{\Lambda}$

as the scalar$\varphi(\Lambda)$$\Phi\rho(\Lambda)B^{\Lambda}\beta+\sum_{\beta\gamma<}G^{\rho}(\gamma\Lambda)=\varphi(.\Lambda)\mathrm{I}\mathrm{d}$$G_{\gamma}^{\beta}(\Lambda)=0$$\det B_{\beta}^{\Lambda}$

for$\gamma\not\leq\beta$for, we have that

$(*)$

since

.

.

$G_{\beta}^{\beta\Lambda}(\Lambda)=\Phi_{\beta}(\Lambda)B\beta’\Phi\rho(\Lambda):=(\varphi k,m(\Lambda))_{k,\mathrm{p}\mathrm{r}\rho}m\in \mathrm{a}’ B_{\beta}^{\Lambda\Lambda km}:=(B_{\beta}(F, F))k,m\in \mathrm{p}_{\mathrm{a}}\mathrm{r}\beta$

Here

(cf.

acts as the$z_{\varphi}$$\Lambda\in \mathfrak{h}^{*}\backslash L$$\beta\in Q_{+}$$L$.

$.\mathrm{w}$$\sum_{k,m\in \mathrm{P}\mathrm{a}\mathrm{r}}\gamma F^{k}\varphi_{k,m}\sigma(F^{m})$, wheresuch$\varphi_{k,m}\in \mathcal{F}$is zero if and only if it acts as

by induction on

defined by:

with

.

on the weight space$\leq \mathrm{o}\mathrm{n}\mathfrak{h}^{*}$. Then there exists$U(\mathrm{g}(A))$

with highest weightthe matrix of the oprator$\{F^{s}(v_{\Lambda})\}S\in \mathrm{p}\mathrm{a}\mathrm{r}\rho$$\Lambda\in \mathfrak{h}^{*}\backslash L$and$\gamma<\beta$$G_{\gamma}^{\beta}(\Lambda)$$M(\Lambda)$

of$s\mathrm{m}$is in the center$\varphi_{k,m}\in \mathcal{F}$$k,$ $m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$

Here we use a partial orderingin the basis

with$Z_{F}$

with highest$M(\Lambda)$

the proof of Proposition 1 below). So the element

scalarin$z_{\varphi}= \sum_{\rho\in Q}+\sum_{k},m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta\varphi Fk\sigma k,m(Fm)$$\mathfrak{h}^{*}\backslash L$.

. Moreover,can be written as:,. Here we recall from [KK, Theorem 1] that the determinant

can be writen as:$\det B_{\beta}^{\check{\Lambda}}=.\prod\sim\prod T_{n\alpha}^{\cdot}.(\Lambda)\infty\#(^{\mathrm{p}}\mathrm{a}\mathrm{r}(\rho-n\alpha))$$\alpha\in\tilde{\Delta}+^{n=1}$,

59

up to a nonzero constant factor independent of . Because$\Lambda$$0$, so that$\varphi_{k,m}(\Lambda)$for$\Lambda\in \mathfrak{h}^{*}.\backslash L,$$\Lambda\in \mathfrak{h}^{*}\backslash L$

is determined.

$k,$ $m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$, we have$\det B_{\beta}^{\Lambda}\neq$$\square$Conversely we have the following proposition.
Proposition 1. An$\mathrm{e}le\mathrm{m}en.t_{X}\in\hat{U}_{F}(\mathrm{g}(A))$..

$x= \sum_{\beta\in Q}\sum k,m\in^{\mathrm{p}_{\mathrm{a}}\mathrm{r}}\beta F^{k}+\varphi_{k,m}\sigma(F^{m})$Proof. Letof the Verma module$v_{\Lambda}$

So

$x$

for somewith$x= \sum_{k,m\in \mathrm{p}\mathrm{a}\mathrm{r}}F^{k}\varphi k,m\sigma(F^{m})$constant be an element of the center$M(\Lambda)$

lies in the center$Z_{F}$$\varphi_{k,m}\in \mathcal{F}$$\varphi_{k,m}\in \mathcal{F}$$Z_{F}$

only if

it.is of the form.
and $|\deg(F^{m})-\deg(F^{k})|<$. It is clear that, for a highest weight vector

with highest weightacts as a scalar on each Verma module$\Lambda\in \mathfrak{h}^{*}\backslash L$$M(\Lambda)$, we have$x(v_{\Lambda})\in \mathbb{C}v_{\Lambda}$with highest weight$\Lambda\in \mathfrak{h}^{*}\backslash L$.
.

Note that, in the summation above for the expression of $x,$ $m$ is an element of the set
Par

for

$=\mathrm{U}_{\beta\in Q}\mathrm{p}_{\mathrm{a}}\mathrm{r}\beta+\cdot$$k\not\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$

Let us fix

We will show by induction on$\beta\in Q_{+}$

and$\Lambda\in \mathfrak{h}^{*}\backslash L$$M(\Lambda)_{\Lambda-\beta}$$B_{\beta}^{\Lambda}=(B_{\beta}^{\Lambda}(F^{k}, F^{m}))k,m\in \mathrm{p}\mathrm{a}\mathrm{r}\beta$$v\in M(\Lambda)_{\Lambda-\beta}$

such that$m\neq\sim m_{0}\vee\in$. Par$\beta$. The element

is nonsingular for. Then$\sum$$x$$\sum$$\sum$$c\in \mathbb{C}$EPar 7$\varphi_{k,m_{0}}(\Lambda)=0$, and $\sigma(F^{m})(v)=0$$\sum$$k$$\gamma$with$c\varphi_{k,m_{0}}(\Lambda)F^{k}(v_{\Lambda})$

,

EPar

$\gamma<\beta$

for any

are linearly independent. This means that.

. Because the matrix, we can choose an element$F^{k}\varphi_{k,m}\sigma(F^{m})(v)+$sumption. Therefore, we deduce that$k\not\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$$m_{0}\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$

for some nonzero

for $m\in$ Par$\{F^{k}(v_{\Lambda})\}k\in \mathrm{p}_{\mathrm{a}\mathrm{r}}$$\beta$$\Lambda\in \mathfrak{h}^{*}\backslash L$$F^{k}\varphi_{k,m}\sigma(F^{m})(v)\in M(\Lambda)_{\Lambda-\beta}$for, then $\varphi_{k,m}=0$$\mathrm{w}\mathrm{e}_{:}\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}$$k\in \mathrm{P}\mathrm{a}\mathrm{r}\gamma<\beta m$$\mathcal{F}$$m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$

acts as a scalar (independent of ). Now fix an arbitrary$\sigma(F^{m_{0}})(v)=cv_{\Lambda}$$M(\Lambda)_{\Lambda}-\beta\supset \mathbb{C}_{V}\ni X(V)=$

wherethat if.

on the weight space

for any$\beta$by the inductive as-$k\not\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$$\varphi_{k,m_{0}}=0$

since the vectors
as an element of$\square$From Theorem 1 and Proposition 1, we see that there exists an algebra isomorphism$H:Z_{F}arrow \mathcal{F}$defined byhomomorphism.

$z_{\varphi}\mapsto\varphi=\varphi_{0,0}$. we call this isomrphism$H$

the Harish-Chandra

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2. HOLOMORPHICITY OF THE FUNCTIONS$\varphi_{k,m}$

2.1. The Tits cone of GKM algebras. From now on, we assume that the GKM

algebraover$\mathrm{g}(A)$$\mathbb{C}$$\mathrm{g}(A)=\mathbb{C}\otimes_{\mathrm{R}9}(A)_{\mathrm{R}})$is the complexification of the GKM algebra. So the Cartan subalgebraof the Cartan subalgebra$\{\alpha_{i}\}_{i\in I}$(i.e.,$\mathfrak{h}_{\mathrm{R}}$$\mathfrak{h}=\mathbb{C}\otimes_{\mathrm{R}}\mathfrak{h}_{\mathrm{R}}$$\mathfrak{h}$over$\mathfrak{h}^{*}$is also denoted by$(\cdot|\cdot)$

Here we define the fundamental chamber$C$$\mathbb{R}$(i.e.,is also the complexification

is a linearly independent subset of the algebraic dual

complexification onover), and the set of simple roots $\Pi=$there exits a nondegenerate $W$ -invariant symmetric$\mathrm{g}(A)$$\mathbb{C}$$\mathrm{g}(A)_{\mathrm{R}}$$\mathbb{R}$$\mathfrak{h}_{\mathrm{R}}^{*}$

of$\mathfrak{h}_{\mathrm{R}}$-bilinear form

over$(\cdot|\cdot)$on$\mathbb{R}$. Further$\mathfrak{h}_{\mathrm{R}}^{*}$, whose.
and the Tits cone $X$ of the GKM algebra. We put$C:=${$\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\alpha_{i})\geq 0$

for $i\in I$ },and then $X:=W \cdot C=\bigcup_{w\in W}w\cdot C$ . We denote by
the closure) of $X$ in the usual metric topology of$\mathfrak{h}_{\mathrm{R}}^{*}$$X^{\mathrm{o}}$(resp.

$X^{-}$) the interior (resp.

.

Remark 1. In [B3] and [K1], the fundamental chamber was defined to be the set$C^{re}:=${$\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\alpha_{i})\geq 0$

and the the Tits cone was defined to be

for $i\in I$ with $a_{ii}=2$ },$X^{r\mathrm{e}}:=W\cdot C^{r\mathrm{e}}$. However this definition is not

appropriate for our purpose here.
The proof of the following lemma is almost the same as in the case of Kac-Moody

algebras (see [Kl, Chap. 3] and$[\mathrm{W}$, Chap. 4]).

Lemma 1. (1) The fundamental chamber$W$

on $X$, i.e., any orbit$W\cdot\lambda$

of$\lambda\in X$$C$

is a fun damental domain for the action of

int ersects$C$

in exactly one point. Moreover, $W$

operates simply transitively on chambers.
(2) $X=$ {$\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\alpha)<0$of$\alpha\in\triangle_{+}$}. In particular,

for only a finite number of$\alpha\in\triangle_{+}$}.

for only a fini$t\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{m}ber$

is a convex cone.
(3)$X^{\mathrm{O}}=${$\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\alpha)\leq 0$

$X$

61

Here we prepare some more notation for GKM algebras. Let $\Pi^{re_{j}}=\{\alpha_{i}\in\Pi|a_{ii}=$$2\}$

be the set of real simple roots, and II

simple roots,$im:=\{\alpha_{i}\in \mathrm{I}\mathrm{I}|a_{ii}\leq 0\}$the set of real roots, and$\triangle^{r\mathrm{e}}:=W\cdot\Pi^{re}$imaginary roots. We know from [Kl, Chap. 11] that$N=${$\alpha\in Q_{+}\backslash \{0\}|(\alpha|\alpha_{i})\leq 0$

In particular, the set

for

$i$

the set of imaginary$\triangle^{i\mathrm{m}}\cap\triangle+=W\cdot N$

with $a_{ii}=2$ , andthe set of$\triangle^{im}:=\triangle\backslash \triangle^{re}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\alpha)$is, where$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}$}$\backslash j\geq\cup j\cdot\Gamma \mathrm{I}^{i}2m$.

is W-stable.

$\triangle_{+}^{im}:=\triangle+\cap\triangle^{im}$Now we have the following lemma.
Lemma 2. (1)(2)$X^{\mathrm{o}}\subset${{$X^{-}\subset$$\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\alpha)>0$

Proof. (1) Let $X’:=$ {set$X^{l}$

for all$\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\alpha)\geq 0$

for all$\alpha\in\triangle_{+}^{im}$is a $W$ -stable closed subset of$\mathfrak{h}_{\mathrm{R}}^{*}$

since

}.

}.

for all$\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\alpha)\geq 0$$\alpha\in\triangle_{+}^{im}$$\alpha\in\triangle_{+}^{im}$$\triangle_{+}^{im}$}. Then it is clear that theis $W$-stable. Because $C\subset X’$ fromthe definition, we have $X\subset X’$ , so that $X^{-}\subset X’$ .
(2) Put

such that$\epsilon>0$that$l:=\dim_{\mathrm{R}}\mathfrak{h}_{\mathrm{R}}*$$(v_{i}|\alpha)\neq 0$$\lambda\pm\epsilon v_{i}\in X$. If$(v_{i}|\alpha)>0$

If $(v_{i}|\alpha)<0$ , we haveLet

, and take a basisoffor $1\leq i\leq l$ . For any, we have$\mathfrak{h}_{\mathrm{R}}^{*}$. Let$\alpha\in\triangle_{+}^{im}$$(\lambda|\alpha)\geq\epsilon(v_{i}|\alpha)>0$$(\lambda|\alpha)\geq-\epsilon(v_{i}|\alpha)>0$

since$I\mathrm{f}-=$$X^{\mathrm{o}}+\sqrt{-1}\mathfrak{h}_{\mathrm{R}}^{*}$.

$X_{\mathbb{C}}$$\lambda\in X^{\mathrm{O}}$. Then thre exists, there exists somesince$(\lambda-\epsilon vi|\alpha)\geq 0$$(\lambda+\epsilon v_{i}|\alpha)\geq 0$$X_{\mathbb{C}}:=X+\sqrt{-1}\mathfrak{h}_{\mathrm{R}}^{*}=\{x+\sqrt{-1}y|x\in X, y\in \mathfrak{h}_{\mathrm{R}}^{*}\}$and denote by $K$ the interior oft..hat$\{v_{i}\}_{i=1}^{l}$

.

such

by (1).

$\square$be the complexified Tits cone,in the usual metric topology of$\mathfrak{h}^{*}$. It is obvious.

From the lemmas above, we get the following lemma which will be used
Lemma 3. (1) Let$v_{i}$$\alpha\in\triangle_{+}^{im}$and$n\in \mathbb{Z}\geq 1$. Then the affie hyperplane

lat.e .
$\mathrm{r}$$T_{n\alpha}(\cdot)=0$

does

not intersect the $domain-\rho+K$ .
(2) Let$\alpha\in\triangle_{+}^{r\mathrm{e}}$. (1) Let$P\tau Oof$assume that

and$n\in \mathbb{Z}\geq 1\cdot H\lambda\in-\rho+K$$\lambda\in-\rho+K$$\lambda\in-p+X^{\mathrm{O}}$, and suppose that

. We show thatand

$T_{n\alpha}(\lambda)=0$, then $\lambda-n\alpha\in-\rho+K$ .

$2(\lambda+\rho|\alpha)=n(\alpha|\alpha)$$(\alpha|\alpha)\leq 0$. Because. Obviously we may$\triangle_{+}^{im}=W\cdot N$, we may62

assume that$(\alpha|\alpha_{i})\leq 0$$\alpha=\sum_{i\in I}k_{i}\alpha_{i}\in N\subset Q_{+}$

for$\alpha_{i}\in\Pi^{re}$. Then we haveby the definition of $N$ and$( \alpha|\alpha)=\sum_{i\in I}k_{i}(\alpha|\alpha i)\leq 0,$$(\alpha_{j}|\alpha_{i})\leq 0(j\in I)$

for$\mathrm{s}^{\mathrm{b}}1,\mathrm{n}\mathrm{C}\mathrm{e}$$\alpha_{i}\in\Pi^{im}$.

Now the equaity above contradicts part (2) of Lemma 2.
(2) Because$\alpha\in\triangle^{re}=W\cdot \mathrm{I}\mathrm{I}^{re}$, we can write$(\alpha|\alpha)=(\alpha_{i}|\alpha_{i})>0$

is defined by$r_{\alpha}(\lambda):=\lambda-(2(\lambda|\alpha)/(\alpha|\alpha))\alpha$$r_{\alpha}\in W$. Now we havefor some $w\in W$ and. Here note that the reflection

In particularso that$\alpha=w\cdot\alpha_{i}$for$\lambda\in \mathfrak{h}^{*}$$r_{\alpha}$of$\mathfrak{h}^{*}$$\alpha_{i}\in\Pi^{r\mathrm{e}}$with respect to.

$\alpha$and can be written as $r_{\alpha}=wr_{i}w^{-1}$ ,$r_{\alpha}(\lambda+\rho)=\lambda+\rho-(2(\lambda+\rho|\alpha)/(\alpha|\alpha))\alpha=\lambda+\rho-n\alpha$by the assumption. Since $K$ is $W$ -stable, we deduce that $\lambda-n\alpha\in-\rho+IC$ .

2.2 Holomorphicity of the functions$\varphi_{k,m}$on the$\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}-\rho+K$$\square$. We first recall

the following elementary lemma in [K2].
Lemma 4 ([K2, Lemma 2]). Let $B=(b_{ij})$ and $C=(c_{ij})$ be two $N\cross N$ -marices, where$b_{ij}$$U$

and$\mathrm{c}_{ij}$are holomorphi fun ctions in the varia bles$c$$z_{1},$$\ldots,$$Z_{N}$

on some neighborhoodof the origin . Put $V:=U\cap\{(z_{1}, \ldots , Z_{N)}\in \mathbb{C}^{N}|z_{1}--0\}$ .
$0$inverti $ble$ on$U\backslash V$

and that on

$V$

(c)

Here

$\dim(\mathrm{K}\mathrm{e}\mathrm{r}B)\equiv s$$\mathrm{K}\mathrm{e}\mathrm{r}B\subset \mathrm{K}\mathrm{e}\mathrm{r}C$

ppose that $B$ is

one has:(a) $\det B$ has zero of multiplicity(b)$S\mathrm{u}$$s\in \mathbb{Z}\geq 1$

;

;

.

$\mathrm{K}\mathrm{e}\mathrm{r}B=\{x\in \mathbb{C}^{N}\}Bx=0\}$(which, in general, depends on$(z_{1}, \ldots, Z_{N)}\in \mathbb{C}^{N})$Then the entries of the matrix $CB^{-1}$ can be extended to holomorphic functions on

We remark that the classification theorem (of indecomposable$\mathrm{G}\mathrm{G}\mathrm{C}\mathrm{M}\mathrm{S}$$[\mathrm{K}1$

$U$

.

.

, Theorem 4.3]) holds also in the case:(1) GGCMs of finite type are exactly GCMs of finite type;(2) GGCMs of affine type are GCMs of affine type plus the zero$1\cross 1$

matrix.

(3) If $A=(a_{ij})_{i,j\in I}$ is a GGCM of indefinite type, then there exists a positive

imaginaryroo.t

GKM algebra$\alpha=\sum_{i\in I}k_{i}\alpha_{i}$$\mathrm{g}(A)$such that $k_{i}>0$ and$(\alpha|\alpha_{i})<0$(cf. the proof of [Kl, Theorem 5.6 ) .
$\mathrm{c}$$]\rangle$

for all $i\in I$ for the63

From now on we assume that the GGCM $A=(a_{ij})_{i,j\in I}$ is lndecomposable, hence is$\dot{\mathrm{H}}$either a

GCM- of finite type, a GCM of affine type, the zeromatrix, or a GGCM

$1\cross 1$(possibly GCM) of indefinite type.

Here we recall the following well-known facts about the (ordinary) Kac-Moody alge-bras

associated to a GCM $A=(a_{ij})_{i,j\in I}$ :$\mathrm{g}(A)$(1) if$A$

is a GCM of finite type, then(2) if$A$

is a GCM of affine type, thenunique (up to acons.t$X=\mathfrak{h}_{\mathrm{R}}^{*}$;$X^{\mathrm{O}}=\{\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\delta)>0\}$actor) element of Q. such$\mathrm{a}\mathrm{n}\mathrm{t}:..\mathrm{f}$, wherefor$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}’(\delta|\alpha_{i})=0$$\delta$

is the$\mathrm{a}.11i\in I$. In

particular, we have $IC-Q_{+}=I\zeta$ in both of these cases.
In addition, if

is the GKM algebra associated to a GGCM $A=..(a_{ij})_{i,j\in I}$ such$\mathrm{g}(A)$

,

that

$a_{ii}\leq 0$

for all$W=\{1\}$ , and$i\in I$

$Q_{+}= \sum_{\alpha_{i\in\Pi\dot{\cdot}m}}\mathbb{Z}\geq 0\alpha i$$K-Q_{+}.=\dot{I}\mathrm{f}\mathrm{i}’ \mathrm{n}:.\mathrm{h}\mathrm{t}$

We
2]).

a..re

, then obviously we have $X-\beta\subset X$ for.

now in a

Theorem 2. Leton the domain

case.

is

(

$\mathrm{i}\mathrm{n}\mathrm{c}.\mathrm{l}.\mathrm{u}$ding$\beta.\in.Q_{+}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{e}X=C$. Hence we have $IC-\beta\subset IC$ for$\mathrm{t}\ddot{\mathrm{h}}\mathrm{e}$case where

$A$

is

$\cdot$t.h$\dot{\mathrm{e}}\mathrm{z}\dot{\mathrm{e}}$ro$\beta\in Q_{+}$

m.a$1\cross 1$

state our main theorem ( .ompare$\acute{\mathrm{t}}\mathrm{r}\mathrm{i}\mathrm{X}$$\mathrm{c}$$\mathrm{p}_{\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}}\mathrm{i}\mathrm{o}\mathrm{n}.\mathrm{t}\mathrm{o}$$\varphi\in \mathcal{F}$$-\rho+I\acute{\iota}$

element of the center(1) If all the functions$z_{\varphi}= \sum_{\beta\in Q}\sum_{k,m}\in^{\mathrm{p}_{\mathrm{a}\mathrm{r}}}\rho\varphi+mF^{k}k,\sigma(Fm)$such that$\varphi_{k,m}$$H(z_{\varphi})=\varphi$$T_{n\alpha}(\lambda)=0$$\varphi$).

Theorembe the ( nique)$u$.

can be extended to holomorphic functions on the domain$- \rho+IC-Q_{+}=\bigcup_{\beta\in Q}(+-p+I\acute{\iota}-\beta)$(2) Let the function’, so thatbe a function that can be extended to a holomorphic fun ction, and$Z_{F}$$\mathrm{w}\mathrm{i}\mathrm{t}.\mathrm{h}..[\mathrm{K}2-.$

,

with

, then we have for$\lambda\in-\rho+I\iota’$

impli es$\alpha\in\triangle_{+}^{re}$and$n\in \mathbb{Z}\geq 1$$\varphi(\lambda)=\varphi(\lambda-n\alpha)$satisfy the condi tion that for$\alpha\in\triangle_{+}^{re}$

and

,

.

$n\in \mathbb{Z}\geq 1$,$\prime l\backslash :$lies$T_{n\alpha}(\lambda)=0wit\dot{h}\lambda\in-\rho+I\zeta imp$

Then,

$f\dot{o}r$$\varphi_{k,m}\in \mathcal{F}$each$\beta\in\dot{Q}_{+}$

with$\varphi(\lambda)=\varphi(\lambda-n\alpha)$.

, there exists a nonempty domain $M\rho\subset K$ such that the functions$k,$ $m\in P\mathrm{a}r\beta$

can be extended to holomorph$\mathrm{i}c$functions on the domain

64$-\rho+M\rho$all. If the GGCM

is of finite or affie type, then we can take$A$

. In the case of indeffiite type, as$\beta\in Q_{+}$$\mu\rho+K\subset K$

for some$M\rho$$M_{\beta}--I\mathrm{f}$, we can take a domain of the form$\mu_{\beta}\in V:=K\cap(-\sum_{\alpha\cdot\in 11^{\Gamma}}e\mathbb{R}_{>}*0\alpha i)$

.
:

’

. (1)

$\mathrm{F}\mathrm{i}\sim \mathrm{r}\mathrm{S}\mathrm{t}\backslash \mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}-$$Pr\dot{o}of’$$K-Q_{+}=I\mathrm{f}$that$\mathrm{i}\mathrm{f}\ldots \mathrm{t}\prime_{\dot{\mathrm{h}}}\mathrm{e}\prime \mathrm{G}\mathrm{G}\dot{\mathrm{C}}\mathrm{M}$

$A$

is

$\mathrm{n}\mathrm{o}\mathrm{t}$of$\mathrm{i}\mathrm{n}\dot{\mathrm{d}}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}$type, thethat$(\beta\in Q_{+})$$-\rho+K-\grave{Q}_{+}\mathrm{i}_{\mathrm{S}}$. The connectedness

itself and the fact that

high

$:$

the scalar$\varphi(\lambda)$ve. In fact it is$\mathrm{s}\mathrm{e}\mathrm{t}_{\mathrm{S}}-\rho+I\mathrm{f}-\beta$$\beta\in Q_{+}$. The latter fact is because$K$.

.ill show$M.(\lambda)_{\mathrm{W}}.\mathrm{i}\mathrm{t}\mathrm{h}$$\mathrm{h}^{-}\dot{\mathrm{a}}$$\mathfrak{h}^{*}=\mathfrak{h}_{\mathrm{R}}^{*}+\sqrt{-1}\mathfrak{h}_{\mathrm{R}}^{*}$$\lambda\in-\rho+.K.\cdot\wedge \mathrm{W}.\mathrm{e}\mathrm{w}.$modulefor any

wefollows from the connectedness of $K$$\mathrm{o}\mathrm{f}-\rho+K-Q_{+}$$K\cap(I\zeta-\beta)\neq\emptyset$

is an open convex cone inLetan open set since it is the union of open$\mathfrak{h}^{*}$$\mathrm{n}$th.e case offrom the remarks above. Second we remark that even inindefinite type, the set $-\rho+K-Q_{+}$ is really a connected open set in$\dot{\mathrm{o}}\mathrm{b}\mathrm{v}\mathrm{i}_{\mathrm{o}\mathrm{u}\mathrm{s}}$for$\mathrm{t},\mathrm{h}\mathrm{a}_{\wedge}.\mathrm{t}..$:theele.m.ent

e.st weight as the scalar$\lambda$

on each weight space$z_{\varphi}\in\hat{U}_{F}(\mathrm{g}(.A))$$\varphi(\lambda)$for$M(\lambda)_{\lambda-\beta}$, oruivalently, that$z_{\varphi}$$\mathrm{e}\mathrm{q}\backslash$$\beta\in Q_{+}$

that the equation $(*)$ (is well-defined and) holds for thiscan act on the Vermaacts as. It clearly suffices to show$\lambda\in-\rho+I\mathrm{f}$(see the proof of

Theorem 1).
Here theentrie.s of the matrix$\Phi_{\beta}(\cdot)=(.\varphi_{k,m}(\cdot)).k,m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$

by assumption, so are the entries of the matrix

that for any$\gamma<\beta$

Let $\lambda\in-\rho+K,$
so that, the entries of the matrix$v\in M(\lambda)_{\lambda-\beta}$

, and

$s,$

$G_{\beta}^{\beta}(\cdot)=\Phi_{\beta}(\cdot)B_{\beta}$$G_{\gamma}^{\beta}(\cdot)$$t\in \mathrm{P}\mathrm{a}\mathrm{r}\gamma$

entries of the matrix$(\mathfrak{h}^{*}\backslash L)\cap(-\rho+K)$$\lambda\in \mathfrak{h}^{*}\backslash L$$\mathfrak{h}^{*}\backslash L$

such that

entries of the matrices$G_{\beta}^{\beta}(\cdot),$

above.

$\sigma(F^{t})v\in M(\lambda)_{\lambda-(\beta}-\gamma)$,, where $\lambda-(\beta-\gamma)\in-\rho+K-Q_{+}$ .

are holomorphic at any

On the other hand, for each
Theorem 1. Since the set$\mathrm{o}\mathrm{n}-\rho+I\mathrm{f}$are holomorphic on $-\rho+K-Q_{+}$ by assumption, the$\varphi_{s,t}(\cdot)$$G_{\gamma}^{\beta}(\cdot)$$\mathrm{o}\mathrm{n}-\rho+K$. Moreover we show

are holomorphic. Then we have$F^{s}\varphi_{s,t}\sigma(F^{t})v=\varphi_{s,t}(\lambda-(\beta-\gamma))Fs\sigma(F^{t})v-$

Because the functions

are holomorphic.

, the equation $(*)$ holds by (the proof of)is dense in$\mathfrak{h}^{*}$$\lim_{marrow\infty}\lambda_{m}=\lambda$$G_{\gamma}\beta(\cdot)$$\lambda\in-\rho+I\mathrm{f}$, we can take a sequence$\{\lambda_{m}\}_{m=1}^{\infty}$

infor each $\lambda\in-\rho+K$ . Because all theare holomorphic at $\lambda\in-\rho+K$ , by taking the limit

as $marrow\infty$ , we have the equation $(*)$ for this $\lambda\in-\rho+K$ .

65

Now let $\Lambda\in-\rho+K$ be such that..

for some$T_{n\alpha}(\Lambda)=0$

Then we have an embedding $M(\Lambda-n\alpha)arrow M(\Lambda)$ by [$z_{\varphi}$$\mathrm{K}\mathrm{K}$$\alpha\in\triangle_{+}^{r\mathrm{e}}$, Prop. 4.1$(\mathrm{b})$and$n\in \mathbb{Z}\geq 1$.

]. The element

obviously acts on the highest weight vector $v_{\Lambda-n\alpha}\neq 0\in M(\Lambda-n\alpha)$ as the scalar$\varphi(\Lambda-n\alpha)$. Thus we have the equality$T_{n\alpha}(\Lambda)=0$$\varphi(\Lambda)=\varphi(\Lambda-n\alpha)$

for $\Lambda\in-\rho+I\zeta$ with.

(2) First of all we remark that, in the case of indefinite type,$\alpha=\sum_{i\in I}k_{i}\alpha_{i}\in\triangle_{+}^{im}$such that $k_{i}>0$ and$(\alpha|\alpha_{i})<0$

since there exists$V\neq\emptyset$

for all $i\in I$ (see the comment

above for the classification theorem of GGCMs).
Now we will take domainthat $K-\alpha_{j}\subset K$ for

by induction on$M_{\beta}$$\alpha_{j}\in\Pi^{im}$$\beta\in Q_{+}$. We first take$M_{0}=I\zeta$. Noteby part (3) of Lemma 1. Let us take $\beta\neq 0\in Q_{+}$ .

Suppose that we have already taken domains $M_{\gamma}=\mu_{\gamma}+K\subset K$ with $\mu_{\gamma}\in V=$
such that$K \cap(-\sum_{\alpha:}\in 11^{\gamma}e\mathbb{R}_{>0}\alpha_{i})$$M_{\gamma}-\alpha_{j}\subset M_{\gamma}(\alpha_{j}\in\Pi^{\grave{\iota}m})$$(M_{\gamma}+\eta)$$M_{\beta}’:= \bigcap_{\gamma<\beta}$$\bigcap_{\eta\leq\beta}$$\gamma\in Q+\eta\in\Sigma_{\alpha_{*}\in}.\mathrm{n}\mathrm{r}e\mathbb{Z}\geq 0\alpha$For$\alpha_{j}\in\Pi^{im}$, we haveassumption. Forany$\gamma<\beta$$M_{\beta}’-\alpha_{j}\subset M_{\beta}’$$\eta\in\sum_{\alpha.\in 11^{\prime\epsilon}}.\mathbb{Z}\geq 0\alpha_{i}$. Hence we have

$M_{\beta}’= \bigcap_{i=1}^{m}(v_{i}+K)$

for

$i$

with

$M_{\beta}’-\eta\subset M_{\gamma}$$\sum_{\alpha.\in\Pi^{\gamma}}.\epsilon \mathbb{R}\alpha_{i}$$V-V= \sum_{\alpha.\in\Pi^{\gamma}}.\mathrm{e}\mathbb{R}\alpha i$:$M_{\gamma}-\alpha_{j}\subset M_{\gamma}$$\eta\leq\beta$.

for$\gamma<\beta$, we obviously havefor any $Q_{+}\ni\gamma<\beta$ and$v_{i} \in\sum_{\alpha:\in 11}r\epsilon \mathbb{R}\alpha_{i}$is an open convex cone ineach , since

since

for $Q_{+}\ni\gamma<\beta$ . Put. Because the set$\varphi_{k,m}\in \mathcal{F}$$-\rho+M_{\beta}$

with$k,$ $m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$. We have. We write$x_{i},$$y_{i}\in V$

for. Then we have$\mu_{\beta}:=\sum_{i=1}^{m}x_{i}\in V$,, and $M_{\beta}:=\mu_{\beta}+K\subset K$ .

is really a nonempty open connected set in

We proceed by induction on

for$V=K \cap(-\sum\alpha.\cdot\in 11re\mathbb{R}>0\alpha_{i})$, we can write $v_{i}=x_{i}-y_{i}$ withsince $K(\supset V)$ is a convex set. So we put$M_{\beta}$$M_{\beta}’-\eta\subset M_{\gamma}$$Q_{+}\ni\eta\leq\beta$$M_{\rho}’= \bigcap_{i1}m(=+K)v_{i}\supset \mathrm{n}^{m}i=1(x_{i}+K)\supset K+\sum_{i=1}^{m}X_{i}$It is obvious that the set

by the inductive$\beta\in Q_{+}$. Let us fix$\beta\in Q_{+}$$\mathfrak{h}^{*}$.

and show that the functions

can be extended to holomorphic functions on the domain$M\rho-\eta\subset M_{\gamma}$

for any$\gamma<\beta$

and$\eta\leq\beta$. Therefore the entries of the

66

matricesfor$G_{\gamma}^{\beta}(\cdot)$$\gamma<\beta$

are holomorphic

are holomorphic$s,$ $t\in \mathrm{P}\mathrm{a}\mathrm{r}\gamma$$\mathrm{o}\mathrm{n}-\rho+M_{\gamma}$$\mathrm{o}\mathrm{n}-\rho+M_{\beta}$for$\gamma<\beta$, since the functions$\varphi_{s,t}(\cdot)$

with. Hence, by the equation $(*)$ in the

proof of Theorem 1, we have only to show that the functions$\varphi_{k,m}$

with$k,$ $m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$can

be holomorphically extended on $-\rho+M\rho$ across the finitely many affine hyperplanes$T_{n\alpha}(\cdot)=0$

for$\alpha\in\triangle+,$we may assume that$\alpha\in\triangle_{+}^{r\mathrm{e}}$Let us fix arbitrary$\dim_{\mathbb{C}}\mathrm{g}_{\alpha}=1$$\Lambda\in-\rho+M_{\beta}$

with. Furthermore, by part (1) of Lemma 3,

and

with

$n\in \mathbb{Z}\geq 1$for$T_{n\alpha}(\Lambda)=0$

with$\alpha\in\triangle_{+}^{re}=W\cdot\Pi^{re}$, we may assume thatis dense in$\mathfrak{h}^{*}$

module$M(\Lambda)$

for $k\in$ Par

Since$J(\Lambda)$$J(\Lambda)$$\mathrm{K}\mathrm{K}$with$\beta,$$m\in$

for all, Prop.

.
are holomorphic$\alpha’\neq\alpha\in\triangle_{+}$$4.1\sim(\mathrm{b})$$\varphi(\Lambda-n\alpha)$and$\mathrm{o}\mathrm{n}-\rho+M_{\beta}\subset$(recall that$m\in \mathbb{Z}\geq 1$

and the formula (4.2) on p. 106], weof the contravariant bilinear form

$B^{\Lambda}(\cdot, \cdot)$on the Verma$B^{\Lambda}(F^{k}v\Lambda, Fmv\Lambda)=\delta_{\beta,\gamma}B_{\beta}^{\Lambda}(F^{k}, F^{m})$Par . Let $R:=M(\Lambda)_{\Lambda-\beta}\cap J(\Lambda)\cong M(\Lambda-n\alpha)(\Lambda-n\alpha)-(\beta-n\alpha)$ .
$\gamma$is the kernel of the contravariant bilinear form
$z_{\varphi}$$s=\#(\mathrm{P}\mathrm{a}\mathrm{r}(\beta-n\alpha))$). So we will show that for any$\gamma<\beta$

is isomorphic to $M(\Lambda-n\alpha)$ , whereof the operator
scalar$G_{\gamma}^{\beta}(\cdot)$$T_{m\alpha’}(\Lambda)\neq 0$). Then, by [can deduce that the kernel$B=B_{\beta}^{\Lambda}$, we have

Because the entries of the matrices$\mathfrak{h}^{*}\backslash L$, and consider the set

and$N=\dim_{\mathbb{C}}M(\Lambda)\Lambda-\beta$$\varphi(\Lambda)I_{N}=\sum_{\gamma<\beta}G^{\beta}(\gamma\Lambda)$$-\rho+K$$n\alpha\leq\beta$. We now want to apply Lemma 4 to the case where$C= \varphi(\Lambda)I_{N}-\sum\gamma<\beta\gamma G^{\beta}(\Lambda)$(remark that$n\alpha\leq\beta$.

$\alpha\in\triangle_{+}^{re}$$\{\Lambda\in-\rho+M_{\beta}|T_{n\alpha}(\Lambda)=0\}$

and

with

$n\in \mathbb{Z}\geq 1$

on

$R$

is

$\sum_{\gamma<\rho\gamma}c^{\beta}(\Lambda)$$B^{\Lambda}(\cdot, \cdot)$on$M(\Lambda)$, the matrix. We will show that the operator acts as theon $R$ . As in the proof of part (1), it suffices to show that the following

equation (is well-defined and) holds for this $\Lambda\in-\rho+M\rho$ :.

$(**)$$\Phi_{\beta-n\alpha}(\Lambda-n\alpha)B_{\beta n\alpha}^{\Lambda}-+\sum-n\alpha.G^{\beta}.-n\alpha(\Lambda-n\alpha)=\varphi(\Lambda-n\alpha)\gamma.\mathrm{I}\gamma<^{\rho}-n\alpha \mathrm{d}$(Note that $(\Lambda-n\alpha)-(\beta-n\alpha)--\Lambda-\beta.$ ) Here we have$(\beta-n\alpha).+\gamma)F^{s}\sigma(Ft)v$

for$v\in M(\lambda)_{\lambda-}(\rho-n\alpha)$$F^{s}\varphi_{s,t}\sigma(F^{t})v=\varphi_{s,t}(\lambda-$with $\lambda\in-\rho-n\alpha+M\rho$ and$s,t\in \mathrm{P}\mathrm{a}\mathrm{r}\gamma$

67

with

$\gamma\leq\beta-n\alpha$(including. So, for each $\gamma\leq\beta-n\alpha<\beta$ , the entries of the matrix$\Phi_{\beta-n\alpha}(\cdot)$) are holomorphic$\mathrm{o}\mathrm{n}-\rho-n\alpha+M\rho$$G_{\gamma}^{\beta-n\alpha}(\cdot)$by the inductive assumption,

since $\lambda\in-\rho-n\alpha+M_{\beta}$ implies $\lambda-(\beta-n\alpha)+\gamma=\lambda+n\alpha-(\beta-\gamma)\in-\rho+M_{\gamma}$ . On
the other hand, for each$\lambda\in \mathfrak{h}^{*}\backslash L$, the equation $(**)$ with$\Lambda$

replaced with$\lambda$

holds by(the proof of) Theorem 1. Hence, by taking the limit, we have the equation $(**)$ forabove. Thus the operator

acts on$z_{\varphi}$$R\cong M(\Lambda-n\alpha)_{\Lambda\beta}-$

Due to Lemma 4 above, we deduce that the functions

as the scalar$\varphi_{k,m}$with$\varphi(\Lambda-n\alpha)$$k,$ $m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$a removable singularity at any $\Lambda\in\{\Lambda\in-\rho+M\rho|T_{n\alpha}(\Lambda)=0$ , and
$0$

for

$\alpha’\neq\alpha\in\triangle_{+}^{re},$ $m\in \mathbb{Z}\geq 1$

with$m\alpha’\leq\beta$$\Lambda$.
have$T_{m\alpha’}(\Lambda)\neq$}. Then we quote the theorem (cf.

$[\mathrm{G}\mathrm{R}$,Theorem 1.8]) which asserts that a function of at least two complex variables can be

holomorphically extended across the intersection of finitely many (but at least two)

affine hyperplanes. Therefore we have proved that the functions
can be extended to holomorphic functions

Remark 2. Let$f\in S(\mathfrak{h})$$\mathrm{o}\mathrm{n}-\rho+M_{\beta}$.

$\varphi_{k,m}$

with$k,$ $m\in \mathrm{P}\mathrm{a}\mathrm{r}\beta$$\square$be $W$ -invariant. Then the function$\varphi(\cdot)\in \mathcal{F}$defined bysatisfies the conditions of Theorem 2 (see the proof of part$\varphi(\lambda):=f(\lambda+\rho)(\lambda\in \mathfrak{h}^{*})$(2) of Lemma 3).

Finally we consider the domain$\bigcap_{\beta\in Q}M_{\beta}+$$-\rho+I\backslash ^{\nearrow}-Q_{+}$

in part (1) and the domain $-\rho+$in part (2) of Theorem 2 above in the case of indefinite type.

We prepare the following lemma, which can be proved almost in the same way as in
the case of Kac-Moody algebras (cf. the proof of [Kl, Proposition 5.8 ) .
$\mathrm{c}$Lemma 5. Let

Then we$\mathrm{g}(A)$$])$

be the $GKM$ algebra associa $ted$ to a GGCM of indefinite type.

$h\mathrm{a}\mathrm{v}e$$X^{-}=${$\lambda\in \mathfrak{h}_{\mathrm{R}}^{*}|(\lambda|\alpha)\geq 0$

for all$\alpha\in\triangle_{+}^{im}$}.

We now have the following proposition.

Proposition 2. Let$\mathrm{g}(A)$be the $GKM$ algebra associated to a GGCM $A=(a_{ij})_{i,j\in I}$

of indefinite type with $a_{ii}=2$ for some $i\in I.$ Then we have $K\neq\subset K-Q_{+}$ , and$\bigcap_{\beta\in Q\beta}+M=\emptyset$.

68

Proof. We first show that there exists a positive imaginary root
simple rootthat$\alpha_{i_{0}}\in\Pi^{re}$$(\alpha’|\alpha_{i})<0$

such that$(\alpha|\alpha_{i_{0}})>0$

for all $i\in I$ . Take$i_{0}\in I$$(\alpha|\alpha_{i_{0}})=(r_{i_{0}}(\alpha’)|\alpha_{i0})=-(\alpha’|\alpha_{i_{0}})>0$

If$K-\alpha_{i_{0}}\subset I\mathrm{f}$$X^{\mathrm{O}}+\sqrt{-1}\mathfrak{h}_{\mathrm{R}}^{*}$

for this$\alpha_{i_{0}}$. Then we have

$K\ni x$$\alpha\in\triangle_{+}^{im}$$X^{-}-\alpha_{i_{0}}\subset X^{-}$

This is a contradiction. Hence we have$x \in\bigcap_{\beta\in Q\rho}+M$, and$a_{i_{0}i_{0}}=2$, and putsince$-\alpha_{i_{0}}\in X^{-}$$\triangle_{+}^{im}$$(X^{\mathrm{O}})^{-}=X^{-}$. We havesince

by Lemma 5.

$K_{\neq}^{\subset}K-Q_{+}$. Then we have $x-\beta\in M_{\beta}-\beta\subset K$ for all.

$\beta\in Q_{+}$. Because

is an open convex cone, we can easily deduce that $K-\beta\subset K$ for all

which contradicts the fact that

$K_{\neq}^{\subset}K-Q_{+}\mathrm{j}\mathrm{u}\mathrm{S}\mathrm{t}$proved above.

$I\acute{\iota}=$from the convexity$(-\alpha_{i_{0}}|\alpha)\geq 0$, so thatsuchis W-stable.

$X^{\mathrm{o}}-\alpha_{i_{0}}\subset X^{\mathrm{o}}$, so that$K-\alpha_{i_{0}}\not\subset I\mathrm{f}$$\alpha’\in\triangle_{+}^{im}$$\alpha:=r_{i_{0}}(\alpha’)$since the set, then we obviously haveof the set $X$ . Because $0\in X^{-}$ , we get

Let. We know that there exists

withand a real$\alpha\in\triangle_{+}^{im}$$\beta\in Q_{+}$,$\square$

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