数理解析研究所講究録
1190 巻 2001 年 136-152

136

TWINING CHARACTER FORMULAFOR DEMAZURE MODULES
筑波大学

. 数学系

内藤聡 (SATOSHI NAITO)INSTITUTE OF MATHEMATICSUNIVERSITY OF TSUKUBA$0$. INTRODUCTION.

Let be a finite-dimensional complex semi-simple Lie algebra with Cartan subalgebra
and Borel subalgebra. Let
be the set of roots of relative to . We choose
the set of positive rootsthat the roots ofLet
be the set$\{h_{i}|i\in I\}$ the set of simple coroots in$A=(a_{ij})_{i,j\in I}$ the Cartanof simple roots in, andmatrix with
the Weyl group. We take andfix a Chevalley basisof , and let.
$\mathfrak{g}$$\triangle\subset \mathfrak{h}^{*}$$\mathrm{b}\supset \mathfrak{h}$$\mathfrak{h}$$\mathfrak{h}$$\mathfrak{g}$$\triangle_{+}\mathrm{s}\mathrm{u}\mathrm{C}\mathrm{h}$$\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{e}-\triangle+\cdot$$\Delta_{+},$$\{\alpha_{i}|i\in I\}$$\mathfrak{h},$$a_{ij}=\alpha_{j}(h_{i})$$W=\langle r_{i}|i\in I\rangle\subset GL(\mathfrak{h}^{*})$$\{e_{\alpha}, f_{\alpha}|.\alpha\in\Delta_{+}\}\cup\{h_{i}|i\in I\}$$\mathfrak{y}_{\mathbb{Z}}--\sum i\in I\mathbb{Z}h_{i}$$\mathfrak{g}$A bijection (of order $N$ ) of the index set such thatfor all $i,j\in I$
induces a unique automorphism , called a (Dynkin) diagram automorphism, of the Liealgebra such thatandfor $i\in I$ . We denotebythe cyclic subgroup (of order $N$ ) ofgenerated by the diagram automorphism. The restriction of to induces a transposed map, which stabilizes theintegral weight lattice
for all $i\in I$ }. We set{$I$$\omega$$a_{\omega(i),\omega(j)}=a_{ij}$$\omega$$\omega(e_{\alpha_{i}})=e_{\alpha_{\omega(i}},$$\omega()f_{\alpha_{i}})=f_{\alpha_{y()}}i$$\mathfrak{g}$$\langle\omega\rangle$$\omega(h_{i})=h_{\omega(i)}$’$\mathrm{A}\mathrm{u}\mathrm{t}(\emptyset)$$\omega$$\omega$$\omega^{*}:$$\mathfrak{h}$$\lambda\in \mathfrak{y}*|\lambda(h_{i})\in \mathbb{Z}$$\mathfrak{h}_{\mathbb{Z}}^{*}=$$\mathfrak{g}^{0}=\{x\in \mathfrak{g}|\omega(x)=x\},$$\simeq \mathrm{H}\mathrm{o}\mathrm{m}(\mathfrak{h}\mathbb{Z}, \mathbb{Z})$$\mathfrak{h}^{0}=\{h\in \mathfrak{h}|\omega(h)=h\},$$(\mathfrak{h}^{*})^{0}=\{\lambda\in \mathfrak{h}^{*}|\omega^{*}(\lambda)=\lambda\}\simeq(\mathfrak{h}^{0})^{*}$, and$\mathfrak{h}^{*}arrow \mathfrak{h}^{*}$$W^{\omega}--\{w\in W|\omega^{*}w=w\omega^{*}\}$$(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}=\{\lambda\in \mathfrak{h}_{\mathbb{Z}}^{*}|\omega^{*}(\lambda)=\lambda\}$,.

Let be the orbit Lie algebra, which is the dual complex semi-simple Lie algebra of the
fixed point (semi-simple) subalgebra of , i.e., a complex semi-simple Lie algebra withthe opposite Dynkin diagram to that of . Let be the Cartan subalgebra
theBorel subalgebra, andthe set of positive roots chosen so that the roots of are. Let
be the set of simple roots in
andthe Weyl group, where the index set is a set of representatives of the -orbits in .
It is known that there exist an isomorphism of groups $$ :and a -linear)
isomorphism:
such that if
is the transposed map of ,. We setthen
for all. In
forparticular,
forms a Coxeter system.
$\wedge \mathfrak{g}$$\mathfrak{g}^{0}$$\mathfrak{g}$$9^{0}$$\wedge \mathfrak{h}$$\mathrm{o}\mathrm{f}_{9}^{\wedge},\mathrm{b}\wedge\supset\wedge \mathfrak{h}$$\triangle\wedge+\subset\wedge \mathfrak{h}^{*}$$-\hat{\Delta}_{+}$$\wedge \mathrm{b}$$\{\hat{\alpha}_{i}|i\in I\}\wedge$$\overline{W}=\langle r_{i}\wedge|i\in I\rangle\wedge\subset GL(^{\wedge}\mathfrak{y}*)$$\hat{\Delta}_{+}$$I\wedge$$I$$\omega$$\overline{W}arrow W^{\omega}$$P_{\omega}$$\mathfrak{h}^{0}arrow\wedge \mathfrak{h}$$(\hat{w})|(\mathfrak{h}^{*})^{0}=P_{\omega^{\circ\hat{w}}}^{*}\circ(P_{\omega}^{*})^{-1}$$(W^{\omega}, \{w_{i}|i\in I\})\wedge$$P_{\omega}^{*}:\mathfrak{h}^{*}$$\wedgearrow\sim\underline{(\mathfrak{h}}^{0}$$\hat{w}\in W$$\mathbb{C}$$*\simeq(\mathfrak{h}^{*})^{0}$$w_{i}=(r_{i})\wedge\in W^{\omega}$$P_{\omega}$$i\in I\wedge$

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, let
For dominant
be the simple -module of highest weight . It
admits a unique -linear-action such that $\omega\cdot(xv)=\omega(x)(\omega\cdot v)$ for each,$v\in L(\lambda)$ and such that
, whereis a (nonzero) highest weight vector of. So therefore does its dual module
withthe longest element
$W$
. Let
in
be the universal enveloping algebra of , and for each $w\in W^{\omega}$ , letbe Joseph’s module of highest weight, withina (nonzero) weight vector inof weight. Since, the weightvectorturns out to be fixed by the action of, and hence Joseph’s
moduleis-invariant. In the talk we will prove a formula of Demazure
pe for the twining characterof
defined by$L(\lambda)$$\lambda\in(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$$\mathbb{C}$$\lambda$$\mathfrak{g}$$\langle\omega\rangle$$x\in \mathfrak{g}$$\omega\cdot v_{\lambda}=v_{\lambda}$$v_{\lambda}$$L(\lambda)$$L(\lambda)^{*}\simeq L(-w_{\mathrm{o}(\lambda)})$$w_{0}$$\mathrm{b}$$\mathrm{U}(\mathrm{b})$$-w(\lambda)$$J_{w}(\lambda)=\mathrm{U}(\mathrm{b})vw*(\lambda)\subset L(\lambda)^{*}$$w\in W^{\omega}$$-w(\lambda)$$L(\lambda)^{*}$$v_{w(\lambda)}^{*}$$L(\lambda)^{*}$$v_{w(\lambda)}^{*}\in L(\lambda)^{*}$$\langle\omega\rangle$$J_{w}(\lambda)\subset L(\lambda)^{*}$$\langle\omega\rangle$$\mathrm{c}\mathrm{h}^{\omega}(J_{w}(\lambda))$$\mathrm{t}\mathrm{y}\lrcorner$$J_{w}.(\lambda)$$e(\mu)$$\mathrm{c}\mathrm{h}^{\omega}(J_{w}(\lambda))=\sum_{)\mu\in(\mathfrak{y}*\mathrm{z}0}\mathrm{T}\mathrm{r}(\omega|_{J(\lambda}w)_{\mu})$.

in the group algebraover
we will find a striking relation:$\mathbb{C}$$\mathbb{C}[(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}]$of$(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$with basis $e(\mu),$.ch$\mathrm{c}\mathrm{h}^{\omega}(J_{w}(\lambda))=P^{*}\omega($

where $\hat{w}=^{-1}(w)\in\overline{W},$character of Joseph’s module$J_{\hat{w}}(\wedge\wedge\lambda))$, and ch$\wedge\lambda=(P_{\omega}^{*})^{-}1(\lambda)\in\wedge \mathfrak{h}^{*}$$\hat{J}_{\hat{w}}(\lambda)\wedge$of$\mu\in(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$. As a corollary,,is the ordinary
over the orbit Lie algebra .

$J_{\hat{w}}(\lambda)\wedge\wedge\in \mathbb{C}[\mathfrak{h}^{*}]\wedge$$\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{e}\mathrm{S}\mathrm{t}_{\mathrm{W}},\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}-\hat{w}(^{\wedge}\lambda)$$\wedge \mathfrak{g}$Although our problem can be stated purely algebraically as above, it seems very
difficult (at least for me) to solve it only by algebraic methods. Hence we resort to(algebro-) geometric methods. For that purpose, we introduce more notation. Letbea connected, simply connected semi-simple linear algebraic group over with maximal
torus and Borel subgroup $B\supset T$ such that Lie, and Lie.
Then the character groupof may be identified withby taking the$i\inI$
differential at the identity element, i.e., by the map. For each,and, wherewe will write
is the coroot of. Thereexists an automorphism of whose differential at the identity element coincides with thediagram automorphismof above. By abuse of notation, we will denote by thisautomorphism of and by
the cyclic subgroup (of order $N$ ) ofgenerated by. We will also denote the induced action of
on A by the same letter , and set
for all $i\in I$ }.
{$G$$\mathbb{C}$$T$$(G)=\mathfrak{g},$$\Lambda=\mathrm{H}\mathrm{o}\mathrm{m}(T, GL_{1})$$(B)=\mathrm{b}$$\mathrm{L}\mathrm{i}\mathrm{e}(T)=\mathfrak{h}$$T$$\mathfrak{h}_{\mathbb{Z}}^{*}$$\lambda\mapsto \mathrm{d}\lambda$$\lambda\in\Lambda$$\alpha_{i}^{}\in \mathrm{H}\mathrm{o}\mathrm{m}(GL1, T)$$\langle\lambda, \alpha_{i}^{\vee}\rangle=(\mathrm{d}\lambda)(h_{i})$$\alpha_{i}\in\Lambda$$G$$\omega$$G$$\omega$$\mathfrak{g}$$\mathrm{A}\mathrm{u}\mathrm{t}(G)$$\langle\omega\rangle$$\omega\in\langle\omega\rangle$$\omega$$\Lambda^{\omega}=\{\lambda\in\Lambda|\omega\cdot\lambda=\lambda\},$$\Lambda_{+}^{\omega}=$$\lambda\in\Lambda^{\omega}|\langle\lambda,$$\omega$$\alpha_{i}^{\vee}\rangle\geq 0$

By a-module $M$ , we will mean a finite-dimensional rational -module that
-action such that $\omega\cdot(gm)=\omega(g)(\omega\cdot m)$ for each $g\in G$ and $m\in M$ .
admits a -linear
Regarding the semi-direct productof andas a linear algebraic group, thisis the same as a rational-module. Likewise for-and-modules. Let
be the group algebra over ofwith basis $e(\mu),$. For a-module$G$$G\rangle\triangleleft\langle\omega\rangle$$\mathbb{C}$$\langle\omega\rangle$$G\rangle\triangleleft\langle\omega\rangle$$G\rangle\triangleleft\langle\omega\rangle$$\mathbb{C}[\Lambda^{\omega}]$$G$$\langle\omega\rangle$$B\rangle\triangleleft\langle\omega\rangle$$\mathbb{C}$$\Lambda^{\omega}$$\mu\in\Lambda^{\omega}$$T\rangle\triangleleft\langle\omega\rangle$$T\rangle\triangleleft\langle\omega\rangle$

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$V$

, we define the twining character$\mathrm{c}\mathrm{h}^{\omega}(V)\in \mathbb{C}[\Lambda^{\omega}]$

of

to be

$V$

,

$\mathrm{c}\mathrm{h}^{\omega}(V)=\sum \mathrm{T}\mathrm{r}(\omega|V_{\mu}.)e(\mu\in\Lambda\omega\mu)$where$V_{\mu}=${ $v\in V|tv=\mu(t)v$

for all $t\in T$ } is the -weight space of
$\mu$

$V$

.

Recall that $W\simeq N_{G}(T)/T$ , where $N_{G}(T)$ is the normalizer of in . Fix,
and let $X(w)$ be the associated Schubert variety over , which is the Zariski closure in theflag variety $G/B$ of the Bruhat cell $B\dot{w}B/B$ , where denotes a right coset representativeof in $N_{G}(T)$ fixed by. If $M$ is a-module, then the B-equivariant$M$-moduleassociated tocarries a structure of “
-equivariant”(.
-equivariant) sheaf, so that its cohomology groups $H^{\cdot}(X(w), \mathcal{L}_{x}(w)(M))$ are-modules. (For the precise definition of a-equivariant sheaf, see ourpreprint [KN].)
$T$

$G$

$w\in W^{\omega}$$\mathbb{C}$$\dot{w}$$w$$\omega\in \mathrm{A}\mathrm{u}\mathrm{t}(G)$$\mathcal{L}_{X(w)}(M)$$\mathcal{O}_{X(w)}$$=$$B\rangle\triangleleft\langle\omega\rangle$$(B, \langle\omega\rangle)$$B\rangle\triangleleft\langle\omega\rangle$$B\rangle\triangleleft\langle\omega\rangle$$(B, \langle\omega\rangle)$

For each, we letdenote the one-dimensional-module on whichacts via through the quotient $Barrow T$ andtrivially. We callfora Demazure module. Joseph’s module
admits a structure ofmodule, and we have an isomorphism of-modules$\lambda\in\Lambda^{\omega}$$\mathbb{C}_{\lambda}$$B\rangle\triangleleft\langle\omega\rangle$$\lambda$$B$$H^{0}(X(w), \mathcal{L}x(w)(\mathbb{C}_{\lambda}))$$\langle\omega\rangle$$J_{w}(\lambda)$$\lambda\in\Lambda_{+}^{\omega}$$B\rangle\triangleleft\langle\omega\rangle-$$B\rangle\triangleleft\langle\omega\rangle$$J_{w}(\lambda)*\simeq H0(X(w), \mathcal{L}\mathrm{x}(w)(\mathbb{C}_{\lambda}))$where$J_{w}(\lambda)*\mathrm{i}\mathrm{s}$the dual$B\rangle\triangleleft\langle\omega\rangle$-module of$J_{w}(\lambda)$, we define the -Demazure operatorForsuch that$i\in I\wedge$$\omega$,.
$\hat{D}_{i}$to be a$\mathbb{C}$-linear endomorphism of$\mathbb{C}[\Lambda^{\omega}]$for$\hat{D}_{i}(e(\mu))=\frac{e(\mu)-e(-si\beta i)e(w_{i}(\mu))}{1-e(-si\beta i)}$where
-orbit of $i\in I$ .

$\beta_{i}=\sum_{k=0^{1}}^{N_{i}-}\alpha_{\omega^{k}(}i$) and$s_{i}=2/ \sum_{k=}^{N_{i}}-10i,\omega(aki)$with$N_{i}$$\mu\in\Lambda^{\omega}$,

the number of elements of the$\omega$

The following is our main result.
THEOREM 0.1. Let $M$ be a finite-dimensional rational-module,letbe a reduced expression in the Coxeter system
Then we have in,$B\rangle\triangleleft\langle\omega\rangle$$w\in W^{\omega}$

, and
.

$(W^{\omega}, \{w_{i}|i\in I\})\wedge$$w=w_{i_{1}}wi_{2}\ldots w_{i_{n}}$$\mathbb{C}[\Lambda^{\omega}]$$\chi_{w}^{\omega}(M)--\sum(-1)j\mathrm{h}\omega(\mathrm{C}H^{j}(X(w), \mathcal{L}_{x}(w)(M)))j\geq 0$$=\hat{D}_{i_{1}}\hat{D}_{i_{2}}\cdots\hat{D}_{i_{n}}(\mathrm{c}\mathrm{h}^{\omega}(M)\mathrm{I}\cdot$

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In particular, for$\lambda\in\Lambda_{+}^{\omega}$, we have$\mathrm{c}\mathrm{h}^{\omega}(H^{0}(X(w), \mathcal{L}x(w)(\mathbb{C}\lambda)))=\hat{D}_{i_{1}}\hat{D}_{i_{2}}\cdots\hat{D}in(e(\lambda))$.

There is thus revealed a striking relation between twining characters for and ordinary.
andcharacters for the orbit Lie algebra . Let, letbe the simple -module of highest weight , and for each
For dominant,
be Joseph’s module of highest weight, let$\mathfrak{g}$$\wedge \mathfrak{h}_{\mathbb{Z}}^{*}=\mathrm{H}\mathrm{o}\mathrm{m}(\mathfrak{h}_{\mathbb{Z}}\wedge, \mathbb{Z})\subset\wedge \mathfrak{h}^{*}$$\wedge \mathfrak{h}_{\mathbb{Z}}=\sum_{i\in I}\wedge \mathbb{Z}^{\wedge}h_{i}$$\wedge \mathfrak{g}$$\wedge\lambda$$\hat{L}(\lambda)\wedge$$\wedge\lambda\in\wedge \mathfrak{h}_{\mathbb{Z}}^{*}$$\wedge \mathfrak{g}$$\hat{w}\in\overline{W}$$-\hat{w}(\lambda)\wedge$$J_{\hat{w}}(\lambda)\wedge\wedge=\mathrm{u}(\mathrm{b})v_{\hat{w}()}\wedge\wedge*\wedge\lambda\subset\hat{L}(\lambda)^{*}\wedge$

with$\wedge v_{\hat{w}(\lambda)}^{*}\wedge\in\hat{L}(\lambda)^{*}\wedge \mathrm{a}$(nonzero) weight vector of

COROLLARY 0.2. Le$\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}-\hat{w}(\lambda)\wedge$

be dominant and,. Then we have in$w\in W^{\omega}$$t\lambda\in(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$$and\wedge\lambda=(P_{\omega}^{*})^{-}1(\lambda)\in\wedge \mathfrak{h}_{\mathbb{Z}}^{*}$$P_{\omega}^{*}$$\mathbb{C}of\mathfrak{h}_{\mathbb{Z}}^{*}\wedge$,-algebra isomorphism
defined byover, of the group algebra
for each basis elementon the right-hand side is a

$P_{\omega}^{*}(e(^{\wedge}\mu))=e(P_{\omega}^{*}(\mu)\wedge)$. We set $\hat{w}=^{-1}(w)\in\overline{W}$$\mathbb{C}[(\mathfrak{h}_{\mathbb{Z}}*)^{0}]$$\mathrm{c}\mathrm{h}^{\omega}(J_{w}(\lambda))=P^{*}\omega(\mathrm{c}\mathrm{h}^{\wedge}J_{\hat{w}}(^{\wedge}\lambda))$

where

.

$\mathbb{C}[\mathfrak{h}_{\mathbb{Z}}^{*}\wedge]\simarrow \mathbb{C}[(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}]$$\mathbb{C}$$e(\mu)\wedge,$$\mathbb{C}[^{\wedge}\mathfrak{h}_{\mathbb{Z}}*]$$\mu\wedge\in\wedge \mathfrak{h}_{\mathbb{Z}}^{*}$.

1. PRELIMINARIES.
1.1. Diagram automorphisms. Let be a finite-dimensional complex semi-simple. Let
be the set
Lie algebra with Cartan subalgebra and Borel subalgebrasuch that the roots ofof roots of relative to . We choose the set of positive roots$\{h_{i}|i\in I\}$ the set of simple. Let
be the set of simple roots in
are
, and
coroots in $A=(a_{ij})_{i,j\in I}$ the Cartan matrix with
the Weyl group. We take and fix a Chevalley basis.
of , and let$\mathfrak{g}$$\mathrm{b}\supset \mathfrak{h}$$\mathfrak{h}$$\triangle\subset \mathfrak{h}^{*}$$\mathrm{b}$$\triangle_{+}$$\mathfrak{h}$$\mathfrak{g}$$\{\alpha_{i}|i\in I\}$$-\triangle_{+}$$\triangle_{+},$$\{e_{\alpha}, f_{\alpha}|\alpha\in\triangle_{+}\}\cup\{h_{i}|i\in I\}$

$GL(\mathfrak{h}^{*})$$\mathfrak{g}$$W=\langle r_{i}|i\in I\rangle\subset$$a_{ij}=\alpha_{j}(h_{i})$$\mathfrak{h},$$\mathfrak{h}_{\mathbb{Z}}=\sum_{i\in I}\mathbb{Z}h_{i}$We fix a bijection $\omega:Iarrow I$ of the index set$a_{\omega(i),\omega(j)}=a_{ij}$$I$such thatfor all$i,$$j\in I$.

the number of elements of the .orbit of $i\in I$ . This
Let $N$ be the order of , andcan be extended in a unique way to an automorphism (also denoted by ) of order $N$ of
the Lie algebra in such a way that$\omega$$N_{i}$$\omega-$$\omega$$\omega$$\mathfrak{g}$$\omega(e_{\alpha_{i}})=e\alpha i)\omega$($\{$ $\omega(f_{\alpha_{i}})=f_{\alpha_{\omega}}(i)$$\omega(h_{i})=h\omega(i)$

,

’

’

$i\in I$,

$i\in I$,

$i\in I$.

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Note that the restriction ofsuch tha$\omega^{*}:$$\mathfrak{h}^{*}arrow \mathfrak{h}^{*}$

to the Cartan subalgebra$\omega$$\mathrm{t}$$\cdot\omega^{*}.(\lambda)(h)=\lambda(\omega(h))\mathrm{f}\mathrm{o}.\mathrm{r}\lambda\in \mathfrak{h}^{*},h\backslash \in \mathfrak{h}$induceS a transposed map$\mathfrak{h}$. We setjand$(\mathfrak{h}^{*})^{0}=\{\lambda\in \mathfrak{h}^{*}|\omega^{*}(\lambda)=\lambda\}$,$(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}=\{\lambda\in \mathfrak{h}_{\mathbb{Z}}^{*}|\omega^{*}(\lambda)=\lambda\}$

.

where
$\rho=(1/2.

$\mathfrak{h}_{\mathbb{Z}}^{*}=\{\dot{\lambda}\in \mathfrak{h}^{*}|\lambda(,\dot{h}_{i})\in \mathbb{Z} \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}:.

)\cdot\sum_{\alpha\in\Delta}+\alpha$is in$(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$.

.

$\cdot$1.1i\in I\}\simeq \mathrm{H}\mathrm{o}\mathrm{m}\backslash (\mathfrak{h}_{\mathbb{Z}\mathrm{r}}\mathbb{Z})$.

$\backslash \cdot$

1.2. Orbit Lie algebras. We choose and fix a set
in , and set, whereis given by$I$$\hat{A}=(\wedge a_{ij})_{i,j}\in I\wedge$$\wedge a_{ij}..\cdot=s_{j}\cross\sum_{k.=0}^{N_{j}-1}$. Note that the Weyl vector$I\wedge$of representatives of the -orbits$\omega$$\wedge a_{ij}$$a_{i},\omega(kj)$for$i,j\in I\wedge$

.

.

with
$-$

$s_{j}= \frac{2}{\sum_{k=0^{1}}^{N_{j}}-a_{j},k\omega(j)}$for.

Set for each $i\in II_{i}\wedge,=\{\omega^{k}(i)|0\leq k\leq N_{i}-1\}\subset I$ . We know that for each$\sum_{k\in Ii}a_{ik}=1$.

$j\in I\wedge$,$i\in I\wedge$or 2.

Moreover, there are only two possibilities:, thenif
is even and the subgraph of the Dynkin diagram correis of type(whereappears. . sponding to the subset.
. times);(b) if, then the subgraph of the Dynkin diagram corresponding to thesubsetis totally disconnected and of type(whereappears
times).
$(\mathrm{a})\mathrm{J}$$\sum_{k\in I_{i}}a_{ik}=1$$N_{i}$$I_{i}\subset I$$A_{2}\cross\cdots\cross A_{2}$$N_{i}/2$$A_{2}$$\sum_{k\in I}\dot{.}a_{ik}=2$$I_{i}\subset I$$A_{1}\cross\cdots\cross A_{1}$$A_{1}$$N_{i}$The orbit Lie algebra associated to the diagram automorphism
Aut(g) is defined
to be the complex semi-simple Lieassociated to the Cartan matrix
with the Cartan subalgebra , the Borel subalgebra, the set of positive roots
so that the roots, the set of simple roots,
the set of simple coroots, and the Weyl group.
$\omega\in$$\hat{A}=(\wedge a_{ij})_{i,j\in I}\wedge$$\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathfrak{g}\wedge$$\wedge \mathfrak{h}$$\triangle\wedge+\subset\wedge \mathfrak{h}^{*}\mathrm{c}\mathrm{h}_{\mathrm{o}\mathrm{s}}\mathrm{e}\mathrm{n}$$\wedge \mathrm{b}\supset\wedge \mathfrak{h}$$\mathrm{o}\mathrm{f}\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{e}\wedge-\triangle\wedge+$$\{h_{i}\wedge|i\in I\}\wedge\subset\wedge \mathfrak{h}$$\{\hat{\alpha}_{i}|i\in I\}\wedge\subset\wedge \mathfrak{h}^{*}$$\overline{W}=\langle r_{i}\wedge|i\in I\rangle\wedge\subset GL(^{\wedge}\mathfrak{y}*)$Remark 1.2.1. We can easily deduce that the orbit Lieis the dual complex semisimple Lie algebra of the fixed point (semi-simple) subalgebraof, i.e., a complex semi-simple Lie algebra which has the opposite Dynkin diagram to that.
$\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}_{9}\wedge$$\mathfrak{g}^{0}=\{x\in \mathfrak{g}|\omega(x)=x\}$$\mathfrak{g}$$\mathrm{o}\mathrm{f}\mathfrak{g}^{0}$

141

We setgiven by$\mathfrak{h}^{0}=\{h\in \mathfrak{h}|\omega(h)=h\}$. Then there exists a linear isomorphism

for each$P_{\omega}(_{k\in I_{i}} \sum h_{k)}=N_{i}h_{i}\wedge$

This map$P_{\omega}$:$\mathfrak{h}^{0\sim}arrow \mathfrak{h}\wedge$

induces a transposed map$P_{\omega}^{*}(\lambda\wedge)(h)=\wedge\lambda(P_{\omega}(h))$

Note that, if$\wedge \mathfrak{h}_{\mathbb{Z}}=\sum_{i\in I}\wedge \mathbb{Z}h_{i}\wedge$

andWe now define the subgroup$\wedge\lambda\in\wedge \mathfrak{h}^{*},$$\wedge \mathfrak{h}_{\mathbb{Z}}^{*}=\mathrm{H}\mathrm{o}\mathrm{m}(^{\wedge}\mathfrak{h}_{\mathbb{Z}}, \mathbb{Z})\subset\wedge \mathfrak{h}^{*}$$W^{\omega}$:$\mathfrak{h}^{0}arrow\wedge \mathfrak{h}$.

$i\in I\wedge$$P_{\omega}^{*}:\mathfrak{h}*\simarrow(\mathfrak{y}^{0} \wedge)*\simeq(\mathfrak{h}^{*})^{0}$for$P_{\omega}$$h\in \mathfrak{h}^{0}$, thensuch that.

$P_{\omega}^{*}(\wedge \mathfrak{y}_{\mathbb{Z}}*)=(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$.

of $W$ by$W^{\omega}=\{w\in W|\omega w=w\omega^{*}\}*$.

It is known that there exists an isomorphism of groups $$ :from the Weyl
groupof the orbit Lie algebra onto the groupsuch that the following diagram:
commutes for each:$\overline{W}arrow W^{\omega}$$\overline{W}$$W^{\omega}$$\wedge \mathfrak{g}$$\hat{w}\in\overline{W}$,$\iota\backslash$$\wedge \mathfrak{h}^{*}arrow P_{\omega}^{*}(\mathfrak{h}^{*})^{0}$$\downarrow(\hat{w})|(\mathfrak{y}*)0$$\hat{w}\downarrow$$\wedge \mathfrak{h}^{*}arrow P_{\omega}^{*}(\mathfrak{h}^{*})^{0}$For each, set$i\in I\wedge$$w_{i}=(r_{i})\wedge\in W^{\omega}$.

. Explicitly,$\prod_{k=}^{N_{i/-}}210(rk(\omega i)r_{\omega^{k+}}N_{i}/2(i)r(\omega^{k}i))$if$\sum_{k=0}^{N_{i}-1}a_{i,\omega^{k}}(i)=1$,$\prod_{k=0}^{N_{i}-1}r_{\omega^{k}(i})$

if$\sum_{k=0}^{N_{i}-1}a_{i,\omega^{k}}(i)=2$.

$w_{i}=\{$of the Weyl group $W$ generated
Hence eachis the longest element of the subgroup. Furthermore,by theforforms a Coxeter system as
is. We will denote the length function of the Coxeter system $(W, \{r_{i}|i\in I\})$$\{w_{i}|i\in I\})$ ) by(resp.
).
(resp.
$W_{I_{i}}$$w_{i}$$(W^{\omega}, \{w_{i}|i\in I\})\wedge$$k\in I_{i}$$r_{k}’ \mathrm{s}$$(\overline{W},$$\{r_{i}\wedge|$$i\in I\})\wedge$$\wedge$$(W^{\omega},$$\ell:Warrow \mathbb{Z}_{\geq 0}$$\ell.W^{\omega}\wedge.arrow \mathbb{Z}_{\geq 0}$. In fact, we canRemark 1.2.2. Note that the longest element $w_{0}\in W$ belongs tomaps the longest element
to theeasily show that the isomorphism
longest element $w_{0}\in W$ .
$W^{\omega}$$:\overline{W}\simarrow W^{\omega}$

1.3. The -Demazure operators.
$i\in I$ on the group ring$\omega$$\hat{w}_{0}\in\overline{W}$

Recall the ordinary Demazure operator:$\mathbb{Z}[\mathfrak{h}_{\mathbb{Z}}^{*}]=\mathrm{I}\mathrm{I}_{\lambda\in \mathfrak{h}^{*}}\mathbb{Z}e(\mathrm{z}\lambda)$$D_{i}$:$e( \lambda)\vdasharrow\frac{e(\lambda)-e(-\alpha_{i})e(r_{i}(\lambda))}{1-e(-\alpha i)}$.

$D_{i}$

for142
be the group algebra overLet
, on
Demazure operatorby$\mathbb{C}[\mathfrak{h}_{\mathbb{Z}}^{*}]\wedge$$i\in I\wedge$$\mathbb{C}$. Define likewise theofbasis
to be the -linear endomorphism of
given$\mathbb{C}[\mathfrak{h}_{\mathbb{Z}}^{*}]\wedge$$D_{r_{i}}\wedge,$$e(\lambda),\lambda\wedge\wedge\in\wedge \mathfrak{h}_{\mathbb{Z}}^{*}$$\wedge \mathfrak{h}_{\mathbb{Z}^{\mathrm{W}\mathrm{i}}}^{*}\mathrm{t}\mathrm{h}$$\mathbb{C}[^{\wedge}\mathfrak{h}_{\mathbb{Z}}*]$$\mathbb{C}$.

$D_{r_{i}} \wedge(e(^{\wedge}\lambda))=\frac{e(\lambda)-e(\wedge-\hat{\alpha}i)e(^{\wedge}r_{i}(^{\wedge}\lambda))}{1-e(-\hat{\alpha}_{i})}$

Then transfer
operator$D_{r_{i}}\wedge$

via$P_{\omega}^{*}$(1.3.1)

onto the group algebra$\hat{D}_{i}=P_{\omega}^{*}\circ D_{r_{i^{\circ}}}\wedge(P_{\omega}^{*})^{-1}$

to define the w-Demazure$\mathbb{C}[(\mathfrak{h}_{\mathbb{Z}}*)^{0}]$for.

$i\in I\wedge$Thus we can easily check the following.

LEMMA 1.3.1. Let. For each$i\in I\wedge$$\lambda\in(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$, we have$\hat{D}_{i}(e(\lambda))=\frac{e(\lambda)-e(-s_{i}\beta i)e(w_{i}(\lambda))}{1-e(-S_{i}\beta i)}$,and moreover$e(\lambda)+e(\lambda-S_{i}\beta i)+\cdots+e(w_{i}(\lambda))$$\hat{D}_{i}(e(\lambda))=\{$ $0$$-(e(\lambda+si\beta i)+e(\lambda+2s_{i}\beta_{i})+\cdots+e(w_{i}(\lambda+s_{i}\beta_{i})))$

Remark 1.3.2. Let
be a reduced expression of, . .
, i.e., $\ell(w)\wedge=n$ . We set
systemwhich does not depend on the choice of the reduced expression of

if
if
if

$\lambda(h_{i})\in \mathbb{Z}_{\geq 0}$$\lambda(h_{i})=-1$$(W^{\omega}, \{w_{i}|i\in I\})\wedge$$\hat{D}_{w}=\hat{D}_{i_{1}}\hat{D}_{i_{2}}$,$\lambda(h_{i})\in \mathbb{Z}_{\leq-2}$$w\in W^{\omega}$$w=w_{i_{1}}w_{i_{2}}\cdots wi_{n}$,.

in the Coxeter$\hat{D}_{i_{n}}\in \mathrm{E}\mathrm{n}\mathrm{d}_{\mathbb{C}}(\mathbb{C}[(\mathfrak{h}^{*}\mathbb{Z})0])$$w\in W^{\omega}$,.

1.4. Twining characters. Let be a connected, simply connected semi-simple linear
group overwith maximal torus
and Borel subgroup $B\supset T$ such that
Lie
, and Lie. Then the character groupof may be identified withby taking the differential at the identity element, i.e., by
the map. For each $i\in I$ and, we will write, where. Let
is the coroot of
for all $i\in I$ }{ A
be the set of dominant weights of A.
$G$

$T$

$\mathbb{C}$$\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{c}|$$(G)=\mathfrak{g},$$(B)=\mathrm{b}$$\mathrm{L}\mathrm{i}\mathrm{e}(T)=\mathfrak{h}$$T$$\Lambda=\mathrm{H}\mathrm{o}\mathrm{m}(T, GL_{1})$$\mathfrak{h}_{\mathbb{Z}}^{*}$$\lambda-\rangle$$\lambda\in\Lambda$$\mathrm{d}\lambda$$\alpha_{i}^{}\in \mathrm{H}\mathrm{o}\mathrm{m}(GL1, T)$$\alpha_{i}\in\Lambda$$\langle\lambda, \alpha_{i}^{\vee}\rangle=(\mathrm{d}\lambda)(h_{i})$$\Lambda_{+}=$$\lambda\in$$|\langle\lambda,$$\alpha_{i}^{\vee}\rangle\geq 0$

There exists an automorphism of whose differential at the identity element coincides
with the diagram automorphism of . By abuse of notation, we will denote still by
this automorphism of and by
the cyclic subgroup (of order $N$ ) ofgeneratedby the . Whenever there can be ambiguity, we will write
for the automorphism
may be identified with $N_{G}(T)/T$ ,of . Recall also that the Weyl group$N_{G}(T)$ the normalizer ofin . Each
lifts to an element of $N_{G}(T)$ fixed by$G$$\omega$$G$$\omega$$\mathfrak{g}$$\mathrm{A}\mathrm{u}\mathrm{t}(G)$$\langle\omega\rangle$$\mathrm{d}\omega$$\omega$$W\subset GL(\mathfrak{h}^{*})$$\mathfrak{g}$

$T$

$G$

$w\in W^{\omega}$

143, which will be denoted by . We will also denote the induced action of on. Note that,A
A by the same letter , and set},{, this action of on A coincides with the restriction
under the identification
to .
of$\dot{w}$$\omega\in \mathrm{A}\mathrm{u}\mathrm{t}(G)$$\Lambda^{\omega}=$$\omega$$\omega$$\lambda\in$$|\omega\cdot\lambda=\lambda$$\omega$$\Lambda\simeq \mathfrak{h}_{\mathbb{Z}}^{*}\subset \mathfrak{h}^{*}$$((\mathrm{d}\omega)^{-1})^{*}=((\mathrm{d}\omega)^{*})^{-1}$By athat admits a$\Lambda_{+}^{\omega}=\Lambda^{\omega}\cap\Lambda_{+}$$\mathfrak{h}_{\mathbb{Z}}^{*}$-module $M$ , we will always mean a finite-dimensional rational G-module-linear-action such that$G\mathrm{x}\langle\omega\rangle$$\mathbb{C}$$\langle\omega\rangle$

for all $g\in G,$ $m\in M$ .

$\omega\cdot(gm)=\omega(g)(\omega\cdot m)$of andas a linear algebraic group, thisRegarding the semi-direct productand-module. Likewise foris the same as a finite-dimensional rational.
with basis-modules. Let
be the group algebra over of-module, and let
Let $M$ be a$G$

$G\rangle\triangleleft\langle\omega\rangle$$\langle\omega\rangle$$B\rangle\triangleleft\langle\omega\rangle-$$G\rangle\triangleleft\langle\omega\rangle$$T\mathrm{x}\langle\omega\rangle$$e(\lambda),$$\Lambda^{\omega}$$\mathbb{C}$$\mathbb{C}[\Lambda^{\omega}]$$\lambda\in\Lambda^{\omega}$$Tn\langle\omega\rangle$$M= \prod_{\lambda\in\Lambda}M_{\lambda}$with$M_{\lambda}=${ $m\in M|tm=\lambda(t)m$ for all $t\in T$ }be the weight space decomposition with respect toof $M$ to be$T$. Now we define the twining character$\mathrm{c}\mathrm{h}^{\omega}(M)$.

$\mathrm{c}\mathrm{h}^{\omega}(M)=\sum_{\Lambda\lambda\in\omega}\mathrm{T}\mathrm{r}(\omega|_{M_{\lambda}})e(\lambda)\in \mathbb{C}[\Lambda^{\omega}]$

Remark 1.4.1. It easily follows that for each $t\in T$ ,$\mathrm{T}\mathrm{r}((t, \omega)$since$\omega\cdot M_{\lambda}=M_{\omega(\lambda)}$for$\lambda\in\Lambda$; $M$ )$= \sum_{\omega\lambda\in\Lambda}\mathrm{T}\mathrm{r}(\omega|_{M_{\lambda}})\lambda(t)\in \mathbb{C}$.

the simple rational G-moduleand1.5. An important example. Let
be a-module as follows. Let
into aof highest weight . We can make. Ifis the universal enveloping algebra of ,(nonzero) highest weight vector of
there is an isomorphism of U(g)-modules$L(\lambda)$$\lambda\in\Lambda_{+}^{\omega}$$L(\lambda)$$\lambda$$L(\lambda)$$G\mathrm{x}\langle\omega\rangle$$\mathrm{U}(\mathfrak{g})$$\mathrm{J}\mathrm{J}(\mathfrak{g})/2(\lambda)\simeq L(\lambda)$

where$\tau \mathrm{J}(\sim\lambda)$

is the left ideal of$\mathrm{U}(\mathfrak{g})$$v_{\lambda}$$\mathfrak{g}$via$x\mapsto xv_{\lambda}$,

given by$\tilde{\mathrm{J}}(\lambda)=\sum_{i\in I}(\mathrm{u}(\mathfrak{g})e_{i}+\mathfrak{U}(\mathfrak{g})(h_{i}-\langle\lambda,\alpha^{\vee}\rangle i)+\mathrm{u}(\mathfrak{g})f_{i}^{\langle\lambda}’ i\rangle+1)\alpha \mathrm{v}$is -invariant, i.e.,of, the left ideal
Since-module such that
admits a structure of$3(\lambda)$$\lambda\in\Lambda^{\omega}$$\mathrm{U}(\mathfrak{g})$$\omega$$\mathrm{d}\omega$-invariant, and hence$\langle\omega\rangle$$\omega\cdot(xv_{\lambda})=((\mathrm{d}\omega)(x))v_{\lambda}$

for all.

$x\in \mathrm{U}(\mathfrak{g})$.

$L(\lambda)$

144
Therefore, theadmits a structure of-module such thatthat a-module structure onsuch thatis unique since
cyclic -module generated by .
$L(\lambda)$$Gn\langle\omega\rangle$$L(\lambda)$$G\rangle\triangleleft\langle\omega\rangle$$G$$\omega\cdot v_{\lambda}=v_{\lambda}$$\omega\cdot v_{\lambda}=v_{\lambda}$. Noteis a$L(\lambda)$$v_{\lambda}$, we haveOn the other hand, for each
. Henceis dominant integral. If
is the simple -module of highest weight ,we know that$i\in I\wedge$$(P_{\omega}^{*})^{-}1(\lambda)\in\wedge \mathfrak{h}^{*}$.

$\wedge\lambda$$\hat{L}(\lambda)\wedge$$\wedge \mathfrak{g}$(1.5.1),$\mathrm{c}\mathrm{h}^{\omega}(L(\lambda))=P_{\omega}^{*}(\mathrm{c}\mathrm{h}\hat{L}(^{\wedge}\lambda))$$\iota \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}P_{\omega}*\mathrm{o}\mathrm{n}$$\hat{\lambda}=$$(P_{\omega}^{*})^{-1}(\lambda)(\wedge h_{i})=\lambda(h_{i})$the right-hand!

is$\mathrm{s}_{l}\backslash \mathrm{i}\mathrm{d}\mathrm{e}$d.efined by$\mathrm{a}.\mathbb{C}-\mathrm{a}- \mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}..\mathrm{a}\mathrm{i}_{\mathrm{S}\mathrm{o}\mathrm{m}\mathrm{o}}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}_{\mathrm{S}}\mathrm{m}‘.\cdot \mathbb{C}[\mathfrak{h}_{\mathbb{Z}}*]^{\sim}arrow \mathbb{C}[(\wedge.\mathfrak{y}_{\mathbb{Z}}*)^{0}]$for$P_{\omega}^{*}(e(^{\wedge}\mu))=e(P_{\omega}^{*}(\mu)\wedge)$$\mu\wedge\in \mathfrak{h}_{\mathbb{Z}}^{*}\wedge$

.

.

:. 1

$\tau$

Assume now that $J=I_{i}=\{\omega^{k}(i)|0\leq k\leq N_{i}-1\}\subset I,$, and letbe thestandard parabolic subgroup of associated to . Let(hencewithfor all $j\in J$ ). If
is the simple rational -module of highest weight, then it remains simple as a rational module over the Levi factorofwith theunipotent radicalofacting trivially. We can makeinto a-modulein the same way as
above.
$i\in I\wedge$

$G$

$J$

$l\text{ノ}\in\Lambda^{\omega}$$L_{J}(\nu)$$\langle_{\mathcal{U}\alpha_{j}^{\vee}},\rangle\geq 0$$P_{J}$$\langle_{l\text{ノ},\alpha_{i}}\vee\rangle\geq 0$$P_{J}$$L_{J}$$\nu$$U_{J}$$P_{J}$$L_{J}(I^{\text{ノ}})$$P_{J}$$P_{J}\rangle\triangleleft\langle\omega\rangle$$L(\lambda)$

The following lemma is a first (but important) step towards our main result (Theorem
0.1).
LEMMA 1.5.1. With the notation and assumption as above, we have in$\mathrm{c}\mathrm{h}^{\omega}(L_{j}(\nu))=\hat{D}i(e(\nu))$2.

$\mathrm{P}\mathrm{R}\mathrm{o}\mathrm{O}\mathrm{F}$$\mathbb{C}[\Lambda^{\omega}]$,.

OF THE MAIN RESULT.

Since the proof of our main result is so simple and clear modulo some algebrogeometric arguments, we give a “detailed outline” of it in this section. Fix
and let $X(w)$ be the associated Schubert variety over , i.e., the Zariski closure of the
Bruhat cell $B\dot{w}B/B$ in the flag variety $G/B$ . For a-module $M$ , the w-Euler
characteristic
is defined to be$w\in W^{\omega}$$\mathbb{C}$$B\mathrm{x}\langle\omega\rangle$$\chi_{w}^{\omega}(M)$$\chi_{w}^{\omega}(M)=\sum_{j\geq 0}(-1)j\mathrm{h}^{\omega}(\mathrm{C}H^{j}(X(w), \mathcal{L}X(w)(M)))\in \mathbb{C}[\Lambda^{\omega}]$.

-module, theHere recall that, since $M$ is a-moduleassociated$M$-equivariant ( .
-equivariant) sheaf, and hence the cohomology
tois agroups $H^{j}(x(w), \mathcal{L}X(w)(M)),$ $j\geq 0$ , are-modules.
$B\rangle\triangleleft\langle\omega\rangle$$(B, \langle\omega\rangle)$$=$$\mathcal{O}_{X(w)}$$B\rangle\triangleleft\langle\omega\rangle$$B\mathrm{x}\langle\omega\rangle$$\mathcal{L}_{X(w)}(M)$145
Let $w=w_{i_{1}}\cdots w_{i_{n}}$ be a reduced expression of $w\in W^{\omega}$ in the Coxeter system, i.e., $\ell(w)\wedge=n$ . Note that we have $\ell(w)=\ell(w_{i_{1}})+\cdot\cdot:+\ell(w_{i_{n}})$ .
We want to show that$(W^{\omega}, \{w_{i}|i\in I\})\wedge$$\chi_{w}^{\omega}(M)=\hat{D}_{i_{1}}\cdots\hat{D}_{i_{n}}(\mathrm{C}\mathrm{h}^{\omega}(M))$,

is the -Demazure operator defined in \S 1.3. In particular, wefor $j=i_{1},$wherewill obtain a twining character formula of the Demazure moduletheonis the one-dimensional, wherefor.
trivially.
weight through the quotient $Barrow T$ and$\hat{D}_{j}$$i_{n}$$\omega$$\ldots,$$H^{0}(x(w), \mathcal{L}_{x(w)}(\mathrm{c}_{\lambda}))$$\mathbb{C}_{\lambda}$$\lambda\in\Lambda_{+}^{\omega}$$B\rangle\triangleleft\langle\omega\rangle- \mathrm{m}\mathrm{o}\mathrm{d}.\mathrm{u}\mathrm{l}\mathrm{e}$$\mathrm{w}.$hi.ch-B$\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{s}\mathrm{b}\mathrm{y}:.$$\langle\omega\rangle$$\lambda$

2.1. Formula for theto show$\omega$-Euler characteristics.

(2.1.1)Set$\chi_{w}^{\omega}(M)=\hat{D}_{w}(\mathrm{c}\mathrm{h}^{\omega}(M))$$\hat{D}_{w}=\hat{D}_{i_{1}}\cdots\hat{D}_{i_{n}}$. Then we are.

Let us first make some reductions. Since both sides of (2.1.1) are additive in $M$ , we mayon which is acting by a scalarassume that $M$ is one-dimensional of weight.
and $k\in$ Z. We will denote such $M$ by
for a primitive N-th root of unity inThus we are reduced to showing that$\mu\in\Lambda^{\omega}$$\zeta^{k}$$\omega$$\mathbb{C}$$\mathbb{C}_{\mu,k}$$\zeta$$\chi_{w}^{\omega}(\mathbb{C}_{\mu,k})=\hat{D}_{w}(\mathrm{c}\mathrm{h}^{\omega}(\mathbb{C}_{\mu,k}))$where$\mathrm{c}\mathrm{h}^{\omega}(\mathbb{C}_{\mu,k})=\zeta k(e\mu)$Put for simplicity.

$z_{j}=w_{i_{j}},$$1\leq j\leq n$. We have an isomorphism of$B\rangle\triangleleft\langle\omega\rangle$-modules,(2.1.2)$H^{\cdot}(X(w), c_{X(w)(\mathbb{C}))(}\mu,k\simeq H^{\cdot}x(z_{1}, \ldots, z_{n}), \mathcal{L}_{X(z,\ldots,z)}(1n\mathbb{C}_{\mu,k}))$and for each
(2.1.3),$s$with $1\leq s\leq n-1$ , a$B\rangle\triangleleft\langle\omega\rangle$-equivariant spectral sequence.

$H^{i}(X(z_{s}), \mathcal{L}(H^{j}(x(Z_{s+}1, \ldots , z_{n}), \mathcal{L}(\mathbb{C}_{\mu,k}))))\Rightarrow H^{i+j}(x(z_{s}, \ldots, z_{n}), \mathcal{L}(\mathbb{C}_{\mu},k))$for 1 $\leq s\leq t\leq n$ is the so-called Bott-Samelson variety, and.
-module-modules associated to theis the sheaf of$\in N_{G}(T)$and their right coset representatives,$
Note that, since-invariant sub, the Bott-Samelson variety $X(z_{S}, \ldots, z_{t})$ is an
are fixed byvariety of $(G/B)^{t-S}+1$ . (The proofs of (2.1.2) and (2.1.3) are not so difficult, but rather
long. For details, see our preprint [KN].)

Here$X(z_{S}, \ldots, z_{t})$$B\rangle\triangleleft\langle\omega\rangle$$O_{X(z_{S},\ldots,z)}t$$c_{x()}z_{S},\ldots,zt(\mathbb{C}_{\mu,k})$$z_{S},$$z_{t}\in W^{\omega}$$\ldots$$\dot{z}_{s},$$\mathbb{C}_{\mu,k}$$\ldots$$\langle\omega\rangle$$\omega\in \mathrm{A}\mathrm{u}\mathrm{t}(G)$Remark 2.1.1. There are several equivalent (or inequivalent) definitions of a Bott-Samelson
variety, but in the talk, we stick to that of [Ja]:$X(y_{1}, \ldots, y_{n})=$$y_{n}\in W$ . Ifforthe subgroup$y_{1},$$\ldots,$$J_{i}${$(g_{1}B,$$g_{n}B)\in(G/B)^{n}|g_{i-1}^{-1}g_{i}\in\overline{B\dot{y}_{i}B}$$\ldots,$

for all$i$}is the longest element offor $1\leq i\leq n$ is a subset of andof the Weyl group $W$ , then the Bott-Samelson variety$W_{J_{i}}=\langle r_{k}|k\in J_{i}\rangle$$I$$z_{J_{i}}$146,
is smooth. Moreover, if we assume thatthen the restriction of the n-th projection: $(G/B)^{n}arrow G/B$ to the Bott-Samelson
variety $X(Z_{J_{1}}, \ldots , z_{J_{n}})\subset(G/B)^{n}$ gives a Demazure-Hansen desingularization$\ell(z_{J_{1}j_{n}}\ldots Z)=\ell(z_{J_{1}})+\cdots+\ell(zj_{n})$$X(Z_{j_{1}}, \ldots , z_{J_{n}})$$\phi$$\pi_{n}$$\phi:X(_{Z}j_{1}, \ldots, z_{J_{n}})arrow X(z_{J_{1}}\cdots z_{J_{n}})$of the Schubert variety
open and dense subvarieties.

$X(z_{j_{1}Z_{J_{n}}}\ldots)$. Note that the$\phi$

induces an isomorphism of suitable

Now it follows thatby (2.1.2)$\chi_{w}^{\omega}(\mathbb{C}_{\mu},k)=\sum_{j\geq 0}(-1)j\mathrm{C}\mathrm{h}^{\omega}(Hj(x(Z1, \ldots , z_{n}), \mathcal{L}_{x}(z1,\ldots,z_{n})(\mathbb{C}k)\mu,))$$= \sum_{j\geq 0}(-1)^{j}(_{i\geq}\sum_{0}(-1)i\mathrm{h}^{\omega}(Hi(X(Z_{1}), c(Hj(\mathrm{C}X(Z_{2}, \ldots, z_{n}), c(\mathbb{C}k)\mu,))))\mathrm{I}$$= \sum_{j\geq 0}(-1)^{j}x_{z}(\omega_{1}H^{j}(x(Z_{2}, \ldots, z_{n}), c_{x(}z_{2},\ldots,z_{n})(\mathbb{C}k)\mu,))$by (2.1.3).

By induction on , we may assume that $w=w_{i}$ for somein proving (2.1.1). So put$J=I_{i}$ and let $P=P_{J}$ be the standard parabolic subgroup of
associated to . We are
to show$i\in I\wedge$

$n$

$G$

(2.1.4)$\chi_{w_{i}}^{\omega}(\mathbb{C})\mu,k=\hat{D}_{i}(\zeta^{k}e(\mu))$$J$.

(and hence that
Assume first thatfor all $k\in J$ ). Let
be the simple rational -module of highest weight admitting an-action as in \S 1.5,
and letbe the one-dimensional trivial -module with acting by the scalar .
$\langle\mu, \alpha_{i}^{\vee}\rangle\geq 0$$\langle\mu,$$P_{J}$$\langle\omega\rangle$$\mu$$P_{J}$$\zeta^{k}$$L_{J}(\mu)$$\alpha_{k}\rangle\geq 0$$\zeta^{k}$$\omega$

LEMMA 2.1.2. Let the notation and assumption be as above. Then we have the following
isomorphism of-modules.
$P_{J}\rangle\triangleleft\langle\omega\rangle$.

$H^{0}(P_{J}/B, \mathcal{L}_{P_{J}/B}(\mathbb{C}_{\mu,k}))\simeq L_{J}(\mu)\otimes_{\mathbb{C}}\zeta k$(This lemma is, in a sense, crucial to the proof of our main result. Although no one
doubts the truth of this lemma, its complete proof would be rather long.)

147
Now we deduce that$\chi_{w_{i}}^{\omega}(\mathbb{C})\mu,k=\mathrm{c}\mathrm{h}^{\omega}(H0(p/B, cP/B(\mathbb{C}_{\mu},k)))$$=\mathrm{c}\mathrm{h}^{\omega}(L_{J}(\mu)\otimes_{\mathbb{C}}\zeta^{k})$by Kempf’s vanishing theorem

by Lemma 2.1.2$=\zeta^{k}\mathrm{c}\mathrm{h}^{\omega}(LJ(\mu))$by Lemma 1.5. 1$=\zeta^{k}\hat{D}_{i}(e(\mu))$$=\hat{D}_{i}(\zeta^{k}e(\mu))$Ifvanish.

$\langle\mu, \alpha_{i}\rangle=-1$.

(and hence$\langle\mu,$$\alpha_{k}\rangle=-1$

for all $k\in J$ ), then both sides of (2.1.4)

for all $k\in J$ ). Set(and hence that
Assume finally thatthe positive root system of . By direct
with, we see-module isomorphism Liechecking, using the-modules,that as$\langle\mu,$$\langle\mu, \alpha_{i}\rangle\leq-2$$\alpha_{k}\rangle\leq-2$$P_{J}$$\Delta_{J}^{+}=\Delta_{+}\cap\sum_{k\in J}\mathbb{Z}\alpha_{k}$$\rho_{J}=\frac{1}{2}\sum_{\alpha\in\Delta_{J}^{+}}\alpha$$(P)/\mathrm{L}\mathrm{i}\mathrm{e}(B))^{*}\simeq\oplus_{\alpha\in\triangle_{J}^{+}}\mathbb{C}f_{\alpha}$$($$T\rangle\triangleleft\langle\omega\rangle$$B\rangle\triangleleft\langle\omega\rangle$$\bigwedge_{\mathbb{C}}^{\ell(w_{i}})(\mathrm{L}\mathrm{i}\mathrm{e}(P)/\mathrm{L}\mathrm{i}\mathrm{e}(B))*\simeq \mathbb{C}_{-20}\rho J,\otimes_{\mathbb{C}}(-1)^{l(w_{i})-1}$

where $\ell(w_{i})=\dim_{\mathbb{C}}(P/B)$ andacting trivially and by the scalarduality reads$(-1)^{\ell(w_{i}})-1$$B$$\omega$is the one-dimensional. Then the$(-1)^{\ell(w_{i}})-1$,-module with-equivariant Serre$B\rangle\triangleleft\langle\omega\rangle$$B\rangle\triangleleft\langle\omega\rangle$(2.1.5)$H^{j}(P/B, \mathcal{L}_{P/}B(\mathbb{C}_{\mu,k}))^{*}\simeq H\ell(w_{i})-j(P/B, \mathcal{L}_{P/}B(\mathbb{C}_{-}\mu-2\rho_{j},-k^{\otimes_{\mathbb{C}}()^{\ell(w_{i}}}-1)-1))$

if$H^{0}(P/B, \mathcal{L}P/B(\mathbb{C}_{-}-2-k)\mu\rho_{J},)\otimes \mathbb{C}(-1)^{\ell(w_{i}})-1$$\simeq\{$$j=^{p_{(w_{i})}}$,

otherwise (by Kempf).

$0$(The use above of theproof of our main result.)$B\rangle\triangleleft\langle\omega\rangle$-equivariant Serre duality is the most essential part of theRemark 2.1.3. Put $X=P/B$ and $m=\dim_{\mathbb{C}}X$ . Let. The
module that is locally free of finite rank over(see our preprint on Naito’s home page) asserts that, as$\mathcal{M}$be a$B\rangle\triangleleft\langle\omega\rangle$$\mathcal{O}_{X}$$H^{i}(X, \mathcal{M}^{\mathrm{v}}\otimes X\Omega m)X(\simeq Hm-ix, \mathcal{M})^{*}$-equivariant-equivariant Serre duality$(B, \langle\omega\rangle)$$B\rangle\triangleleft\langle\omega\rangle$$O_{X^{-}}$-modules,for all $0\leq i\leq m$ ,is the canonicalis the dual sheaf ofwhere. This Serre-module ofis the dual
sheaf on $X$ , and-action on the one-dimensional
duality will be a consequence of the triviality of the. Since the triviality of the -action on it is known, it remains
vector space-action. There are many ways to show it, but the way
to show the triviality of the$\mathcal{M},$$\mathcal{M}^{\vee}=\mathcal{H}om_{X}(\mathcal{M}\cdot, O_{x})$$H^{m-i}(X, \mathcal{M})^{*}$$\Omega_{X}^{m}=\bigwedge_{X}^{m}\Omega_{X}^{1}$$H^{m-i}(x, \mathcal{M})$$B\mathrm{x}\langle\omega\rangle$$Bn\langle\omega\rangle$$B$$H^{m}(X, \Omega^{m})X$$\langle\omega\rangle$

148
we take here is (I think) purely algebro-geometric and elementary: first take a$P\rangle\triangleleft\langle\omega\rangle-$

equivariant closed immersion
for sufficiently dominant
then
use the fact that the full automorphism group $PGL(L(\lambda))$ of
acts trivially on the
one-dimensional vector space, where (though not so trivial)with$\iota:Xarrow \mathrm{P}=\mathrm{P}(L(\lambda))$$\lambda\in\Lambda_{+}^{\omega};$$\mathrm{P}(L(\lambda))$$H^{l}(\mathrm{P},$$l=\dim_{\mathbb{C}}\mathrm{P}$$\Omega_{\mathrm{P}}^{\iota_{)}}$$H^{l}(\mathrm{P}, \Omega_{\mathrm{p}}^{\iota})\simeq H^{m}(\mathrm{P}, \iota*\Omega^{m})X\simeq H^{m}(X, \Omega^{m})X$

as$P\rangle\triangleleft\langle\omega\rangle$-modules.

The proof of the following lemma is easy.

LEMMA 2.1.4. Let be an
groupof , and let
following isomorphism of$J$$W_{J}$$P_{J}$$\omega$-invariant subset of $I,$
be such that-modules.

$\nu\in\Lambda^{\omega}$$w_{J}$$\langle\nu, \alpha_{k}^{}\rangle\geq 0$

the longest element of the Weylfor all $k\in J$ . Then we have the$P_{J}\mathrm{x}\langle\omega\rangle$$L_{J}(\nu)*\simeq L_{J}(-w_{J}(_{\mathcal{U}}))$$\langle\omega\rangle$.

The isomorphism (2.1.5) together with Lemmas 2.1.2 and 2.1.4 implies that, as-modules,$H^{\ell(w_{i}})(P/B, \mathcal{L}_{P}/B(\mathbb{C}_{\mu,k}))\simeq(L_{j}(-\mu-2\rho J)^{*k}\otimes \mathbb{C}\zeta)\otimes_{\mathbb{C}}(-1)^{\ell}(wi)-1$$.(2.1.6^{\cdot})‘ \mathfrak{t}$$\simeq L_{J}(w_{i}(\mu+2\rho j))\otimes_{\mathbb{C}}\zeta k_{\otimes \mathbb{C}}(-1)^{\ell(}w_{i})-1$

Then, setting$\mu\wedge=(P_{\omega}^{*})^{-}1(\mu)$,$\chi_{w_{i}}^{\omega}(\mathbb{C}_{\mu},k)=(-1)^{\ell}(wi)_{\mathrm{C}\mathrm{h}^{\omega}(L(}Jw_{i}(\mu+2\rho_{J}))\otimes_{\mathbb{C}}\zeta^{k}\otimes_{\mathbb{C}}(-1)^{\ell()-1}w_{i})$by (2.1.6)$=-\zeta^{k}\mathrm{c}\mathrm{h}\omega(LJ(w_{i}(\mu+2\rho_{J})))$$=-\zeta^{k}\hat{D}_{i}(e(w_{i}(\mu+2\rho_{J})))$by Lemma 1.5.1$=-\zeta^{k}(P_{\omega}^{*}\mathrm{o}D_{r_{i}}\wedge\circ(P_{\omega}*)-1\mathrm{I}(e(w_{i}(\mu+2\rho j)))$$=-\zeta^{k}(P_{\omega}^{*}\circ D\wedge \mathrm{I}r_{i}(e(r_{i}\wedge(\mu\wedge+\hat{\alpha}_{i})))$

since$(P_{\omega}^{*})^{-1}(2\beta j)=\hat{\alpha}_{i}$$=-\zeta^{k}P_{\omega}^{*}(-D_{r}\wedge(ie(^{\wedge}\mu)))$$=\zeta^{k}(\hat{D}_{i}\circ P_{\omega}*)(e(^{\wedge}\mu))$$=\zeta^{k}\hat{D}_{i}(e(\mu))$$=\hat{D}_{i}(\zeta^{k}e(\mu))$.

Thus in all cases (2.1.4) holds, and we are done.
$,\mathrm{I}\mathrm{f}\lambda\in\Lambda_{+}^{\omega}$.

, then for any Schubert variety $X(w)$ ,$H^{j}(X(w), c_{x}(w)(\lambda))=0$

for all $j\geq 1$$B\rangle\triangleleft$

149
by the Demazure vanishing theorem of Andersen et al. Hence we have proved

THEOREM 2.1.5. Let $M$ be a finite-dimensional rational
Then we have in,$B\rangle\triangleleft\langle\omega\rangle$-module and$w\in W^{\omega}$.

$\mathbb{C}[\Lambda_{\omega}]$,$\chi_{w}^{\omega}(M)=\sum_{j\geq 0}(-1)j\mathrm{h}^{\omega}(\mathrm{C}H^{j}(X(w), \mathcal{L}_{x(}(M)))=\hat{D}(_{\mathrm{C}}\mathrm{h}^{\omega}(M)w)w)$

where
the Coxeter system$\hat{D}_{w}=\hat{D}_{i_{1}}\hat{D}_{i_{2}}\cdots\hat{D}_{in}$for any reduced expression
. In
, for

$w=w_{i_{1}}w_{i_{2}}\cdots w_{in}$$(W^{\omega}, \{w_{i}|i\in I\})\wedge$$partiCu\iota ar$, we have$\lambda\in\Lambda_{+}^{\omega}$of$w\in W^{\omega}$in,$\mathrm{c}\mathrm{h}^{\omega}(H^{0}(X(w), c_{X}(w)(\mathbb{C}\lambda)))=\hat{D}_{w}(e(\lambda))$whereis the one-dimensional
the quotient $Barrow T$ and$\mathbb{C}_{\lambda}$$B\rangle\triangleleft\langle\omega\rangle$$\omega t\dot{n}via\iota\iota_{y}$-module on which$B$

acts by the weight$\lambda$

through.

Theorem 2.1.5 above reveals that there exists a striking relation between the -Euler
characteristic
for and the ordinary Euler characteristic for the orbit
Lie algebra . To state the relation, we need some notation. Recall tha.t the orbit
Lie algebra is the dual complex semi-simple Lie algebra of the fixed point subalgebra$\mathfrak{g}^{0}=\{x\in \mathfrak{g}|\omega(x)=x\}$of . Letbe a connected, simply connected semi-simple
linear algebraic group over with maximal torus and Borel subgroupsuch that
Lie, and Lie
. For
, we take a right coset
representativeof , and define the Schubert variety
over by$\omega$$\chi_{w}^{\omega}(\mathbb{C}_{\lambda})\in \mathbb{C}[(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}]$$\mathfrak{g}$$\wedge \mathfrak{g}$$\wedge \mathfrak{g}$$\hat{G}$$\mathfrak{g}$$\hat{T}$$\mathbb{C}$$(\hat{G})=\mathfrak{g}\wedge,$$(\hat{B})=\mathfrak{h}\wedge$$\mathrm{L}\mathrm{i}\mathrm{e}(\hat{T})=\mathfrak{h}\wedge$$\hat{w}\in N_{\hat{G}}(\hat{T})$$\hat{B}\supset\hat{T}$$\hat{w}\in\overline{W}\simeq N_{\hat{G}}(\hat{T})/\hat{\tau}$$\hat{w}$$\hat{X}(\hat{w})$$\hat{x}(\hat{w})=\overline{\hat{B}\hat{w}\hat{B}/\hat{B}}=\overline{\hat{B}\hat{w}\hat{B}}/\hat{B}\subset\hat{G}/\hat{B}$$\mathbb{C}$.

For each, we denote by
the (locally free) -equivariant sheaf ofmodules associated to the one-dimensional -module
on which acts by the weight
through the quotient.
$\hat{B}$$\wedge\lambda\in\wedge \mathfrak{h}_{\mathbb{Z}}^{*}$$c_{\hat{X}(\hat{w})\lambda}(\mathrm{c}_{\wedge})$$\mathcal{O}_{\hat{X}(\hat{w})^{-}}$$\hat{B}$$\hat{B}$$\mathbb{C}_{\lambda}\wedge$$\hat{B}arrow\hat{T}$$\wedge\lambda$

Now we are ready to state the following

COROLLARY 2.1.6. Let
and. Then we have in the algebra$w\in W^{\omega}$$\lambda\in(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$.

$(P_{\omega}^{*})^{-}1(\lambda)\in\wedge \mathfrak{h}_{\mathbb{Z}}^{*}$We set $\hat{w}=^{-1}(w)\in\overline{W}$,$and\wedge\lambda=$$\mathbb{C}[(\mathfrak{h}_{\mathbb{Z}}*)^{0}]$$\chi_{w}^{\omega}(\mathbb{C}_{\lambda})=\sum(-1)^{j}\mathrm{c}\mathrm{h}^{\omega}(Hj(x(w), \mathcal{L}X(w)(\mathbb{C}_{\lambda})))j\geq 0$$=P_{\omega}^{*}$

where ch($\sum(-1)^{j}$$H^{j}(\hat{X}(\hat{w}), \mathcal{L}\hat{x}(\hat{w})(\mathbb{C}\wedge)\lambda)\in \mathbb{C}[\mathfrak{h}_{\mathbb{Z}}^{*}]\wedge$

cohomology groupforof$H^{j}(\hat{X}(\hat{w}), \mathcal{L}\hat{\mathrm{x}}(\hat{w})(\mathbb{C}\wedge)\lambda)$ch$H^{j}(\hat{x}(\hat{w}), \mathcal{L}(\hat{X}(\hat{w})\mathbb{C}\wedge)\lambda)$$j\in \mathbb{Z}_{\geq 0}$$\hat{X}(\hat{w})$.

),is the ordinary character of the j-th

150(This immediately follows from Theorem 2.1.5 and the ordinary Demazure character formula for the orbit Lie algebra )$\wedge \mathfrak{g}.$, with
2.2. Joseph’s modules. Let us finally return to Joseph’s module. Thus let
andbe a (nonzero) lowest weight vector of the dual module), and let(which is the dual element of a (nonzero) highest weight vectorof. Since. Joseph’s module
is fixed by , so isrepresentingof highestinis defined to be$w\in W^{\omega}$$J_{w}(\lambda)$$\lambda\in\Lambda_{+}^{\omega}$$L(\lambda)^{*}$$v_{\lambda}^{*}$$L(\lambda)$$v_{\lambda}$$w\in W^{\omega}$$\dot{w}\in N_{G}(T)^{\omega}$$J_{w}(\lambda)$$\omega$$v_{\lambda}^{*}$$\dot{w}v_{\lambda}^{*}$$L(\lambda)^{*}$$\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}-w(\lambda)$$J_{w}(\lambda)=\mathrm{u}(\mathrm{b})(\dot{w}v^{*})\lambda\subset L(\lambda)^{*}$,. Note that, sinceis the universal enveloping algebra of
where. Moreover, since, Joseph’s module
is a-submodule of(nonzero) highest weight vector of
fixed by , there is an isomorphism ofmodules$\omega\cdot(\dot{w}v_{\lambda}^{*})=$$\mathrm{b}=\mathrm{L}\mathrm{i}\mathrm{e}(B)$$\mathrm{U}(\mathrm{b})$$J_{w}(\lambda)$$\dot{w}v_{\lambda}^{*}$$L(\lambda)^{*}$$B\rangle\triangleleft\langle\omega\rangle$$L(\lambda)^{*}$(2.2.1)$\omega$$L(\lambda)^{*}\simeq L(-w_{\mathrm{o}(\lambda)})$which enables us to regarda short exact sequence of$J_{w}(\lambda)$

as a-modules$B\rangle\triangleleft\langle\omega\rangle$$\dot{w}_{0}v_{\lambda}^{*}$

is a$G\rangle\triangleleft\langle\omega)-$,-submodule of$L(-w_{\mathrm{o}(\lambda)})$. Then we obtain$B\lambda\langle\omega\rangle$,$0arrow J_{w}(\lambda)^{*}arrow L(-w_{0}(\lambda))^{*}arrow J_{w}(\lambda)^{\perp}arrow 0$

with $J_{w}(\lambda)^{\perp}=\{\phi\in L(-w_{0}(\lambda))^{*}|\phi(J_{w}(\lambda))=0\}$ . On the other hand, Lemma 2.1.2 forthe case $J=I$ combined with (2.2.1) yields an isomorphism of-modules$G\mathrm{x}\langle\omega\rangle$$H^{0}(G/B, c_{G/B}(\mathbb{C}_{\lambda}))\simeq L(-w_{0}(\lambda))*$.

Since the restriction map$H^{0}(G/B, c_{G/B}(\mathbb{C}_{\lambda}))arrow H^{0}(X(w), \mathcal{L}x(w)(\mathbb{C}_{\lambda}))$is known to be a (modules$B\rangle\triangleleft\langle\omega\rangle$-equivariant) surjection, we obtain an isomorphism of$J_{w}(\lambda)^{*}\simeq H^{0}(x(w), c_{X}(w)(\mathbb{C}\lambda))$,$J_{w}(\lambda)\simeq H^{0}(X(w), \mathcal{L}\mathrm{x}(w)(\mathrm{c}\lambda))^{*}$.

$B\rangle\triangleleft\langle\omega\rangle-$or equivalently(2.2.2)

We now define a$\mathbb{C}$-linear conjugation$-:\mathbb{C}[\Lambda^{\omega}]arrow \mathbb{C}[\Lambda^{\omega}]$$\sum_{\mu\in\Lambda^{\omega}}a\mu e(\mu)=\sum_{\mu\in\Lambda^{\omega}}a_{\mu}e(-\mu)$Then we obtain the following theorem from thewith$a_{\mu}\in \mathbb{C}$$B\lambda\langle\omega\rangle$

by

for

$\mu\in\Lambda^{\omega}$.

-module isomorphism (2.2.2).

151

THEOREM 2.2.1. Let$\lambda\in\Lambda_{+}^{\omega}$and$w\in W^{\omega}$. Then we have in$\mathbb{C}[\Lambda^{\omega}]$$\mathrm{c}\mathrm{h}^{\omega}(J_{w}(\lambda))=\overline{\mathrm{C}\mathrm{h}\omega(H^{0}(X(w),\mathcal{L}x(w)(\mathbb{C}\lambda)))}$,.

By combining Theorems 2.1.5 and 2.2.1, we obtain the following

COROLLARY 2.2.2. Let$\lambda\in\Lambda_{+}^{\omega}$and$w\in W^{\omega}$. Then we have in$\mathrm{c}\mathrm{h}^{\omega}(J_{w}(\lambda))=\hat{D}w(e(\lambda))$$\mathbb{C}[\Lambda^{\omega}]$,.

Finally, by combining Corollary 2.1.6 and Theorem 2.2.1, we obtain a remarkable
relation between the twining characterof Joseph’s module
for andthe ordinary character of Joseph’s module for the orbit Lie algebra , which is the dual
complex semi-simple Lie algebra of . For each, let
$J_{w}(\lambda)$$\mathrm{c}\mathrm{h}^{\omega}(J_{w}(\lambda))$$\mathfrak{g}$$\wedge \mathfrak{g}$$\hat{w}\in\overline{W}$$\mathfrak{g}^{0}$$J_{\hat{w}}(\lambda)\wedge\wedge\wedge=\mathrm{u}(\mathrm{b})(\hat{w}.v_{\lambda}^{*}\wedge\wedge)\subset\hat{L}(\lambda)^{*}\wedge$

be Joseph’s module of highest weight$\hat{L}(\lambda)^{*}\wedge$$-\hat{w}(\lambda)\wedge$, with$\wedge v_{\lambda}^{*}\wedge\in\hat{L}(\lambda)^{*}\wedge$a lowest weight vector of.

COROLLARY 2.2.3. Let$and\wedge\lambda=(P_{\omega}^{*})^{-}1(\lambda)\in\wedge \mathfrak{h}_{\mathbb{Z}}^{*}$

be dominant and. Then we have in,$\lambda\in(\mathfrak{h}_{\mathbb{Z}}^{*})^{0}$$w\in W^{\omega}$. We set $\hat{w}=^{-1}(w)\in\overline{W}$$\mathbb{C}[(\mathfrak{h}_{\mathbb{Z}}*)^{0}]$$\mathrm{c}\mathrm{h}^{\omega}(J_{w}(\lambda))=P^{*}\omega(\mathrm{c}\mathrm{h}J_{\hat{w}}\wedge\wedge(\lambda)\wedge(\lambda))$.

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INSTITUTE OF MATHEMATICS, UNIVERSITY OF TSUKUBA, TSUKUBA, IBARAKI 305-8571, JAPAN-mail address:$E$sukuba. ac. jp$\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{t}\mathrm{o}\emptyset \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{t}$