数理解析研究所講究録
1183 巻 2001 年 102-119

102

A Quantization of Conjugacy Classes of Matrices
大島利雄 (Toshio Oshima)
東京大学大学院数理科学研究科
Graduate School of Mathematical Sciences, University of Tokyoの
概要: 一般の放物型部分群のスカラー表現から誘導されたGeneralized Verma 加群の零化イデアルの具体的な生成元を、行列における小行列式や単因子の「量子化」 として構成する。「量子化」のパラ
メータ を とした「古典極限」では、構成した生成元は正方行列の相似類の作る集合の定義イデアルとなる。
$\mathfrak{g}\mathfrak{l}(n, \mathbb{C})$$\epsilon$$0$

1. Introductionof size with
be an element of the spaceof squareLetcomponents in C. Then the conjugacy class containing is the algebraic variety. Under theby denoting $G=GL(n, \mathbb{C})$ and, we will study a quantization ofinterpreted as follows:-action onin theor the -invariant defining ideal of
For the defining equations of, we will associate left invariant differential
ring of polynomial functions ofof the ring of the left invariant differential operators
operators on or an ideal
and we identify theis identified withon . The Lie algebra ofleft invariant differential operators on with the universal enveloping algebra
to a suitable
is a-homomorphism ofof . Then our quantization of$M$
becomes a representation spacemodule . Note that the quantization of$M$
is a function space on a homogeneous space oforof a real formof if-homogeneous vector bundle.
a space of sections of a$A$

$M(n, \mathbb{C})$$\mathrm{m}\mathrm{a}\mathrm{t}_{}\mathrm{r}\mathrm{i}\mathrm{C}\mathrm{e}\mathrm{s}$

$7l$

$A$

$\mathrm{A}\mathrm{d}(g)A=gAg^{-1}$$V_{A}= \bigcup_{g\in G}\mathrm{A}\mathrm{d}(g)A$$G$$M(n, \mathbb{C})$$V_{A}$$G$$V_{A}$$V_{A}$$M(n, \mathbb{C})$$G$$J_{A}$$G$$GL(n, \mathbb{C})$$\mathfrak{g}$$M(n, \mathbb{C})$$G$$V_{A}$$\mathfrak{g}$$U(\mathfrak{g})$$U(\mathfrak{g})/J_{A}$$U(\mathfrak{g})$$V_{A}$$U(\mathfrak{g})$$G$$G_{\mathbb{R}}$$G_{\mathbb{R}}$$G_{\mathbb{R}}$$V_{A}= \bigcup_{g\in c^{\mathrm{A}}}\mathrm{d}(g)A$$arrow G$ -invariant defining ideal of.

Representations of$V_{A}$$\downarrow \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{Z}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$$U(\mathfrak{g})$orIdeal of

$G_{\mathbb{R}}arrow$$U(\mathfrak{g})$In \S 2 we introduce a homogenized universal enveloping algebra
to study
our quantization together with “the classical limit” $(\epsilon=0)$ . We construct generators of
ffom the generalized Capelli operators introduced by [O2] which canbe considered as quantizations of minors and we show in Theorem 2.8 that theygenerate the annihilator of a generalized Verma module induced from a characterand is a nilpotent matrix, the correof a parabolic subalgebra of . Whensponding result is Tanisaki’s conjecture [Ta], which is solved by Weyman [We]. In
particular, if is a regular nilpotent matrix, the result is due to Kostant [Ko].
$U^{\epsilon}(\mathfrak{g})$$J_{A}$$\mathfrak{g}$$A$$\epsilon=0$

$A$

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In \S 3 we examine how the annihilator determines the difference between thegeneralized Verma module and the Verma module, which is important for applications. For example, the theorem on boundary value problems for symmetric spaces
studied in [O2, Theorem 5.1] is improved by the generator system defined in thisnote.
ffom which we can construct
We can also quantize the minimal polynomial ofanother generator system of the annihilator. This is valid for other general reductive
Lie algebras and is studied in another paper [O3].
$V_{A}$

2. Elementary divisors
and also with
The Lie algebra of $G=GL(n, \mathbb{C})$ is identified withthe space of left $G$-invariant holomorphic vector fields on $G$ . Then is spanned byis the fundamental matrix unit whose
for $1\leq i\leq n$ and $1\leq j\leq n$ where$(p, q)$ -component equals
and$M(n, \mathbb{C})$$\mathfrak{g}$$\mathfrak{g}$$E_{ij}$$E_{ij}$$\delta_{i,p}\delta_{j,q}$(2.1)$E_{ij}= \sum_{\nu=1}x_{\nu}ni\frac{\partial}{\partial x_{\nu j}}$-module.
with the coordinate $(x_{ij})\in G$ . Then is naturally a
Using the non..degenerate symmetric bilinear form X,
wewith its dual . The dual basis
given by $E_{ij}^{*}=E_{ji}$ . For simplicity, we will denote $E_{i}=E_{ii}$ and$(\mathfrak{g}, G)$$\mathfrak{g}$$\langle$$M(n, \mathbb{C})\cross M(n, \mathbb{C})$$Y\rangle$$\{E_{ij}^{*}\}$$\mathfrak{g}^{*}$$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}6r\emptyset$$=\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{C}\mathrm{e}(X\mathrm{Y})$of.

$\{E_{ij}\}$onis$e_{i}=E_{ii}^{*}$

DEFINITION 2.1. The homogenized universal enveloping algebra
defined by(2.2)$U^{\epsilon}(\mathfrak{g})$of$\mathfrak{g}$is$U^{\epsilon}( \mathfrak{g})=(\sum_{k=0}^{\infty}\otimes^{k}\mathfrak{g})/\langle X\otimes Y-\mathrm{Y}\otimes X-\epsilon[X, Y];x, Y\in \mathfrak{g}\rangle$. Here is a
is denoted byand the subalgebra of -invariants incomplex number (or an element commuting with g) and the denominator is thespan as a two-sided ideal of the numerator, the tensor algebra of .
$G$$U^{\epsilon}(\mathfrak{g})^{G}$$U^{\epsilon}(\mathfrak{g})$$\epsilon$$\mathfrak{g}$-module induced ffom the tensor algebra.
is naturally aNote thatand the symmetric
are the universal enveloping algebra
andgives an, the map defined byof , respectively. If
algebra.
onto
algebra isomorphism ofinwill
The residue class of the element
is denoted byinbe denoted by $X_{1}X_{2m}\ldots x$ and the image of.
$(\mathfrak{g}, G)$$U^{\epsilon}(\mathfrak{g})$$U^{0}(\mathfrak{g})$$U^{1}(\mathfrak{g})$$S(\mathfrak{g})$$U(\mathfrak{g})$$\epsilon\neq 0$$\mathfrak{g}$$U^{\epsilon}(\mathfrak{g})$$E_{ij}\vdash+\epsilon E_{ij}$$U(\mathfrak{g})$$X_{1}\otimes X_{2}\otimes\cdots\otimes X_{?n}(X_{j}\in \mathfrak{g})$$\sum_{k=0}^{m}\otimes^{k}\mathfrak{g}$$U^{\epsilon}(\mathfrak{g})^{(}?n)$$U^{\epsilon}(\mathfrak{g})$$U^{\epsilon}(\mathfrak{g})$104

For an ordered partition
integers put$\{n_{1}’, \ldots, n_{L}’\}$

of a positive integer

$n$

into

$L$

positive(2.3)

of strictly increasing positive
The ordered partition of is expressed by the set
withby the span of
andintegers ending at . Define Lie subalgebras.
, respectively, and put
and,We denoteandis a parabolic. Thenandsubalgebra containing the minimal parabolic subalgebra . We remark that$\Theta$$n$$n$$\iota\ominus(i)>\iota\ominus(j),$$\mathfrak{n}_{\ominus},\overline{\mathfrak{n}}_{\ominus}$$\iota_{\ominus}(i)=\iota_{\ominus}(j)$$\iota\ominus(i)<\iota_{\ominus}(j)$$\mathrm{m}_{\ominus}^{k}=\sum_{L\circ(i)}=\iota \mathrm{e}(j)=k\mathbb{C}E_{i}j,$$a– \sum_{j=1}^{n}\mathbb{C}E_{i}$$E_{ij}$$\mathrm{m}_{\ominus}$$\mathfrak{p}_{\ominus}=\mathrm{m}_{\ominus}+\mathfrak{n}_{\ominus}$$\mathfrak{n}=\sum_{1\leq j<i\leq n}\mathbb{C}E_{ij},\overline{\mathfrak{n}}=\sum_{1\leq i<j}\leq?\mathrm{t}\mathbb{C}Eij$$\mathrm{m}_{\ominus}=\mathrm{m}_{\ominus}^{1}\oplus\cdots\oplus \mathrm{m}_{\ominus}^{L}$$\mathfrak{p}=a+\mathfrak{n}$$\mathfrak{p}_{\ominus}$$\mathfrak{p}=$$\mathfrak{p}$$\{X\in \mathfrak{g};\langle X,\mathrm{Y}\rangle=0(\forall Y\in \mathfrak{n}\ominus)\}$Fix$\lambda=(\lambda_{1}, \ldots, \lambda_{L})\in \mathbb{C}$.

and define a closed subset of :$\mathfrak{p}$$A_{\ominus,\lambda}= \sum_{j=1}\lambda)Ej+\mathfrak{n}_{\ominus}n\iota \mathrm{e}(j$

$=\{$

(2.4)

;

$A_{ij}\in M(n_{i}, n ;j \mathbb{C}\prime\prime)\}$.

denotes the identity matrix of size $m$ and $M(k,l;\mathbb{C})$ denotes the space
Herewith components in C. The generic element ofof matrices of size’s canonical form and any Jordan’s canonical form
corresponds to a unique
is obtained by this correspondence with a suitable choice of and .
because any
is a closed algebraic variety of
The setcan be transformed into an element in under the Ad-action
element of
we have, forof the unitary group $U(n)$ . Then if$I_{m}$$k\cross p$$A_{\ominus,\lambda}$$\mathrm{J}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{a}\dot{\mathrm{n}}$$\Theta$$\lambda$$M(n, \mathbb{C})$$\bigcup_{g\in G}\mathrm{A}\mathrm{d}(g)A\ominus,\lambda$$M(n, \mathbb{C})$$\mathfrak{p}$$\epsilon=0$$f\in U^{0}(\mathfrak{g})=S(\mathfrak{g})$$(\forall g\in c)$$f(\cup \mathrm{A}\mathrm{d}(g)A\ominus,\lambda g\in G)=0\Leftrightarrow(\mathrm{A}\mathrm{d}(g)f)(A\ominus,\lambda)=0$$\Leftrightarrow \mathrm{A}\mathrm{d}(g)f\in J_{(\lambda}^{\epsilon})$$\Leftrightarrow f\in \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))$$(\forall g\in c)$

105

where
$J_{\ominus}^{\epsilon}( \lambda)=\sum X\in \mathfrak{p}_{\ominus}U^{\epsilon}(9)(X-\lambda\ominus(X))$(2.5)$M_{\ominus}^{\epsilon}(\lambda)=U^{\epsilon}(\mathfrak{g})/J_{\ominus}^{\epsilon}(\lambda)$

A..nn

,

,
,

$(M_{\ominus}^{\epsilon}(\lambda))=\{D\in U^{\epsilon}(\mathfrak{g});DM_{\ominus}\epsilon(\lambda)=0\}$$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{}^{\epsilon}(\lambda))=\{D\in U^{\epsilon}(\mathfrak{g});\mathrm{A}\mathrm{d}(g)D\in \mathrm{A}\mathrm{n}\mathrm{n}(M_{\ominus}^{\epsilon}(\lambda))(\forall g\in G)\}$and the character(2.6)$\lambda_{\ominus}$of$\mathfrak{p}_{\ominus}$

is defined by$\lambda_{\ominus}(Y+\sum_{k=1}^{L}x_{k})=\sum_{1k=}^{L}\lambda_{k}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(x_{k})$Whencharacter$\epsilon=1,$$\lambda_{\ominus}$

of

for

$X_{k}\in \mathrm{m}_{\ominus}^{k}$and$Y\in \mathfrak{n}_{}$.

is a generalized Verma module induced ffom the, which is a quotient of the Verma module$M_{\ominus}(\lambda)=M_{\ominus}1(\lambda)$$\mathrm{m}_{\ominus}$(2.7)$M(\lambda\ominus)=U(9)/J(\lambda_{\ominus})$

with

(2.8)

$J^{\epsilon}( \lambda_{\ominus})=\sum_{X\in \mathfrak{p}}U\epsilon(\mathfrak{g})(x-\lambda_{(X)})\mathrm{a}$In general we will omit the superfix$\epsilon$

if$\epsilon=1$.nd$J(\lambda\ominus)=J^{1}(\lambda\ominus)$.

.

PROPOSITION 2.2.
(2.9)$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))=\mathrm{A}\mathrm{n}\mathrm{n}(M_{\ominus}^{\epsilon}(\lambda))$(2.10)

Annc

if$(M_{\ominus}^{\epsilon}( \lambda))=\bigcap_{g\in G}\mathrm{A}\mathrm{d}(g)J_{\ominus}^{\epsilon}(\lambda)$$\epsilon\neq 0$,.

to prove the proposition.
$(XD-Dx)v=$
and. Then for
Let $D\in$ Ann.
Since XD-DX$X(Dv)-D(xv)=0$ and therefore
and thereforeinfor $g\in G$ .
.
. Since
Annc
Put$IP=PI\equiv 0$ mod,
because is a two-sided ideal of
For
.
Ann
which means

Proof. We may assume$\epsilon\neq 0$$(M_{\ominus}^{\epsilon}(\lambda))$$X\in \mathfrak{g}$$v\in M_{\ominus}^{\epsilon}(\lambda),$$XD-DX\in \mathrm{A}\mathrm{n}\mathrm{n}(M_{\ominus}^{\epsilon}(\lambda))$$\epsilon \mathrm{a}\mathrm{d}(X)D$$U^{\epsilon}(\mathfrak{g}),$$\mathrm{A}\mathrm{n}\mathrm{n}(M_{\ominus}\epsilon(\lambda))\subset J_{\ominus}^{\epsilon}(\lambda),$$J_{}^{\epsilon}(\lambda)$$P\in U^{\epsilon}(\mathfrak{g}),$$I\subset$$\mathrm{A}\mathrm{d}(g)D\in \mathrm{A}\mathrm{n}\mathrm{n}(M_{\ominus}^{\epsilon}(\lambda))$$\mathrm{a}\mathrm{d}(X)D\in \mathrm{A}\mathrm{n}\mathrm{n}(M_{\ominus}^{\epsilon}(\lambda))$$I= \bigcap_{g\in c^{\mathrm{A}\mathrm{d}}}(g)J_{()}^{\epsilon}\lambda$$I$$(M_{\ominus}^{\epsilon}(\lambda))\subset I$$U^{\epsilon}(\mathfrak{g})$$\square$$(M_{\ominus}^{\epsilon}(\lambda))$DEFINITION 2.3. Define the polynomials and an integer(2.11)$=$

106

by putting

if $P>0$ ,if $l\leq 0$$z(z-\epsilon)\cdots(z-(P-1)\epsilon)$(2.12)$z^{(\ell)}=\{$

1

and $\{e_{m}^{\epsilon}(X);1\leq m\leq n\}$ the characteristic polynomial, theand call, respectively.
minimal polynomial and the elementary divisors of$d_{n}^{\epsilon}(x),$$q^{\epsilon}(x)$$M_{\ominus}^{\epsilon}(\lambda)$

REMARK 2.4. i) The set$\epsilon)$$\{e_{n}^{\epsilon},(x)\}$recovers$\{d_{n}^{\epsilon},(x)\}$

because$e_{m}^{\epsilon}(x)\in \mathbb{C}[x]e_{\tau}^{\epsilon},-1(\mathrm{L}x^{-}$.

, the greatest common divisor of $?n$-minorsii) For the generic element of, the above definition
and therefore whenof the matrix $xI_{n}-A$ equals.incides with that in the linear algebra.
will be clear in [O3].
foriii) The meaning of the minimal$A$$J_{\ominus}^{0}(\lambda)$$\epsilon=0$$d_{m}^{0}(x)$$\mathrm{c}\mathrm{o}$$\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}’ 1$$\epsilon\neq 0$

Now we introduce quantized minors.

$i_{\mu},$

DEFINITION 2.5. For set of indices $I=\{i_{1}, \ldots, i_{m}\}$ and $J=\{j_{1}, \ldots,j_{?n}\}$ with, define a generalized Capelli operator (cf. [O2]).-.,$j_{\nu}\in\{1,$(2.13)

in$n\}$$\det^{\epsilon}(x;EIJ)=\det((x+(\mathrm{t}^{\text{ノ}}-m)\epsilon)\delta ij\mu\mu-E_{i_{\mu}j_{\nu}})11\leq\nu\leq\mu\leq?n\leq’ n$$U^{\epsilon}(\mathfrak{g})[x]$(2.14)

by the column determinant:$\det(A_{\mu\nu})_{1\leq}\mu\leq?n(\mathrm{n}\sigma)A11\leq\nu\leq?n=\sum_{\in\sigma \mathfrak{S}_{m}}\mathrm{S}\mathrm{g}\sigma(\mathrm{I}1A_{\sigma(2)2}\cdots A\sigma(?n)m\cdot$

PROPOSITION 2.6. The Capelli operators satisfy(2.15)$\det^{\epsilon}(x;EI)\sigma(\mathrm{I}^{\sigma’(J})=\mathrm{s}\mathrm{g}\mathrm{n}(\sigma)$(2.16)

sgn(a’)$\det^{\epsilon}(x;E_{IJ})$

for a,$\sigma’\in \mathfrak{S}_{?n}$,$\mathrm{a}\mathrm{d}(E_{ij})\det\epsilon(X;E_{IJ})=D1-D_{2}$where$a(I)=\{i\sigma(1), \ldots, i_{\sigma}(m)\}$,$\sigma’(J)=\{j\sigma’(1)’\ldots,j_{\sigma’}(m)\}$,$\det^{\epsilon}(x;E\{i_{1},\ldots,i_{\mu-}1,j,i_{\mu}+1,\ldots,im\}J)$

if there exists only one$0$$otherwise\rangle$$\det^{\epsilon}(x;E_{I\{\nu-1}i,+j_{m}\})j_{1},\ldots,j,j_{\nu}1,\ldots,$

if there exists only one$0$otherwise.

$D_{1}=\{$$D_{2}=\{$$i_{\mu}$$j_{\nu}$with $i_{\mu}=j$ ,with $j_{\nu}=i$ ,$(2.15)$ and (2.16) are proved by [O2, Lemma 2.2 andWhen, we have the propoProposition 2.4]. Combining this with the definition ofsition.

Proof.

$\epsilon=1,$$U^{\epsilon}(\mathfrak{g})$$\square$

DEFINITION 2.7. Under Definition 2.3 and Definition 2.5, put(2.17)$\det^{\epsilon}(x;E_{I}J)=hIJ(X)d^{\epsilon}(mx)+r_{IJ}^{d_{m}-}X1d_{m}-1+\cdots+r_{IJ}^{1}x+r_{IJ}^{0}$

107

with
in
define the two-sided ideal ofand$h_{IJ}[x]\in U^{\epsilon}(\mathfrak{g})[x]$$U^{\epsilon}(\mathfrak{g})[x]$(2.18)$U^{\epsilon}(\mathfrak{g})$$r_{IJ}^{j}\in U^{\epsilon}(\mathfrak{g})^{(}m-j)$

for $j=0,$$\ldots,$$d_{?n}-1$and:$I_{\ominus}^{\epsilon}( \lambda)=,\sum_{n=1I=}^{n}\sum_{\#\# J=m}\sum_{j=0}^{d-1}Um\epsilon(\mathfrak{g})r_{I}^{j}J$

Note that if $m \leq n-\max\{n_{1’ L}’\ldots, n’\}$ the summand equals because $d_{m}=0$ .
Moreover note thatwith $\neq I=n$ are in. In particular, if $\Theta=$$\{1,2, \ldots, n\}$ , thenandis generated by suitable elements in.
Now we can state the main result in this section and we callquantized$d_{m}^{0}(X;\Theta, 0)=$Tanisaki generators of. In the case when
and the generators are introduced by [Ta].
$0$$\{r_{IJ}^{j}\}$$U^{\epsilon}(\mathfrak{g})^{G}$$n$$I_{}^{\epsilon}(\lambda)$$\mathfrak{p}_{\ominus}=\mathfrak{p}$$U^{\epsilon}(\mathfrak{g})^{G}$$\dot{d}_{IJ}$$\epsilon=\lambda=0,$$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))$$x^{d_{m}}$THEOREM 2.8. Under the notation (2.5) and (2.18)$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))=I_{}^{\epsilon}(\lambda)$

If all the roots of$d_{n}^{\epsilon}(x)=0$

tesimal character of(2.19)$M_{}^{\epsilon}(\lambda)$.

are simple, which is equivalent to say that the infini-is regular (cf. Remark 2.14), then.

$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{}^{\epsilon}(\lambda))=\sum L$$k=1 \# I=\# J=n+1-x\sum_{?k}U\epsilon(_{9})D_{I}^{\epsilon_{J(+}}’\lambda_{k}n_{k-}1\epsilon)$

Here for $I=\{i_{1}, \ldots, i_{m}\}$ and $J=\{j_{1}, \ldots,j_{m}\}$ we put(2.20) $D_{IJ}^{\epsilon}(x)=(-1)^{m}\det\epsilon(X;EIJ)--\det(E_{i_{\mu}j_{\nu}}-(x+(\nu-m)\epsilon)\delta_{ij\mu\nu}))_{1\leq\leq}1\leq\nu\leq m\mu m$.

If all the roots ofby$d_{n-1}^{\epsilon}(X)=0$

are simple, (2.19) holds modulo the ideal generated.
holds if$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{}^{\epsilon}(\lambda))\cap U^{\epsilon}(\mathfrak{g})^{G}$

When
above holds if$\epsilon=0,$$(2.19)$$\lambda_{i}\neq\lambda_{j}$$\lambda_{i}\neq\lambda_{j}$for $1\leq i<j\leq L$ and the last statement

for $1\leq i<j\leq L$ satisfying

and $n_{j}’>1$ .

$n_{i}’>1$

REMARK 2.9. Let
be the set of the roots ofbe the multiplicity of the root
. Here $d_{?n}=m_{1}+\cdots+m_{k}$ and. Then$d_{m}^{\epsilon}(x)=0$$\{\lambda_{1}’, \ldots, \lambda_{k}’\}$$m_{k}$$\lambda_{k}’$and letif$\lambda_{\mu}’\neq\lambda_{\nu}’$$1\leq\mu<\iota \text{ノ}\leq k$(2.21)$d_{m}-1 \sum \mathbb{C}r_{IJ}^{j}=\sum\sum \mathbb{C}(\frac{d^{j-1}}{dx^{j-1}}Dkm_{i}I\epsilon_{J(}X))|_{x=\lambda_{i}}$,$i=1j=1$$j=0$

for $\neq I=\neq J=m$ .
The rest of this section will be devoted to the proof of this theorem. First we
will examine the image of our mino.rs under the Harish-Chandra homomOrp.h.
to $S(a)=U^{\epsilon}(a)$ by
Define the map of$\mathrm{i}_{\mathrm{S}\mathrm{m}}$$\omega$(2.22)$U^{\epsilon}(\mathfrak{g})$.

$D-\omega(D)\in U^{\epsilon}(\mathfrak{g})\mathfrak{n}+\overline{\mathfrak{n}}U^{\epsilon}(\overline{\mathfrak{n}}+\mathfrak{a})$108

Fix $I=\{i_{1}, \ldots,i_{m}\}$ and $J=\{j_{1}, \cdots,j_{m}\}$ with $1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n$ and$1\leq j_{1}<j_{2}<\cdots<j_{m}\leq n$ . Then [O2, Corollary 2.11] in the caseshows$\epsilon=1$$\omega(D_{Ij(X)}^{\epsilon})=$(2.23)

under the notation in Theorem 2.8. Introducing the algebra isomorphism$-:S(\mathfrak{a})arrow S(a)$(2.24)

with $E_{j}^{\infty}=E_{j}-(- \frac{n-1}{2}+(j-1))\epsilon$ for $j=1,$$\ldots,$$n$(cf. Remark 2.14), put(2.25)$\overline{\omega}(P)=\overline{\omega(P)}$.

Then defines the Harish-Chandra isomorphism of
onto the algebra $S(a)^{W}$of-invariants in $S(a)$ . Here we note that if $I=\{i_{1}<i_{2}<\cdots<i_{m}\}$ ,$U^{\epsilon}(\mathfrak{g})^{G}$$\overline{\omega}$$\mathfrak{S}_{n}$(2.26)$\overline{\omega}(D_{II}^{\epsilon}(x))=\prod_{\nu=1}^{m}(E_{i_{\nu}}-X+(\frac{n-1}{2}+\nu-i_{\nu})\epsilon)$

Since
ficients of.

(cf. Proposition 2.6), it is clear that the coefas a polynomial of generate the algebra.

$D_{\{1,\ldots,n\}\{n\}}^{\epsilon}(1,\ldots,x)\in U^{\epsilon}(\mathfrak{g})^{G}[x]$$x$$D_{\{1,\ldots,n\}\{n\}}^{\epsilon}(1,\ldots,x)$$U^{\epsilon}(\mathfrak{g})^{G}$be a triangular decomposition of a reductive
LEMMA 2.10. Let
Lie algebra over C. Here and are nilpotent subalgebras of and a is a Cartan
is a Borel subalgebra of . For an element $D$ of thesubalgebra of anduniversal enveloping algebraof , we define $\omega(D)\in S(a)$ so that$\mathfrak{g}=\overline{\mathfrak{n}}\oplus a\oplus \mathfrak{n}$$\overline{\mathfrak{n}}$$\mathfrak{n}$$\mathfrak{g}$$\mathfrak{g}$$\mathfrak{p}=\mathfrak{a}\oplus \mathfrak{n}$$\mathfrak{g}$$U(\mathfrak{g})$(2.27)$\mathfrak{g}$$\mathfrak{g}$$D-\omega(D)\in U(\mathfrak{g})\mathfrak{n}+\overline{\mathfrak{n}}U(\overline{\mathfrak{n}}+\mathfrak{a})$For a subspace

$V$

of

$U(\mathfrak{g})$

.

put

.

(2.28)$\langle\omega(V)\rangle_{s_{()}}\mathrm{Q}=\sum_{p\in\omega(V)}S(a)p$Then if(2.29)$\mathrm{a}\mathrm{d}(\mathfrak{g})V\subset V$, we have$\omega(PDQ)\in\langle\omega(V)\rangle_{s()}\alpha$

Proof. Let

for any$P,$$Q\in U(\mathfrak{g})$and any $D\in V$ .

and $\{H_{1}, \ldots, H_{M}\}$ be the basis ofand , respectively. Then {;with $\mathrm{N}=\{0,1,2, \ldots\}$ is abasis$\{X_{1}, \ldots,X_{N}\},$ $\{Y_{1}, \ldots, Y_{N}\}$$a$$\mathbb{N}^{N},$of..

$\beta\in \mathbb{N}^{M},$$U(\mathfrak{g})$$\mathfrak{n},\overline{\mathfrak{n}}$$1M1N$$Y^{\alpha}H^{\beta}X^{\gamma}=Y_{1}^{\alpha_{1}}\cdots Y_{N}\alpha_{N}H\beta_{1}\ldots H\beta MX\gamma 1\ldots X^{\gamma N}$$\gamma\in \mathbb{N}^{N}\}$$\alpha\in$$\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{a}\mathrm{r}\mathrm{e}-\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{k}\mathrm{h}\mathrm{o}\mathrm{f}\mathrm{f}_{-}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{t}’ \mathrm{s}$.

Let $D\in V$ . The assumption implies $PDQ\in U(\mathfrak{g})V$ and therefore we may, weassume $Q=1$ in (2.29). Since $XD=\mathrm{a}\mathrm{d}(X)D+DX\in V+U(\mathfrak{g})\mathfrak{n}$ for. On the other hand,
have
and therefore
if $\alpha=0$$X\in \mathfrak{n}$$X^{\gamma}D\in V+U(9)\mathfrak{n}$$a)+U(\mathfrak{g})\mathfrak{n})\subset\overline{\mathfrak{n}}U(\overline{\mathfrak{n}}+a)+U(\mathfrak{g})\mathfrak{n}$$Y^{\alpha}H^{\beta}D-Y\alpha H\beta\omega(D)\in \mathrm{Y}^{\alpha}H^{\beta}(\overline{\mathfrak{n}}U(\overline{\mathfrak{n}}+$$\omega(Y^{\alpha_{H^{\beta}D)}}=H^{\beta}\omega(D)$

109

and

otherwise. Hence

$0$

and

$\omega(Y^{\alpha}H^{\beta}x^{\gamma}D)\in\langle\omega(V)\rangle_{s_{()}}\mathrm{Q}$for$\omega(PD)\in\langle\omega(V)\rangle_{s(\mathfrak{a})}$.

$P\in U(\mathfrak{g})$$\square$

LEMMA 2.11. Under the notation in Lemma 2.10, fix $H_{}\in$ a so that thewithandconditionmeans $c_{Y}\geq 0$ . Suppose. Let
be the centralizer ofinand letandbesubspaces spanned by the elements in and , respectively, satisfying$c_{Y}Y$ with $c_{Y}\neq 0$ .
Thenbe a Levi decomposition of a parabolic
containing . Let
subalgebra
denote the ce nter of. For an element of
the dualof we define a characterof so thatand. Suppose there exist
inforso that$c_{Y}\in \mathbb{C}$$\mathrm{a}\mathrm{d}(H_{\ominus})Y=C_{Y}Y$$\mathrm{a}\mathrm{d}(H\ominus)\mathfrak{n}\neq\{0\}$$\mathrm{Y}\in \mathfrak{n}\backslash \{0\}$$H_{\ominus}$$m\ominus$$\mathrm{Y}$$\overline{\mathfrak{n}}_{}$$\mathfrak{n}_{\ominus}$$\mathfrak{g}$$\mathrm{a}\mathrm{d}(H)Y=$$\overline{\mathfrak{n}}$$\mathfrak{n}$$\mathfrak{p}_{\ominus}=\mathrm{m}_{\ominus}\oplus \mathfrak{n}_{\ominus}$$\mathfrak{p}_{\ominus}$$\lambda_{}$$\lambda_{\ominus}(H)=\lambda(H)$$D_{1}(\lambda),$for$X\in \mathfrak{g}$Proof.

$\ldots,$$U(\mathfrak{g})[\lambda]$$m$$\ldots,$

,

$m$

.

$D_{k}(\lambda)\in \mathrm{A}\mathrm{n}\mathrm{n}(M_{}(\lambda))$$\mathfrak{g}=\mathfrak{g}\mathfrak{l}(n, \mathbb{C})$$\ldots,$.

Retain the notation in the proof of Lemma 2.10. We may assume
is a basis offor a suitable $N’$ . We note that for$\{Y_{1}, \ldots, Y_{N}’\}$(2.32)$D_{m}(\lambda)$

for $k=1,$

and therefore$m$ under the same notation as in the case

$D_{k}( \lambda)\in\sum_{X\in \mathfrak{p}\ominus^{U(\mathfrak{g}}})[\lambda](x-\lambda_{\ominus}(x))$$\ldots,$

and $k=1,$$D_{k}( \lambda)\in\sum_{X\in \mathfrak{p}}U(9)[\lambda](X-\lambda_{\ominus}(X))+\overline{\mathfrak{n}}U(\mathfrak{g})[\lambda]$

for $k=1,$

Here$\lambda_{(\mathfrak{n}_{\ominus}}+[\mathrm{m}_{},\mathrm{m}_{\ominus}])=0$$\mathfrak{p}_{\ominus}$$H\in a_{\ominus}$$\mathrm{a}\mathrm{d}(X)D_{k}(\lambda)\in\sum_{j=1}^{\mathit{7}n}U(\mathfrak{g})[\lambda]D_{j}(\lambda)$(2.31)

Then$\mathrm{m}_{}$$a_{}$$\mathfrak{a}_{\ominus}^{*}$(2.30)$\lambda$$a_{}$$\mathfrak{p}$$D\in U(\mathfrak{g})[\lambda]$$\overline{\mathfrak{n}}_{}$

mod$c_{\alpha}(D\cdot, \lambda)Y\alpha$$D \equiv\sum_{\alpha\in \mathrm{N}^{N}},$$c_{\alpha}(D;\lambda)\in \mathbb{C}[\lambda]$$\sum_{X\in \mathfrak{p}\ominus}U(\mathfrak{g})[\lambda](X-\lambda_{\ominus}(X))$.

are uniquely determined by $D$ because of the decomposition$U(\mathfrak{g})=U(\overline{\mathfrak{n}}_{\ominus})\oplus U(\emptyset)\mathfrak{p}$.

Putand $I_{\lambda}-- \sum_{H\in a}s(a)[\lambda](H-\lambda(H))$ and sup$m$ and therefore
pose $D\in I$ . Then (2.31) implies
for $k=1,$
by Lemma 2.10 which implies $c_{0}(D;\lambda)=$for $P,$$\omega(D)(\lambda)=0$ .
is a proper -submodule ofHence
for any fixed$I= \sum_{k=1}^{m}U(9)D_{k}(\lambda)U(\mathfrak{g})$$\omega(D_{k}(\lambda))\in I_{\lambda}$$\omega(PD_{k}(\lambda)Q)\in I_{\lambda}$$IM_{\ominus}(\lambda)$

.
Since

$\ldots,$$Q\in U(\mathfrak{g})$$M_{\ominus}(\lambda)$$\mathfrak{g}$$\lambda\in a_{\ominus}^{*}$(if the infinitesimal
is an irreducible -module for a genericcharacter of the Verma module with the highest weight which equals to the weight, thenplus is different from that ofwithis irreducible),$IM\ominus(\lambda)=0$ for a generic. Hence $c_{\alpha}(D;\lambda)=0$ for
and
for
any .
The following remark is clear ffom the argument in the proof of Lemma 2.11.
$Y^{\alpha}$$\lambda$$M_{\ominus}(\lambda)$$\alpha\neq 0$$\mathfrak{g}$$\lambda$$M_{\ominus}(\lambda)$$M_{\ominus}(\lambda)$$\alpha\in \mathrm{N}^{N’}$$\lambda$$IM_{\ominus}(\lambda)=0$$\square$$\lambda$

REMARK 2.12. i) Let be a positive integer and let. Ifvalued in
hnction of.
for any, then$p$$(\lambda, \epsilon)\in \mathbb{C}^{\ell+1}$$(\lambda, \epsilon)$$r(\lambda, \epsilon)\in \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))$$U^{\epsilon}(\mathfrak{g})$$r(\lambda, \epsilon)$

be a polynomial
for generic$r(\lambda, \epsilon)\in \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))$$(\lambda, \epsilon)$

110. ii) Let

. Replacing,
be a suitable polynomial function ofto$p(\mu)$, respectively, in Lemma 2.11, we haveand byandthe same conclusion if
is irreducible for generic.
$\mathbb{C}^{\ell}$$p$$D_{k}(\mu),$$\lambda$$U(\mathfrak{g})[\lambda]$$D_{k}(\lambda)$$a_{\ominus}^{*}$$U(\mathfrak{g})[\mu]$$M_{\ominus}(p(\mu))$$\mu\in \mathbb{C}^{\ell}$

REMARK 2.13. Use the notation in Lemma 2.10. Let
Verma module$M( \lambda)=U(\mathfrak{g})/(U(\mathfrak{g})\mathfrak{n}+\sum_{H\in \mathrm{Q}}U(9)(H-\lambda(H)))$(2.33)and consider the. Then$\lambda\in a^{*}$$P_{\lambda}=\{D\in U(\mathfrak{g});\omega(D)(\lambda)=\omega(\mathrm{a}\mathrm{d}(x)D)(\lambda)=0(\forall X\in \mathfrak{g})\}$

is the annihilator Annof the unique irreducible quotient. Hereof
we identim $S(a)$ with the space of polynomial functions of . This may be also
considered
be a quantization of the conjugacy class of semisimple matrices.
$(L(\lambda))$$M(\lambda)$$L(\lambda)$$a^{*}$$\mathrm{t}.0$

Proof.

Lemma 2.10 proves thatis a two-sided ideal of. Since theassumption means that the projection of
into the highest weight space ofvanishes,
because of the irreducibility of. On the other hand,$DL(\lambda)=0,$
if
and therefore $\omega(D)(\lambda)=0$ .
, we have Ann
is a two-sided ideal ofSince Ann.
$U(\mathfrak{g})$$P_{\lambda}$$P_{\lambda}L(\lambda)$$L(\lambda)$$P_{\lambda^{r}}L(\lambda)=0$$L(\lambda)$$D \in U(\mathfrak{g})\mathfrak{n}+\sum_{H\in \mathfrak{a}}U(\mathfrak{g})(H-\lambda(H))$$(L(\lambda))$$(L(\lambda))\subset P_{\lambda}$$U(\mathfrak{g})$$\square$by
Trace
REMARK 2.14. Define
and $w.\lambda=w(\lambda+\rho)-\rho$
for the elementof the Weyl group $W$ of the pair
. Then the infinitesimalcharacter of the highest weight module
is parametrized by. We say thatthe infinitesimal character is regular if
for any $w\in W$ with $w\neq e$ .
, thenIf$\rho\in a^{*}$$p(X)= \frac{1}{2}$$\mathrm{a}\mathrm{d}(H)|_{\mathfrak{n}}$$w$$(\mathfrak{g}, \mathfrak{a})$$W.\lambda$$M(\lambda)$$w.\lambda\neq\lambda$$\mathfrak{g}=\mathfrak{g}\mathfrak{l}(n, \mathbb{C})$(2.34)$\rho=(-\frac{?\mathrm{z}-1}{2}+(1-1))e_{1}+\cdots+(-\frac{n-1}{2}+(n-1))e_{?x}$$W\simeq \mathfrak{S}_{n}$,and$w( \sum_{j=1}^{n}\mu je_{j})=\sum_{j=1}^{n}\mu_{j}e-1w(j)=\sum_{j=1}^{n}\mu_{w(j)}ej$

for$(\mu_{1}, \ldots,\mu_{n})\in \mathbb{C}^{n}$

and $w\in W$.

Inchanges into
and the infinitesimal character of
equalsthat of. Hence the infinitesimal character is regular if and only if all theroots ofare simple because the set of roots isby putting$U^{\epsilon}(\mathfrak{g}),$$p$$\rho^{\epsilon}=\epsilon\rho$$M_{}^{\epsilon}(\lambda)$$M^{\epsilon}(\lambda_{\ominus})$$d_{n}^{\epsilon}(x)=0$(2.35)$\{\overline{\lambda}_{\nu}+\frac{n-1}{2};\nu=1, \ldots, n\}$$\lambda_{}+\rho^{\epsilon}=\overline{\lambda}_{1}e_{1}+\cdots+\overline{\lambda}e_{n}n$.

LEMMA 2.15. Let $I=\{i_{1}, \ldots, i_{m}\}$ and $J=\{j_{1}, \ldots,j_{m-1}\}$ be sets of positive
numbers with $m>0,$ $i_{1}<i_{2}<\cdots<i_{m}$ and $j_{1}<j_{2}<\cdots<j_{m-1}$ . Then there
exists a positive number $\mu\leq m$ such that $\#\{j\in J;j<i_{\mu}\}=\mu-1$ and.
$i_{\mu}\not\in J$

Proof. Suppose $m>1$ since the lemma is clear when $m=1$ . If $j_{m-1}<i_{ln}$ , wecan put $\mu=m$ . If $j_{?n-1}\geq i_{m}$ , we can reduce to the case when $I=\{i_{1}, \ldots, i_{m-1}\}$and $J=\{j_{1}, \ldots,j_{?n-2}\}$ .
Retain the notation in Theorem 2.8. Fix with $1\leq k\leq L$ and put $m=$
$n+1-n_{k}’$ and $J=\{1,2, \ldots, n\}\backslash \{nk-1+1, nk-1+2, \ldots, nk\}$ . Note that $\neq J=m-1$ .
$\square$

$k$

111

For $I=\{i_{1}, \ldots , i_{m}\}$ with $1\leq i_{1}<=*\cdot<i_{m}\leq n$, choose an integer
as inLemma 2.15. Then $n_{k-1}<i_{\mu}\leq n_{k}$ and $\#\{1,2, \ldots, n_{k-1}\}=\mu-1$ , ffom which wehave $\mu=n_{k-1}+1$ and $\lambda(E_{i_{\mu}})-(\lambda_{k}+n_{k-1}\epsilon)+(\mu-1)\epsilon=0$ and therefore (2.23)
and Proposition 2.6 show$\mu$(2.36)

if $\# I=\neq J=n+1-n_{k}’$ .

$\omega(D_{IJ}^{\epsilon}(\lambda_{k}+n_{k-1}\epsilon))\in\sum_{H\in\alpha}S(a)(H-\lambda(H))$

Denoting(2.37)$J(m,x)= \neq I=\#\sum_{J=m}\mathbb{C}D_{IJ(X)}\epsilon$,

the basis of $J(n+1-n_{k}’, \lambda_{k}+n_{k-1}\epsilon)$ satisfies the assumption in Lemma 2.11 forand therefore$\epsilon=1$(2.38)$J(n+1-n_{k}’, \lambda_{k}+n_{k-1}\epsilon)\subset \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{}^{\epsilon}(\lambda))$for $k=1,$$\ldots,$$L$.

. But this holds for any because of Remark 2.12 i) with the isomorphism
for
betweenand.
Now the Laplace expansions ofwith respect to the first and the lastcolumn show (cf. [O2, Proposition 2.6 )$\epsilon=1$$\epsilon$$U(\mathfrak{g})$$U^{\epsilon}(\mathfrak{g})$$D_{IJ}^{\epsilon}(x)$$\mathrm{i}$(2.39)$])$$J(m+1, \lambda)+J(m+1, \lambda+\epsilon)\subset U^{\epsilon}(\mathfrak{g})J(m, \lambda)$

if $m<n$and therefore(2.40)$J(n+1-n_{k}’+j, \lambda k+(n_{k-1}+i)\epsilon)\in \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{}^{\epsilon}(\lambda))$When$\epsilon=0$, it is obvious by the Laplace expansion of$(_{\frac{d^{i}}{dx^{i}}D_{IJ}^{0}}.(X))|_{x=\lambda_{k}}=0$

Hence ifby denoting$c\in \mathbb{C}$satisfies(2.41)

for $0\leq i\leq j\leq n_{k}’-1$ .
that$D_{IJ(X)}^{0}$

for $\neq I=\neq J=n+1-n_{k}’+j$ with $0\leq i\leq j\leq n_{k}’-1$ .

$d_{?n}^{\epsilon}(c;\lambda)=0$, then$\det_{m}^{\epsilon}(c;E_{I}J)\in I_{}^{\epsilon}(\lambda)’$$I_{}^{\epsilon}( \lambda)’=\sum_{k=1}^{L}U^{\epsilon}(\mathfrak{g})J(n+1-n_{k}’, \lambda_{k}+n_{k-1}\epsilon)$

for $\# I=\neq J=m$.

We have proved(2.42)$I_{}^{\epsilon}(\lambda)^{l}\subset I_{\ominus}^{\epsilon}(\lambda)$

and$I_{\ominus}^{\epsilon}(\lambda)’\subset \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{}^{\epsilon}(\lambda))$if all the root ofare simple for $m=1,$
and(cf. Remark 2.9). Hence it follows ffom Remark 2.12 i) that$d_{m}^{\epsilon}(x;\lambda)=0$$I_{\ominus}^{\epsilon}(\lambda)’=I_{}^{\epsilon}(\lambda)$(2.43)
Note that the element
they are in the center$I_{\ominus}^{\epsilon}(\lambda)\subset \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{}^{\epsilon}(\lambda))$$r_{IJ}^{j}$$U^{\epsilon}(\mathfrak{g})$$n$.

for $\neq I=n$ in (2.17) are contained in
of
and
mod

$U^{\epsilon}(\mathfrak{g})^{G}$$\ldots,$$U^{\epsilon}(\mathfrak{g})^{G}\equiv \mathbb{C}$$J^{\epsilon}(\lambda_{\ominus})$$J^{\epsilon}(\lambda_{})$.

because112

to complete the proof ofThus we have only to show
using the result in the
Theorem 2.8. We can prove this for generic withnext section (cf. [O3]) or Theorem 2.21 but we reduce it to the claim$I_{\ominus}^{\epsilon}(\lambda)\supset \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))$$\epsilon\neq 0$$\lambda$(2.44).

$I_{\ominus}^{0}(0)=\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{0}(0))$

For $\epsilon=\lambda=0$ , this is conjectured by [Ta] and is proved by [We]. In this casewith degree $\neq I-j$ . Here weare of homogeneous polynomials of, whichwith respect tois homogeneous of degreenote thatis well-defined under any choice of Poincare-Birkhoff-Witt basis because of thehomogenized universal enveloping algebra.
with degree $m$ . Thenbe the space of homogeneous elements ofLet, we denote by $a_{m}(D)$ the correand forin (2.17) does not depend on. Note thatsponding element inand . Hence$\dot{d}_{IJ}\in S(\mathfrak{g})$$\mathfrak{g}^{*}$$\neq I$$\det^{\epsilon}(x;E_{IJ})$$(\mathfrak{g}, \epsilon, \lambda)$$S(\mathfrak{g})$$S(\mathfrak{g})_{m}$$U^{\epsilon}(9)^{(m})/U^{\epsilon}(\emptyset)^{(m})\simeq S(\mathfrak{g})_{m}$$D\in U^{\epsilon}(9)^{(m})$$S(\mathfrak{g})_{m}$$\sigma_{\# I-j(r_{I})}jJ$$\lambda$$\epsilon$(2.45)$I_{\ominus}^{0}(0)==n+1- \max\sum_{m\{n\ldots,n1’ L\}\#=}^{n}\sum_{I\prime l\# J=m}\sum_{j=0}^{d-1}S(\mathfrak{g})\sigma j(r^{j})mm-IJ$

Put
We will prove
mod$R^{\epsilon}(\lambda)^{(m)}=\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))\cap U^{\epsilon}(\mathfrak{g})^{()}m$$D\in I_{}^{\epsilon}(\lambda)$

and$D\in R^{\epsilon}(\lambda)^{(m)}\backslash R^{\epsilon}(\lambda)^{(n}’-1)$by the induction on $m$ . Since (2.10) implies, we have.

$\mathrm{A}\mathrm{d}(g)D\equiv 0$$U^{\epsilon}(\mathfrak{g})^{(1}m-)\mathfrak{p}\ominus+U^{\epsilon}(\mathfrak{g})^{(1}m-)$(2.46)$\sigma_{m}(D)(\mathrm{A}\mathrm{d}(g)\mathfrak{n})=0$$(\forall g\in G)$. Hence it follows ffom (2.44) and (2.45) that there exist
andsatisfying $\sigma_{m}(D)=\sum p_{IJ}^{j}\sigma_{\#-j}I(r_{IJ})j$ . Herehomogeneous elements. Let
appeared in (2.17) and $\deg(p_{IJ}^{j})+\# I-j=m$ if
are generators ofwith $\sigma_{m-\# I+}j(p^{j})IJ=p_{IJ}^{j}$ and put $D’= \sum P_{IJIJ}^{j}D^{g}$ . Thenby theand $D-D’\in R^{\epsilon}(\lambda)^{(m}-1)$ and therefore we have
hypothesis of the induction. Thus we have completed the proof of Theorem 2.8.
$a_{m}(D)\in I_{}^{0}(\mathrm{o})$$r_{IJ}^{j}$$p_{IJ}^{j}\in S(.\mathfrak{g})$$p_{IJ}^{j}.\neq 0$$I_{}^{\epsilon}(\lambda)$$P_{IJ}^{j}\in U^{\epsilon}(\mathfrak{g})^{(\#}m-I+j)$$D-D’\in I_{\ominus}^{\epsilon}(\lambda)$$D’\in I_{}^{\epsilon}(\lambda)$$\square$

REMARK 2.16. The procedure to deformis studied by [BK].

$\lambda$

to$0$

under the classical limit$\epsilon=0$

In the proof of Theorem 2.8 we have shown the following, which is proved by[BB] together with the fact that it is not valid for a generalized Verma module of ageneral semisimple Lie algebra induced form a character of a parabolic subalgebra.

COROLLARY 2.17. The graded ring$\mathrm{g}\mathrm{r}(\mathrm{A}\mathrm{n}\mathrm{n}G(M_{(\lambda}^{\epsilon})))=\bigoplus_{m=0}^{\infty}(\mathrm{A}\mathrm{n}\mathrm{n}c(M_{(\lambda}\epsilon))\cap$equals the defining ideal of the closure of
the nilpotent conjugacy class of the generic element $A_{,0}$ of the form (2.4). $In$.
particular it is a prime ideal and does not depend on$U^{\epsilon}(\mathfrak{g})^{(m)})/(\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))\mathrm{n}U^{\epsilon}(\mathfrak{g})^{()}m-1)$$(\lambda, \epsilon)$

113

COROLLARY 2.18. The following two conditions are equivalent.
(2.47)$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))\supset \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M^{\epsilon}\ominus’(\lambda’))$(2.48)$d_{m}^{\epsilon}(x;\Theta, \lambda)\in \mathbb{C}[x]d_{m}^{\epsilon}(X, \Theta’, \lambda’)$.

for $m=1,$$n$$\ldots,$.

Proof. It is obvious that the latter condition implies the former. Hence suppose

the first condition. Let $f_{m}(x)$ be the least common multiple ofand;-,, . Then if $\neq I=\neq J=m,$mod. Applying , to this relation as in the proof of Theorem 2.8, we have$\det^{0}(x;E_{I}J)\in S(\mathfrak{g})x\deg(fm)$modbecause of the homogenuity
with respect to. Let
be the generic element of the form (2.4) andletbe the maximal ideal ofcorresponding to. Considering under, we can conclude that all the $m$-minors of the matrix $(x-A_{\ominus},0)$ aremoduloin $\mathbb{C}[x]X\deg(fm)$ . On the other hand,is the greatest common devisors of$m$-minors of $(x-A_{\ominus},0)$ and therefore $\deg f_{m}(x)\leq d_{m}(\Theta)=\deg d_{m}^{\epsilon}(x;\Theta, \lambda)$ andwe have the latter condition.
$d_{m}^{\epsilon}(x;\Theta, \lambda)$$d_{?n}^{\epsilon}(x$$\det^{\epsilon}(x;E_{IJ})\in U^{\epsilon}(\mathfrak{g})f_{n},(x)$$\lambda’)$$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))$$\mathbb{C}[x]\otimes$$a_{n}$$\mathbb{C}[x]\otimes \mathrm{A}\mathrm{n}\mathrm{n}c(M_{\ominus}^{0}(0))$$(\mathfrak{g}, \epsilon, \lambda)$$A_{\ominus,0}$$S(\mathfrak{g})$$J_{\ominus}$$A_{\ominus,0}$$J_{\ominus}$$x^{d_{m}(\ominus)}$$\square$, Corollary 2.18 gives the closure relation in the conjuREMARK 2.19. If
gacy classes of the matrices.
$\epsilon--0$

REMARK 2.20. The following theorem is a part of a conjecture proposed by[O1] for the general symmetric pair. The case in this note corresponds to the pair$(cL(n, \mathbb{C}),$ $U(n))$ . In the case of the classical limit, the following theorem
is obtained by [DP] and [Ta].
$\epsilon--\lambda=0$

THEOREM 2.21. Let
be the Weyl group ofand let $W=W(\Theta)W_{\ominus}$ bethe decomposition ofso thatbe the set of the representatives ofwith the minimal length. Then the common zeros ofcoincides with the set
counting their multiplicities.
In particular, the space
is naturally a representation
space of $W$ which is isomorphic to$W_{\ominus}$$\mathrm{m}_{\ominus}$$W(\Theta)$$W=\mathfrak{S}_{n}$$W/W_{\ominus}$$\omega(\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda)))$$\{w.\lambda_{\ominus} ; w\in W(\Theta)\}$$S(\mathfrak{a})/\omega(\mathrm{A}\mathrm{n}\mathrm{n}c(M_{\ominus}^{0}(\lambda)))$$\mathrm{I}\mathrm{n}\mathrm{d}_{W}^{W}\mathrm{i}\mathrm{d}\ominus\cdot$

Proof. Under the notation (2.35)$\overline{\lambda}_{\nu}=\lambda_{\iota(\nu)}-\frac{?x-1}{2}+(\nu-1)$

for $\nu=1,$$\ldots,$

$n$

.

and
$\overline{\omega}(D_{II}^{\epsilon})(\lambda_{k}+\prime x_{k-}1\epsilon)=\prod^{n}(Ei_{\mu}\mu \mathit{7}=1-\lambda_{k}+(\frac{n-1}{2}-nk-1+\mu-i\mu)\epsilon)$

Fix.

$K/=\{n_{k-1}+1, \ldots, n_{k}\}$ ,with $1\leq k\leq L$ and $w\in W(\Theta)$ . Put $m=n+1-n_{k},$$K^{c}=\{1, \ldots, n\}\backslash K$ and $J=w(K^{c})$ .
For $I=\{i_{1}, \ldots, i_{ln}\}$ with $1\leq i_{1}<\cdots<$$i_{?n}\leq n$ , choose
as in Lemma 2.15 and put $p=w^{-1}(i_{\mu})$ . Then $\ell\in K$ and$\{\nu\in K^{c}; w(\nu)<i_{\mu}\}=\mu-1$ , which implies $\#\{\nu\in K;w(\nu)<i_{\mu}\}=i_{\mu}-\mu$ .
On the other hand, since the condition $n_{k-1}<\nu<\nu’\leq n_{k}$ means $w(\nu)<w(\nu’)$ ,
$k$

$\mu$

114

we have $\{\nu\in K;w(\nu)<i_{\mu}\}=\{n_{k-1}+1,n_{k-1}+2, \ldots,\ell-1\}$ and therefore$l-n_{k-1^{-}}1=i_{\mu}-\mu$ and$\overline{\lambda}_{\ell}-\lambda_{k}+(\frac{n-1}{2}-n_{k-1}+\mu^{-i_{\mu}})\epsilon=(P-1-n_{k-1}+\mu-i_{\mu})\epsilon=0$.

we can conclude thatcomponent of, which is equivalent, to the condition
vanishes atvanishes. Hence if is generic,vanishes atthatbecause of the continuity. In
for $w\in W(\Theta)$ and therefore for any
atfor generic and therefore for any byparticular,
the same reason.
and [Ta,are generated by homogeneous polynomials ofSince, we have
Theorem 1] shows$\dim
S(a)/\omega(I_{}^{\epsilon}(\lambda))=\neq
W(\Theta)$ and the theorem. Thus we can conclude
is $W$-invariant.
follows ffom this. In fact, the last claim is clear because

Since

is the$\overline{\lambda}_{\ell}$$(\overline{\lambda}_{w(1)}, \ldots,\overline{\lambda}_{w(})n)$$i_{\mu^{-}}\mathrm{t}\mathrm{h}$$\overline{\omega}(D_{II})(\lambda_{k}+n_{k-1}\epsilon)$’$w(\lambda_{\ominus}+\rho^{\epsilon})$$\omega(D_{II})(\lambda_{k}+n_{k-1}\epsilon)$$w.\lambda_{}$$\omega(I_{}^{\epsilon}(\lambda))$$\lambda$$\lambda\in \mathbb{C}^{L}$$w.\lambda_{\ominus}$$\dim S(a)/\omega(I_{\ominus}^{\epsilon}(\lambda))\geq\neq W()$$\lambda$$\lambda$$(a, \lambda, \epsilon)$$\omega(I_{\ominus}^{\epsilon}(\lambda))$$\dim S(\mathfrak{a})/\omega(I_{\ominus}^{0}(0))=\neq W(\Theta)$$\dim S(a)/\omega(I_{\ominus}^{\epsilon}(\lambda))\leq$$\neq W(\Theta)$$\square$$I_{\ominus}^{0}(\lambda)$

3. Generalized Verma modules

In this section we study the necessary andcondit,ion on$\mathrm{s}\iota\iota \mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}$$\lambda\in \mathbb{C}^{L}$

so

that
(3.1)

$J_{\ominus}^{\epsilon}(\lambda)=\mathrm{A}\mathrm{n}\mathrm{n}c(M_{\ominus}^{\epsilon}(\lambda))+J^{\epsilon}(\lambda_{\ominus})$Note that it is clear by the definition that
(3.2)

$J_{}^{\epsilon}(\lambda)\supset \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{}^{\epsilon}(\lambda))+J^{\epsilon}(\lambda_{})$$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))=\mathrm{A}\mathrm{n}\mathrm{n}_{G}(U\epsilon(\mathfrak{g})/(\mathrm{A}\mathrm{n}\mathrm{n}c(M_{\ominus}^{\epsilon}(\lambda))+J^{\epsilon}(\lambda_{\ominus})))$

In general it is proved by [BG] and [Jo] that for
(3.3)

.

{I;

$I$

is the two sided ideal of$\ni I\vdash+I+J(\mu)\in$

is injective if
(3.4)

$\mu$

$\langle\mu+\rho, \alpha\rangle\neq 0$$\mu$$U(\mathfrak{g})$}with $J\supset J(\mu)$ }and$D\equiv\gamma(\mu;D)$mod
we note thatfor any rootlet
. For a basis$D\in U^{\epsilon}(\mathfrak{g})$$J^{\epsilon}(\mu)$$\alpha$for the pair$\epsilon\neq 0,$$\gamma(\mu;D)$$\alpha$for the pair.

$(3.1)$

is valid if$(\mathfrak{n}, \mathfrak{a})$$\lambda_{\ominus}+\rho^{\epsilon}$is regular andfor $1\leq i<j\leq n$ .

denote the unique element inwithof an ad(g)-invariant subspaceof$U^{\epsilon}(\overline{\mathfrak{n}})$$V$$\{R_{j}\}$$U^{\epsilon}(\mathfrak{g})$(3.7)$(\mathfrak{n}, a)$is regular, that is,$\overline{\lambda}_{j}-\overline{\lambda}_{i}\not\in\{0, -\epsilon, -2\epsilon, \ldots\}$$\mu\in \mathfrak{a}^{*}$

the map$I\supset \mathrm{A}\mathrm{n}\mathrm{n}(M(\mu))$for any positive rootand dominant. Hence in our case withdominant:

For.

is dominant:Moreover the map is surjective if

(3.6)

with

{ $J;J$ is the left ideal of$2 \frac{\langle\mu+p,\alpha\rangle}{\langle\alpha,\alpha\rangle}\not\in\{-1, -2, \ldots\}$(3.5)$U(\mathfrak{g})$$\mu\in a^{*}$

and$\gamma(\mu;\sum P_{j}R_{j})\in\sum U^{\epsilon}(\overline{\mathfrak{n}})\gamma(\mu;R_{j})$for$P_{j}\in U^{\epsilon}(\mathfrak{g})$

.

115

Let $R$ -denote the set of weights of$R_{-}=${$\sum_{i=1}^{n}$miei;

with respect to a. Then

$U^{\epsilon}(\overline{\mathfrak{n}})$and$\sum m_{i}=0$$m_{i}\in \mathbb{Z},$$m_{1}\geq m_{2}\geq\cdots\geq m_{n}$}$\backslash \{0\}$.

are weight vectors and. Since$\gamma(\mu;R_{j})=0$
has the weight which equals that ofif the weight ofis not in. Moreover since
has a maximal weight $e_{i}-e_{i+1}$ in -for any integer
with $1\leq i<n$ ,Suppose$\gamma(\mu;R_{j})$$U^{\epsilon}(\mathfrak{g})V+J^{\epsilon}(\mu)\neq U^{\epsilon}(\mathfrak{g})$$R_{j}\in U^{\epsilon}(\mathfrak{g})$$R_{j},$$R_{-}$(3.8)$R_{j}$$R$$E_{ii+1}$$E_{ii+1}\in U^{\epsilon}(\mathfrak{g})V+J^{\epsilon}(\overline{\lambda})\Leftrightarrow \mathbb{C}E_{ii+1}=$$i$$\sum$

the weight of$\mathbb{C}\gamma(\mu;R_{j})$.

$R_{j}=\mathrm{e}_{i}-e_{i+1}$

The key to studying the condtion for (3.1) is the following argument used in[O2, proof of Theorem 5.1].
Fix positive integers
and satisfying $1\leq k\leq L$ and.
$J=\{j_{m},\ldots,j_{1}\}$$I=\{i_{n},,\ldots,i_{1}\}$
Letandbe a set of positive numbers such that$k,$$n_{k-1}<\overline{i}<\overline{j}\leq n_{k}$$\overline{i}$$\overline{j}$$1\leq i_{1}<i_{2}<\cdots<i_{m}\leq n$(3.9)$i_{\nu}=j_{\nu}$if$\nu\neq\ell$,,$i_{\ell}=\overline{i}<j\ell=\overline{j}<i_{\ell+}1$

with a suitable $1\leq P\leq m$ . Define non-negative integers(3.10)$\{$$m’$$=n-m$ ,$a_{j}’$$=n_{j}’-\#\{\nu;nj-1<i_{\nu}\leq n_{j}\}$$a_{j}$,$=n_{j}-\#\{\nu;i_{\nu}\leq n_{j}\}=a_{1}’+\cdot\cdot \mathrm{z}+a_{j}’,$$b$$=\#\{\nu;n_{k-}1<i_{\nu}<\overline{i}\}$$b’$$=\neq\{\nu;\overline{j}<i\nu\leq n_{k}\}$$a_{0}=0$

,

,

.

Then

(3.11)$1\leq a_{L}=m’\leq n-2,1\leq a_{k}’=n_{k}’-b-b’-1$ ,$0\leq a_{j}’\leq n_{j}’-\delta_{kj},$$0\leq b\leq\overline{i}-n_{k-1}+1,0\leq b’\leq n_{k}-\overline{j}$and we have$\det^{\epsilon}(x;E_{IJ})\equiv’\prod_{\nu=\ell+1}^{n}(x-E_{i\mathcal{U}}-(\nu-1)\epsilon)\cdot E_{\overline{i}\overline{j}}$(3.12).

$\prod_{\nu=1}^{\ell-1}(X-Ei_{\nu}-(\nu-1)\epsilon)$$\equiv\frac{\prod_{j=1}^{L}p_{I}^{j}J(_{X)}}{s_{IJ}(x)}E_{\overline{i}\overline{j}}$mod$J^{\epsilon}(\lambda_{\ominus})$mod$U^{\epsilon}(\mathfrak{g})\mathfrak{n}$116

by putting$p_{IJ()=}^{j}X(x-\lambda_{j}-(n_{j}-1^{-}aj-1)\epsilon)(n_{j}^{l}-a_{j})l$(3.13),$\{$$s_{IJ}(x)=X-\lambda k-(n_{k}-1-a_{k-1}+b)\epsilon$.

Hence it follows ffom ffom (2.17) thatmodmod$\mathbb{C}E_{\overline{i}\overline{j}}$(3.14)$\sum_{i=0}^{d_{m}}\mathbb{C}r^{i}-1IJ\equiv\{$$0$$J^{\epsilon}(\lambda)$$J^{\epsilon}(\lambda)$

ifotherwise.

$\prod_{j=}^{L}1p_{I}^{j}J(x)\not\in \mathbb{C}[x]S_{I}J(x)d^{\epsilon}?n(X)$,Since $(n_{j}’-a’j-a_{j}-1)-(n_{j}’-m^{J})=m’-a_{j}\geq m’-a_{L}\geq 0$, we can define polynomials$\overline{p}_{IJ(X)}^{j}=\frac{p_{IJ(_{X)}}^{j}}{(x-\lambda-jnj-1\epsilon)(n^{l}j-?n)\prime}$

Then the conditionof with$\prod_{\mathrm{j}=1}^{L}p_{IJ}^{j}(X)\in \mathbb{C}[x]s_{I}J(x)d_{m}^{\epsilon}(x)$.

is equivalent to the existence$j$(3.15)$\overline{p}_{IJ}^{j}(X)\in \mathbb{C}[x]s_{I}J(X)$.

, the condition (3.15) is equivalent to the condition that
Ifsatisfying$\epsilon\neq 0$(3.16)$0\leq\nu\leq n_{j}’-a_{j}’-1$

and

$\nu$

is an integer( $\nu<a_{j-1}$ or $\nu\geq a_{j-1}+n_{j}’-m’$ )by denoting(3.17)

If$\lambda_{k}+(n_{k-1^{-a}k1}-+b)\epsilon=\lambda_{\mathrm{j}}+(n_{j-1\mathrm{j}-1}-a+\nu)\epsilon$$\epsilon=0$, it is equivalent to(3.18)$1,\overline{j},$.

$\lambda_{j}=\lambda_{k}$and $a_{j}’<m’$ .

Put $I=\{n,n-1, \ldots,n_{k}+1,\overline{i},n_{k}-1,nk-1-1, \ldots, 1\}$ and $J=\{n,$ $n-1,$$n_{k-1},nk-1-1,$. Then$\ldots,$$m’=n_{k}’-1,$$\mathit{7}x_{k}.+$$\ldots,$$1\}$$b=b’=0,$ $a_{k}’=n_{k^{-}}’1,$ $a_{j}’=0$ and $n_{j}’-a_{j}’-1=n_{j}’-1$ if $j\neq k$ .

Suppose (3.15) holds. Then $j\neq k$ because$a_{j-1}-1=-1<0$ and$\{$$a_{\mathrm{j}-1}-1=n_{k}’-2$

and$\overline{p}_{IJ}^{k}(x)=1$. Since$a_{j-1}+n_{j}’-m’=n_{j}’-n_{k}’+1$$a_{j-1}+n_{j}’-m’=n_{j}’>n_{j}’-a_{j}’-1$

the condition (3.16) is equivalent to$\max\{0, n_{jk}’-n’+1\}\leq\nu’\leq n_{j}’-1$$\{$$1-n_{k}’ \leq\nu’\leq\min\{n_{jk}’-n’, -1\}$

if $j<k$ ,if $k<j$with$\nu’=(U-aj-1)-(b-ak-1)=\{$$\nu$$\nu-n_{k}’+1$

if $j<k$ ,if $k<j$ .

if $j<k$ ,if $j>k$ ,117Hence (3.15) is equivalent to the condition (cf. Remark 2.14)(3.19)
and $(\mu\in\Lambda_{j,\mu’}\in\Lambda_{k}\backslash \Lambda_{j}\Rightarrow(\mu’-\mu)(k-j)>0)$$\Lambda_{k}\cap\Lambda_{j}\neq\emptyset,$with$\Lambda_{k}\not\subset\Lambda_{j}$$\Lambda_{i}=\{\overline{\lambda}_{\nu};n_{i-1}<\nu\leq n_{i}\}=\{\lambda_{i}+((\nu-1)-\frac{?\mathrm{z}-1}{2})\epsilon;ni-1<\nu\leq n_{i}\}$and $n_{k}’>1$$\lambda_{j}=\lambda_{k}$if$\epsilon\neq 0$

if$\epsilon=0$,.

Thus we have the following theorem.
THEOREM 3.1. i) Fix$k$.

with $1\leq k\leq L.$ Recall$\mathrm{m}_{}^{k}=\sum_{n_{k1}}‘<i\leq n_{k}\mathbb{C}E\mathcal{R}_{k^{-}1}-<j\mathrm{c}j\leq?\mathrm{t}_{k}i$

Then

Ann

(3.20)$G(M_{\ominus}^{\epsilon}(\lambda))+J^{\epsilon}(\lambda_{\ominus})\supset \mathrm{m}_{\ominus}^{k}\cap\overline{\mathfrak{n}}$.
if and only if (3.19) does not hold for $j=1,$ii) The equality (3.1) is valid if and only if (3.19) does not hold for $j=1,$$L$$\ldots,$

and $k=1,$(3.21)$\ldots,$$L$, which is equivalent to the condition$\min\overline{\Lambda}_{i}>\min\overline{\Lambda}_{j}$$\{$$\lambda_{i}\neq\lambda_{j}$$L$$\ldots,$or$\max\overline{\Lambda}_{i}>\max\overline{\Lambda}_{j}$or$\Lambda_{i}\cap\Lambda_{j}=\emptyset$or$\Lambda_{i}=\Lambda_{j}$or $n_{i}’=n_{j}’=1$

if
if$\epsilon\neq 0$$\epsilon=0$,,

for $1\leq i<j\leq L$ .
Herecharacter of$\overline{\Lambda}_{i}=\{{\rm Re}\mu;\mu\in\Lambda_{i}\}$$M_{\ominus}^{\epsilon}(\lambda)$

etc. In particular (3.1) is valid if the infinitesimalis regular.

We have only to prove that (3.20) is not valid if (3.19) holds for aandwhich satisfies (3.19). Fix suchsuitable . Suppose there exists
and suppose (3.15)
continue the argument just before the theorem. Put$\nu=b$ in (3.17) and since $0\leq b\leq n_{k}’-a_{k^{-1}}’$
does not valid for $j=k$ . Then ifand (3.16) is not valid with.$j=k$ , we have

Proof.

$j=j_{\mathit{0}}$$j$$j_{\mathit{0}}$$\overline{j}=\overline{i}+1$$\epsilon\neq 0,$(3.22). $-m’$ and$0_{k-1},\leq b<a_{k-1}+n_{k}’$$m’>n_{k}’$if$\epsilon\neq 0$.

, we have $a_{k}’=m’$ because . $\leq a_{L}=m’$ .
. PutFirst consider the case when$b=0$
because of (3.22) and it follows. If. Thenandffom (3.19) that.
andOn the other hand, if$\epsilon=0$$a_{h}J$$j_{\mathit{0}}<k$$\overline{j}=\overline{i}+1$$\epsilon\neq 0,$$\ell=\lambda_{k}+n_{k}-1^{-}\lambda j_{\mathit{0}^{-n}}j_{\mathit{0}}-1,$$\overline{i}=n_{k}-1+1$$a_{k-1}.=a_{j_{\mathit{0}}}=0$$0\leq P<n_{j_{\mathit{0}}}’$$p+n_{k}’>?l_{j_{\mathit{0}}}’$in (3.17) we have $\nu=\ell$ and then $0\leq\nu,$ $n_{j}’-n_{k}’+1\leq\nu$
In this case putting. We have this relationin (3.16), which implies
andbecausealso in the case whenif the weight$m’-(m’-a_{h}.)’=a_{k}’>0$ . Thus we can conclude
mod$j=j_{\mathit{0}}$$\overline{p}_{IJ(X)}^{j_{0}}\in \mathbb{C}[x]_{S_{IJ}}(x)$$\nu\leq n_{\dot{7}}’.-1$$\epsilon=0$$\deg\overline{p}_{IJ}^{j\mathrm{o}}(X)=?x’.-joa_{j_{\mathit{0}}}-’(n_{j_{\mathit{0}}}’-?n’)=m’-a_{j_{\mathit{0}}}’\geq$$r_{IJ}^{J}\equiv 0$$J^{\epsilon}(\lambda_{\ominus})$

118

is
. Note that the weight of
isof}
because of (3.8).
Hence, the same argument as in the. If
Lastly consider the case when
. Put
works and therefore we may assume
case when. Then similarly we haveand$r_{IJ}^{j}$$\sum_{\nu}^{??l}=1e_{i\nu}-e_{i\nu}.\cdot$$r_{\{i_{1,\ldots,m}}^{j}i$$e_{\overline{i}}-e_{\overline{i}+1}$$\{j1,\ldots,jm\}$$E_{\overline{ii}+1}\not\in \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))+J^{\epsilon}(\lambda_{\ominus})$$\epsilon=0$$k<j_{\mathit{0}}$$\epsilon\neq 0$$j_{\mathit{0}}<k$$\lambda_{k}-n_{k-1},$ $\overline{i}=n_{k}-1$$\ell=\lambda_{j_{\mathit{0}}}+n_{j_{\mathit{0}}-1}-$$\overline{j}=n_{k}$$1\leq\ell<n_{k}’,$$n_{k}’\leq p+n_{j_{\mathit{0}}}’,$$b’=0,$$a_{k}’=n_{k^{-}}^{J}b-1$

and $a_{k}=a_{k}’+a_{k-1}>(?l’k-b-1)+(b-n_{k}’+m’)=m’-1$ by (3.22). Since$a_{k}\leq a_{L}=m’$ , we havein
. Puttingand(3.17), we have $\nu=-l-a_{k-1}+b+a_{j_{\mathit{0}}-1}=a_{k}’-\ell+b=n_{k^{-}}’\ell-1$ and therefore$0\leq\nu$ and $\nu\leq n_{j_{\mathit{0}}}’-1=n_{j_{0}}’-a’jo-1$ and $\nu<n_{k}’-1\leq m’=a_{j_{\mathit{0}}-1}$ in (3.16). Henceas in the previous
and thus$a_{k}=a_{j_{\mathit{0}}}=a_{j_{\mathit{0}}-1}=m’$$\overline{p}_{I}^{;_{0_{J(X)}}}\in \mathbb{C}[x]S\tau J(x)$$j=j_{\mathit{0}}$$a_{\mathrm{j}_{\mathit{0}}}’=0$$E_{\overline{ii}+1}\not\in \mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))+J^{\epsilon}(\lambda_{\ominus})$case.

$\square$

EXAMPLE 3.2. Suppose $n=3,$ $\Theta=\{2,3\}$ and$d_{1}^{\epsilon}(X)=1,$$d_{2}^{\epsilon}(_{X})=x-\lambda_{1},$$\lambda=(\lambda_{1}, \lambda_{2})$. Then,$d_{3}^{\epsilon}(X)=(X-\lambda_{1})(x-\lambda_{1}-\epsilon)(x-\lambda_{2}-2\epsilon)$$J^{\epsilon}( \lambda_{\ominus})=\mathrm{s}\sum_{\geq i>j\geq 1}U(\mathfrak{g})E_{ij}+U(9)(E1-\lambda 1)+U(\mathfrak{g})(E_{2}-\lambda_{1})+U(9)(E\mathrm{s}-\lambda 2)$$J_{}^{\epsilon}(\lambda)=J^{\epsilon}(\lambda)+U^{\epsilon}(9)E12$

,

.

Since
$D_{IJ}^{\epsilon}(X)=(E_{i}1j_{1}-(x-\epsilon)\delta_{i_{1}j_{1}})(Ei2j2-X\delta_{i}j2)2$$-(E_{i_{2}j_{1}}-(x-\epsilon)\delta_{i_{2}}j\iota)(E_{ij_{2}}1-x\delta_{i_{1}j_{2}})$

for $I=\{i_{1}>i_{2}\}$ and $J=\{j_{1}>j_{2}\}$ , we have

(3.23).

119

Here the above$\equiv \mathrm{i}\mathrm{s}$considered under modulo$J^{\epsilon}(\lambda_{\ominus})$. Note that

(3.24)$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M(\Theta^{\epsilon}(\lambda))=3\geq j_{1>}j3\geq i_{1}>i\geq 1\sum_{2,2}U^{\epsilon}(\mathfrak{g})D_{\{i\}}\epsilon_{i12\{j_{1}j_{2}\}}(\lambda 1)\geq 1$$+ \sum_{D\in U^{\epsilon}(9)^{G}}U\epsilon(\mathfrak{g})(D-\omega(D)(\lambda))$Hence if
Suppose$\lambda_{1}\neq\lambda_{2}+\epsilon$.

which is equivalent to (3.21), we have (3.1).
. Then since, we have$\lambda_{1}=\lambda_{2}+\epsilon$$\mathrm{a}\mathrm{d}(\mathfrak{p})(E32E_{12})\subset J^{\epsilon}(\lambda)$$J_{\ominus}^{\epsilon}(\lambda)=U^{\epsilon}(\overline{\mathfrak{n}})E_{12^{\oplus}}J\epsilon(\lambda_{\ominus})$(3.25)$\neq\supset_{\mathrm{A}\mathrm{n}\mathrm{n}_{G}}(M_{\ominus}(\lambda))+J^{\epsilon}(\lambda_{\ominus})=U^{\epsilon}(\overline{\mathfrak{n}})E_{23}E_{12}\oplus J^{\epsilon}(\lambda_{})_{\neq^{J(\lambda_{\ominus}}}\supset\epsilon)$If
and$\epsilon\neq 0$, the above inclusion relation gives a Jordan-H\"order sequence of(3.26)$M^{\epsilon}(\lambda_{\ominus})$$J_{\ominus}^{\epsilon}(\lambda)/(\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))+J^{\epsilon}(\lambda\ominus))\simeq M_{\ominus}^{\epsilon},(\lambda’)$with $\Theta’=\{1,3\}$ and$(\lambda_{1}-\epsilon, \lambda_{1}, \lambda_{1}),$.

$\lambda_{\ominus}’,$$\lambda’=(\lambda_{1}+\epsilon, \lambda_{1}-\epsilon)$$-\lambda_{}=\epsilon(e_{1}-e_{2}),$. Note that$\rho^{\epsilon}=(-\epsilon, 0, \epsilon),$$(1,2).\lambda_{\ominus}=\lambda_{\ominus}’$, and$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon}(\lambda))=$under the notation in Remark 2.14. Here Ann
unique two-sided proper ideal ofwhich is larger than$\mathrm{A}\mathrm{n}\mathrm{n}_{G}(M_{\ominus}^{\epsilon},(\lambda’))$$\lambda_{\ominus}+\rho^{\epsilon}=$$(M_{\ominus}(\lambda))$is the.

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