数理解析研究所講究録 1410 巻 2005 年 154-172

154

Capelli identities for symmetric pairs
京都大学大学院理学研究科西山享 (Kyo Nishiyama)
Department of Mathematics, Graduate School of Science, Kyoto University

北海道工業大学総合教育研究部和地輝仁 (Akihito Wachi)
Division of Comprehensive Education, Hokkaido Institute of Technology

Introduction

1

Consider a see-saw pair of real reductive Lie groups in the real symplectic youp,

$Sp_{2N}(\mathrm{R})$$M_{0}$$G_{0}$$\cup$$\cup$$\cross$$H_{0}$$K_{0}$,

is called aform dual pairs. The pair
andwhere both
. In addition,
are the commutants of each other inanddual pair, if
is asymmetrica symmetric pair of Hermitian type. Then there are
we assume thatis also
1 [HOw89]. Note thatthree types of such see-saw pairs as in Table 1[HOw89].
$G_{0}$$(G_{0}, H_{0})$$(K_{0}, \mathrm{A}\mathrm{f}_{0})$$(K_{0},M_{0})$$(G_{0}, H_{0})$$Sp_{2N}(\mathrm{R})$$\mathrm{e}\mathrm{a}\mathrm{A}$$H_{0}$$(G_{0}, K_{0})$$(M_{0}, H_{0})$$\mathfrak{F}1$:see-saw pairs with

$G_{0}$

Hermitian type$\frac{Sp_{2N}(\mathrm{R})G_{0}K_{0}M_{0}H_{0}}{\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{R}Sp_{2k(\mathrm{p}+q})(\mathrm{R})Sp_{2k}(\mathrm{R})U_{k}U(p,q)O(p,q)}$

Case
Case

$\mathrm{C}$$\mathrm{H}$$Sp2[p+q$ ) $(r+s)(\mathrm{R})$$U(p, q)$$U_{p}\mathrm{x}U_{q}$$U(r, s)\mathrm{x}U(r, s)$$Sp4k(p+q)(\mathrm{R})$$O^{*}(2k)$$U_{k}$$U(2p, 2q)$

$U$

(r, )
$s$

$Sp(p, q)$a symmetric pair in all the three cases.

and 9 its complexification, and so on. Denote
Let 90 be the Lie algebra of, whereisby the Weil representation (the oscillator representation) of. Then we have the following equation in
the complexified Lie algebra of:
the Weyl algebra on$G_{0}$$\epsilon \mathfrak{p}_{2N}$$\omega$$Sp_{2N}(\mathrm{R})$$V\simeq \mathrm{C}^{N}$$\omega(U(\mathfrak{g})^{K})=\omega(U(\mathrm{m})^{H})$,$\epsilon \mathfrak{p}_{2N}$155

respectively, andandwhere $K$ and $H$ denote the complexifications of.
denotes the set of $K$-invariants in the universal enveloping algebrabe the complexified Cartan decomposition. The subalgebraLet$K$of the -invariants in the symmetric algebra (p) is isomorphic to a polynomial. Let us take a if-linear
be a set of generators of
ring, and let Xi, , ,is $K$-invariant and hence. The image
mapping ::can be expressed in terms of$H_{0}$$K_{0}$$U(\mathfrak{g})$$U(\mathfrak{g})^{K}$$=\epsilon$$\mathfrak{g}$$S(\mathfrak{p})^{K}$$\oplus \mathfrak{p}$$S$$X_{2}$$\mathrm{S}(\mathfrak{p})$$\iota$$\ldots$$S(\mathfrak{p})^{K}$$X_{r}$$\omega(\iota(X_{d}))$$\iota(X_{d})$$arrow U(\mathfrak{g})$$\omega(U(\mathrm{m})^{H})$$(C_{d}\in U(\mathrm{m})^{H})$$\omega(\iota(X_{d}))=\omega(C_{d})$.

We call this formula a Capelli identity for a symmetric pair and$C_{d}$a Capellielement for a symmetric pair.
The Capelli identity depends on the choice of the $K$-linear mapping . We takebe the irreducible decomposition of a K-module.
as follows. Letandare commutative Lie algebras, since 90 is of Hermitian type.
Then both
We therefore have the isomorphism,$\iota$$\iota$$\mathfrak{p}$$=\mathfrak{p}^{-}\oplus \mathfrak{p}^{+}$$\mathfrak{p}^{+}$$\mathfrak{p}^{-}$$S(\mathfrak{p})\simeq S(\mathfrak{p}^{+})\otimes_{\mathrm{C}}S(\mathfrak{p}^{-})=U(\mathfrak{p}^{+})$and define
plication$\iota$$\mathit{7})_{1}$:$S(\mathfrak{p})arrow U(\mathfrak{g})$$\otimes u_{2}\vdasharrow$$U(\mathfrak{p}^{-})$,

by the composite of this isomorphism and the multi-

uiu2 on$\iota(u_{1}u_{2})=$(Sc$U(\mathfrak{g})$

,

uiu2

$(u_{1}\in \mathrm{S}(\mathfrak{p}^{+}), u_{2}\in S(\mathfrak{p}^{-}))$

.

(1.1)

, wherefor every homogeneous element
This satisfies
is the canonical map from the subspace FiUfa) of filter degree: FiUfa)of degree of theto the homogeneous subspaceof the filtered algebra
$K$
$(*)$
a pseudO-symmetrization. We call a -map satisfying
graded algebra$u\in ff\dot{i}(\mathfrak{p})$$(*):\mathrm{g}\mathrm{r}_{i}(\iota(u))=u$$\iota$$arrow S^{:}(\mathfrak{g})$$\mathrm{g}\mathrm{r}_{:}$

$i$

$i$

$S.\cdot(\mathfrak{g})$$U(\mathfrak{g})$$S(\mathfrak{g})$map.
is compact, that is, $M_{0}=U_{n}$ or
We give the Capell identities only whenin this article. We, however, strongly believe that we can obtain theis not compact by using the Fourier
Capelli identities for the cases wheretransform of the Weyl algebra on , due to the suggestion of Hiroyuki Ochiai and$M_{0}$$U_{n}\cross U_{n}$$M_{0}$$V$

Jiro Sekiguchi.
This work is motivated by the harmonic analysis on symmetric spaces [Hua02],[Lee04], for instance. We discuss an application to the harmonic analysis in aforthcoming paper.

2

Case

$\mathrm{R}$In this section, we give the Capell identity for the symmetric pair of Case
, and statein Table 1. We first fix the notation, describe the generators of$\mathrm{R}$$S(\mathfrak{p})^{K}$158

the main theorem for Case R. Before proving the theorem, we demonstrate thecomputation when taking the principal symbols, in order to see the outline of theproof. We prove two key lemmas and they complete the proof of the theorem. Oneof these lemmas is also used for Case and Case H. At the end of this section,
we prove that the Capelli elements are H-invariant.
$\mathrm{C}$Preliminary

2.1

Define a complex Lie algebra , its subalgebras andalgebras.
$t$$\mathfrak{p}^{\pm}$$\mathfrak{g}$$\mathfrak{g}$$\mathrm{t}$$=z\mathfrak{p}_{2k}=\{$

$=\{$

(

)

’

$\mathrm{y}_{H}$$-$$|H\in G,F\mathrm{g}\mathrm{E}_{\mathrm{S}\mathrm{y}\mathrm{m}(k;}^{\mathrm{k}}$$\mathrm{C}$

)

$\}$

,

$\mathfrak{p}^{+}=\{$$(\begin{array}{ll}0 G0 0\end{array})\in \mathfrak{g}\}$$\mathfrak{p}^{-}=\{$$(\begin{array}{ll}0 0F 0\end{array})$,$(\begin{array}{ll}H 00 -{}^{t}H\end{array})\in \mathfrak{g}\}\simeq \mathfrak{g}\mathfrak{l}_{k}$$H_{j}=E_{\dot{l}j}-E_{k+j,k+:}\in$.

t,$G_{ij}=E_{k+\prime j}.\cdot+E_{j,k*}$$\mathrm{E}$$\mathfrak{p}^{+}$,, and elements of these$\mathrm{E}$$\mathfrak{g}\}$,,$F_{\dot{|}j}=E_{k+:_{i}}+E_{k+j,:}\in \mathfrak{p}^{-}$,

denotes the matrix unit and Sym(fc; C) denotes the set of the complex
wherematrices. Define a complex Lie algebra and its subalgebra
symmetric$E_{\dot{|}j}$$k\mathrm{x}k$$[)$$\mathrm{m}$by$\mathrm{m}$$=\mathfrak{g}\mathfrak{l}_{n}$F7,$=0_{n}=\{X\in \mathfrak{g}\mathfrak{l}_{n} ; X+X=0_{n}\}$.

) and denote the linear coordinate functions on
Set $V=$ Mat(yz,corresponding differential operators by$k;\mathrm{C}$,$x_{\dot{\mathrm{r}}}$$\partial_{si}$$(1\leq s\leq n, 1\leq i\leq k)$$V$and the,respectively.
andbe the complex symplectic Lie algebra, in which bothLet
form dual pairs. We have the Weil representation of 6 on the spaceof polynomial functions on , and its explicit forms on 9 and are as follows:$(\mathfrak{g}, \mathfrak{h})$$\mathrm{B}$$=\epsilon \mathfrak{p}_{2kn}$$\mathrm{C}[V]$$\omega$$(\mathrm{t}, \mathrm{m})$$V$$\mathrm{m}$$n$$\omega(G_{\dot{\iota}j})=\sqrt{-1}1^{x}$

si$aj,$\omega(F_{\dot{\mathrm{s}}j})=\sqrt{-1}\sum_{s=1}^{n}\partial_{l}\dot{.}\partial_{sj}$

,

(21)

$\epsilon=1$$\omega(H_{j}.\cdot)=\sum_{s=1}^{n}x_{s;}CJ_{sj}+\frac{n}{2}\delta_{\dot{\tau}j}$,.

$v(E_{st})= \dot{.}\sum_{=1}^{k}x_{si}\partial_{ti}+\frac{k}{2}\delta_{\epsilon t}$

157

We now recall the structure of. Since go is of Hermitian
acts multiplicity-fieely both on the symmetric algebra:$S(\mathfrak{p})^{K}$$GL_{k}(\mathrm{C})$$\mathrm{t}\mathfrak{M}\mathrm{e}_{f}K\simeq$$S(\mathfrak{p}^{+})$and on$S(\mathfrak{p}^{-})$$S(\mathfrak{p}^{+})=\oplus_{\mu}W_{\mu}$,$S(\mathfrak{p}^{-})=\oplus_{\mu}W_{\mu}^{*}$,

where runs over the set of all the even partitions with length at most ,isparametrized by the partition , andthe simple -submodule ofis the,simple submodule of. Thus we have the expression ofdual to$k$$\mu$$S(\mathfrak{p}^{+})$$\epsilon$$S(\mathfrak{p}^{-})$$W_{\mu}$$W_{\mu}^{*}$$\mu$$S(\mathfrak{p})^{K}$$W_{\mu}$$S(\mathfrak{p})^{K}=(S(\mathfrak{p}^{+}) \ \mathrm{c}S(\mathfrak{p}^{-}))^{K}=\oplus_{\mu}(W_{\mu}\otimes_{\mathrm{C}}W_{\mu}^{*})^{K}$.

In fact,
is isomorphic to a polynomial ring with algebraically independent, the dth generator is the basfc vector of the onegenerators. For $d=1,2$ ,dimensional vector spacefor $\mu=(2,2, \ldots, 2, 0, \ldots,0)$ where, 2Oc
appears times and 0 appears $(k-d)$ times. The explicit form of the generators$k$$S(\mathfrak{p})^{K}$$k$$\ldots$$(W_{\mu}$$W_{\mu}^{*})^{K}$

$d$

are
$\mathrm{S}(\mathfrak{p})^{K}$$X_{d}= \sum_{I,J\in \mathrm{I}_{d}^{k}}\det \mathrm{G}_{IJ}\cdot\det \mathrm{F}_{JI}\in$$(d=1,2, \ldots, k=r)$ ,(2.2)

where is the index set defined by $\{I\subset\{1,2, \ldots, k\}|\# I =d\}$ , and $G7J$ denotes
thesubmatrix of thewith the rows and the columns chosen
matrixby I and , respectively. Note that the generators above belong to the symmetric, and thatalgebraappearing in the generators commute withandeach other in this context.
$\mathrm{I}_{d}^{k}$$d\mathrm{x}d$$k\mathrm{x}k$$(G_{j}.\cdot)$$J$$S(\mathfrak{p})$2.2$F_{\dot{1}’j’}$$G_{\dot{\iota}j}$Capelli identity for Case$\mathrm{R}$The pseud0-symmetrization map defined by (1.1) embeds the generators (2.2)
ofinto
without symmetrization. Hence the image of the generator. In theitself, except that the images are inunder looks the same asfollowing theorem, we use the column-determinant for the determinant of a matrix
with non-commutative entries, defined by$\iota$$S(\mathfrak{p})^{K}$$\iota$$X_{d}$$U(\mathfrak{g})$$X_{d}$$\det(Z_{ij})=\sum_{\sigma\in 6_{d}^{\vee}}Z_{\sigma(1)1}Z_{\sigma(2)2}\cdots Z_{\sigma(d)d}$$U(\mathfrak{g})$

.

158

Theorem 2.1. For $1 \leq d\leq\min(k, n)$ , we have the Capelli identities for thesymmetric pair of Case in Table 1:$\mathrm{R}$$\omega(\sum_{I,J\in \mathrm{I}_{d}^{k}}\det \mathrm{G}_{IJ}\cdot\det \mathrm{F}_{JI})$$= \omega((-1)^{d}\sum_{S,T\in \mathrm{I}_{d}^{n}}\det(E_{S()T(j)}+(d-j-1-k/2)\delta_{S(:),T(j)}):i$$\mathrm{x}\det(E_{S(\dot{\iota})T(j)}+(d-j-k/2)\delta_{S(:),T(j)}):,j)$,where $S(i)$ denotes an element of the index set with $5(1)$ $<5(2)$The expression on the right-hand side is the image under of a surn of products.
minors with entries inof twoon the left-hand side is the image under ofNote thatthe generator (2.2), and that it is an element of (g) in particular. There areas (2.2), however the equation above is trivial when $n<d\leq k$generators ofsince the right-hand side becomes an empty sum.
$S$$<\cdot\cdot$

$1$

$<$

$5(\mathrm{i})$$\omega$$d\cross d$$U(\mathrm{m})$$\sum_{I,J}\det \mathrm{G}_{IJ}\cdot$$\iota$$\det \mathrm{F}_{JI}$

$k$

$U$

$S(\mathfrak{p})^{K}$$\square$

As explained in Introduction, the right-hand side of the Capelli identity is Hinvariant in the Weyl algebra, however it is not automatic that its inverse image is$H$-invariant in. In fact, the inverse image is $H$-invariant and we prove thisinvariance at the end of this section.
Before proving Theorem 2.1, we demonstrate the computation when taking theprincipal symbols, in order to see the outline of the proof. This computation forms
a part of the proof of the theorem. We first recall a basic lemma.
$U(\mathrm{m})$Lemma 2.2 (Cauchy-Binet). Let be a commutative ring and $d\leq N$ . For$A\in$ Mat(d, $N;R$ ) and $B\in$ Mat(d, $d;R$ ), we have$R$

$A.,s$ $\det B_{S}$$\det$$AB= \sum_{\mathit{8}\in \mathrm{I}_{d}^{N}}\det$submatrix ofwhere A.ts is thecolumns are chosen by S.
$d\mathrm{x}d$Define$n\mathrm{x}k$matrices $X$ and

$A$

,.,

in which all the rows are chosen and the$\square$a by$X=(_{X_{s}:})_{1\leq s\leq n,1\leq:\leq k}$,$\partial$$=(\partial_{s}\dot{.})_{1\leq\epsilon\leq n,1\leq:\leq k}$.

In the following computation we take the principal symbols, and we write the, the lemma above. For ,
by the same letter
principal symbol of$\partial_{si}$$\partial_{s}\dot{.}$$I$$J\in \mathrm{I}_{d}^{k}$159

yields$\omega(\det(\mathrm{G}_{IJ}))$$= \det(\sqrt{-1}\sum_{s=1}^{n}x_{s,I(i)^{X}s,J(j))_{1\leq:\dot{o}\leq d}}$$=(\sqrt{-1})^{d}$ $\det({}^{t}(X.,I)X.,J)$$\det X_{SJ}$$=( \sqrt{-1})^{d}\sum_{s\in x_{d}^{n}}\det(X_{SI})$.

Similaxly we have$\det\theta_{\Gamma I}$$\omega(\det(\mathrm{F}_{JI}))=(\sqrt{-1})^{d}\sum_{T\in \mathcal{D}_{d}^{1}}\det(b_{J})$,and the equation of matrices$(\omega(E_{S(:)T(j)}))_{1\leq i\dot{p}\leq d}=(X^{t}\partial)_{ST}$$(S,T\in \mathrm{I}_{d}^{n})$,where $E_{S(:)T(j)}$ is an element in . Note here that the contribution of the character
appearing in (2.1) vanishes, since we are taking the principal symbols. Note alsothat elements in the expressions above commute with each other for the same
reason, and we have$\mathrm{m}$$\sum_{I,J\in \mathrm{I}_{\text{\’{e}}}^{k}}\omega(\det \mathrm{G}_{IJ}\cdot\det \mathrm{F}_{JI})=(-1)^{d}\sum_{I,J}\sum_{s,\tau\in \mathrm{I}_{d}^{n}}\det X_{SI})\det X_{SJ}\det b_{J})\det\theta_{\Gamma I}$$=$’$(-1)^{d} \sum_{I,S,T}\det{}^{t}(X_{SI})$$\det(X\partial)_{ST}\det b_{I}$$\det \mathrm{o}\mathrm{e}_{I}$$=(**-1)^{d} \sum_{I,S,T}\det(X^{t}\partial)_{ST}\det{}^{t}(X_{SI})$$=$’$(-1)^{d} \sum_{s,\tau}\det$(a) $s\tau\det$ ($X^{t}$

a) $ST$$X^{t}$$=(-1)^{d} \sum_{s,\tau}\omega(\det \mathrm{E}_{ST}\det \mathrm{E}_{ST})$

.

(2.3)

This is nothing but our desired formula of Theorem 2.1 except that there are no
diagonal shifts in the last expression above. The equalities with and $**$ above
do not hold when we do not take the principal symbols, and we prove the noncommutative analogues of these two equalties with diagonal shifts in the folowing
subsections.
Remark 2.3. First, the non-commutative analogue of the equality with is, infact, the formula which is used for proving the classical Capelli identity. So theformula is known, and there is essentially the same formula in [Ume, Theorem 2],
for instance.
Second, the non-commutative analogue of the equality with $**$ seems a natural.
formula when considering the prehomogeneous vector space
$*$

$*$

$(\mathfrak{g}\mathfrak{l}_{n}, V)$

160

2.3

First lemma for the theorem

We prove the non-commutative analogue of the equality with in (2.3). Remarkthat the goal of this subsection, Lemma 2.6, is not new as mentioned in Remark
2.3, however we give a complete proof using the exterior algebra. This method isvery effective to simplfy the computation involving determinants or permanents,and has been used mainly for constructing central elements of universal enveloping, and for obtaining Capelli
algebras of simple Lie algebras [IUOI], [Wac03],, [Wac04].
,
identities of various types [Ume],
In this subsection we fix , $T\in l_{d}$ .
$*$$[\mathrm{I}\mathrm{t}\mathrm{o}04\mathrm{b}]$$[\mathrm{I}\mathrm{t}\mathrm{o}03]$$[\mathrm{I}\mathrm{t}\mathrm{o}04\mathrm{a}]$$S$

be the standard basis, and form the exDefinition 2.4. Let , e2, ,in the tensor product. Define the elements , andterior algebra$e_{1}$$e_{d}\in \mathrm{C}^{d}$$\ldots$$\wedge \mathrm{C}^{d}$algebra$\wedge \mathrm{C}^{d}\otimes \mathrm{c}$

End

$Qg$

$(\mathrm{C}[V])$$\zeta_{j}$$\zeta_{j}(u)$by,$\eta_{l}=\sum_{\dot{|}=1}^{d}e:x_{S(:),l}$$(1\leq l\leq k)$$\zeta_{\mathrm{j}}=\sum_{\dot{\iota}=1}^{d}e:\omega(E_{S(:)T(j)}-(k/2)\delta_{S(:),T(j)})$$(1 \leq j\leq d, u\in \mathrm{C})$( $1\leq j\leq d,$ tg$\zeta_{j}(u)=\sum_{\dot{|}=1}^{d}e:\omega(E_{S(:)T(j)}+(u-k/2)\delta_{S(:),T(j)})$$\in \mathrm{C}$,).

Note that products of these elements produces determinants. For example,$\eta_{I(1)}\eta_{I(2)}\cdots\eta_{I(d)}=e_{1}e_{2}\cdots e_{d}\det X_{SI}$.

Lemma 2.5. We have the following relations:$(1\leq j\leq d)$ ,(1)$(1\leq j\leq d, 1\leq m\leq k)$ .
(2) $\zeta_{j}(u)\eta_{m}=-$ ?7 ( $j(u$ -1$\zeta_{j}=\sum_{l=1}^{k}\eta_{lb_{(j),l}}$

$)$

$m$

Proof (1)$\zeta_{j}=\sum_{=1}^{d}\dot{.}e_{\dot{l}}\sum z_{=1}x_{S(\dot{\cdot}),l}$b(j),l$= \sum_{l}\eta_{}\phi(j),l$. For (2), we compute as

foUows:$\zeta_{i}\eta_{m}=\sum_{l=1}^{k}\eta_{l}\phi_{(j),l}\sum_{\dot{|}=1}^{d}e.\cdot x_{S(\dot{\cdot}),m}$$= \sum_{l_{\dot{l}}},\eta_{l}e:(X_{S(:),mb_{(j),l}+\delta_{T(j),S(:)}\delta_{lm})}$We add$u \sum_{=1}^{d}.\cdot e:\delta_{S(:)}$

0(f)$\zeta_{i}(u)\eta_{m}=-(j_{m}\zeta_{\mathrm{J}}(u-1).$$\eta_{m}$$=- \eta_{m}\zeta_{j}+\sum_{1}$.

$\eta_{m}e_{i}\delta_{T(j),S(:)}$.

to both sides of the expression above, and we obtain$\square$1Gl

Lemma 2.6. We have the following equation:.

$\sum_{J\in \mathrm{I}_{d}^{k}}\det X_{SJ}\det\phi_{J}=\det(\omega(E_{S(i)T(j)}+(d-j-k/2)\delta_{S(:),T(j)}))_{1\leq i_{\dot{\beta}}\leq d}$

Proof. First we have$\zeta_{j}(u)=\sum_{i=1}^{d}e:\omega(E_{S(i)T(j)}+(u-k/2)\delta_{S(:),T(j)})$from thedefinition, and we therefore obtain$\zeta_{1}(u_{1})\zeta_{2}(u_{2})\cdots$,,$\zeta_{d}(u_{d})=e_{1}e_{2}\cdots$ $e_{d}\det(\omega(E_{S(:)T(j)}+(u_{j}-k/2)\delta s(i),T(j)))_{1\leq:\dot{s}\leq}$$)$

for $u:\in$ C.
Second, using Lemma 2.5 (1) and (2) repeatedly, we have$\zeta_{1}(d-1)\zeta_{2}(d-2)\cdots$ $\zeta_{d}(0)=\zeta_{1}(d-1)\cdots$ $\zeta_{\mathrm{t}-1}(1)\sum_{l=1}^{k}\eta lb(d),l$$=(-1)^{d-1} \sum_{l=1}^{k}\eta_{l}\cdot\zeta_{1}(d-2)\cdots\zeta_{d-1}(0)\cdot b_{(d),l}$.

.

$\cdot$$=$Since ’s are anti-commutative (i.e.
the expression above equals$\eta_{l_{\mathrm{j}}}$$\mathrm{p}$.

$((-1)^{d-1})^{d} \sum_{l_{1\prime}\ldots,l_{d}=1}^{k}\eta_{l_{1}}\cdots/l\mathrm{j}b_{(1),\iota_{1}}\cdots b_{(d),l_{d}}$$\eta_{l_{\mathrm{j}}}\eta_{l_{j}}$,$+$\eta llj’\eta llj $=0$ ),$l_{j}$’s are distinct. Hence$\sum\eta_{J(\sigma(1))}\cdots\eta_{J(\sigma(d))}\cdot b_{(1),J(\sigma(1))}\cdots b_{(d),J(\sigma(d))}$$J\in \mathrm{I}_{d}^{k\sigma\in\tilde{\Theta}_{d}}$$=$$\mathrm{E}$$\eta_{J(1)}\cdots\eta_{J(1)}\mathrm{s}\mathrm{g}\mathrm{n}(\sigma)\cdot b_{(1),J(\sigma(1))}\cdots b_{(d),J(\sigma(d))}$$= \sum_{J}^{J,\sigma}e_{1}\cdots e_{d}\det X_{SJ}\cdot\det\phi_{J}$.

Comparing these two formulas we have the lemma.

2.4$\square$Second lemma for the theorem

We prove the non-commutative analogue of the equality with $**$ in (2.3).
mark that the goal of this subsection, Lemma 2.9, is not new as mentioned in
Remark 2.3, however we give a complete proof using the exterior algebra again.
.
andIn this subsection we fix${\rm Re}$

$S,T\in \mathrm{I}_{d}^{n}$$I\in \mathrm{I}_{d}^{k}$

182

Definition 2.7. Define the elements$\wedge \mathrm{C}^{d}$$\eta_{i}’$,$\mu_{j}$and$\mu_{j}(u)$in the tensor product algebra

O End(C $[V]$ ) by$\mathrm{c}$,$\eta_{\dot{n}}’=\sum_{h=1}^{d}e_{h}x_{S(i),I(h)}$$(1\leq i\leq d)$$\mu_{j}=\sum_{\dot{|}=1}^{d}\eta’\dot{.}\omega(E_{S(:)T(j)}-(k/2)\delta_{S(:),T\mathrm{C})}.)$$(1\leq j\leq d, u\in \mathrm{C})$.

$(1\leq j\leq d, u\in \mathrm{C})$.

$\mu_{j}(u)=.\cdot\sum_{=1}^{d}\eta_{\dot{1}}’\omega(E_{S(:)T(j)}+(u-k/2)\delta_{S(:),T(j)})$Lemma 2.8. We have the following relations:(1)(2)$\mu_{j}\eta_{g}’=-\eta_{g}’\mu_{\mathrm{j}}$$(1\leq j,g\leq d)$,$\mu_{j}(u)=\sum_{=1}^{d}\dot{.}\omega(E_{S(i)T(\dot{g})}+(u-1-k/2)\delta_{S(:),T(j)})\eta_{\dot{1}}’$$(1\leq j\leq d, u\in \mathrm{C})$.

Proof. We have (1) by a direct computation:$d$$\mu_{j}\eta_{g}’=E$ $\eta’.\cdot\omega(E_{S(\cdot)T(j)}.-(k/2)\delta_{S(:),T(j)})\eta_{\mathit{9}}’$$.\cdot=1$$\eta’\dot{.}\sum_{l=1}^{k}x_{S(i),lb_{(\mathrm{j}),l}\sum_{h=1}^{d}e_{h}x_{S(g)I(h)}}$.

$= \sum_{1}$$= \sum_{i,l,h}\eta_{i}’e_{h}x_{S(:),l}(x_{S(g)I(h)}b_{(j),l}+\delta_{S(g)T(j)}\delta_{l,I(h)})$.

.

$=- \eta_{g}’\mu_{j}+\sum$ $\delta_{S}(g),T(\mathrm{j})\eta’.\cdot\eta_{*}’$$=-\eta_{g}’\mu_{j}$.

$\cdot$We have (2) also by a direct computation:$\mu_{j}(u)=.\cdot\sum_{=1}^{d}(’\dot{.}(t)(E_{S(i)T(j)}+(u-k/2)\delta_{S(:),T(j)})$$=$$\mathrm{E}$$\sum_{h=1}^{d}e_{h}x_{S(\dot{\cdot})I(h)}(\sum_{l=1}^{k}x_{S(i),l}b_{(j),l}+u\delta_{S(:),T(j))}$.

$= \dot{.},\sum_{h,l}e_{h^{X}S(:),l(b_{(j),l^{X}S(\cdot)I(h)}-\delta_{S(:),T(j)}\delta_{l,I(h)})+\sum_{i}\delta s(:),T(j)\eta_{}’}$$= \sum.\cdot\omega(E_{S(\dot{l})\tau\mu)}-(k/2)\delta_{S(:),T(j)})\eta_{}’-\sum_{\dot{|}}\delta s(*\cdot),T(j)\eta_{\dot{1}}’+u\sum_{}\delta_{S(\dot{1}),T(j)\eta_{}}’$$\omega(E_{S(\dot{1})T(f)}+(u-1-k/2)\delta_{S(\dot{\iota}),T(\mathrm{j})})\eta_{*}’.$$= \sum_{1}$.

.
$\square$183

Lemma 2.9. We have the following formula for$u_{1}$,$u_{2}$,...,$u_{d}\in$C:$\det$ $X_{SI}$ $\det$ $(\omega(E_{S(i)T(j)}+u_{\mathrm{j}}\delta_{S(i),T(j)}))_{1\leq i,j\leq d}$.

$=\det(\omega(E_{S(i)T(j)}+(u_{j}-1)\delta_{S(i),T(j)}))_{1\leq i\dot{\mathfrak{p}}\leq d}\det X_{SI}$Proof. We compute$\mu_{1}(u_{1})\cdots/\ _{d}(\mathrm{J}_{d})$is equal tohave ffom Lemma 2.8 (1),in two different ways. First the factor
by the definition, and hence we$\mu_{j}(u)$$\sum_{=1}^{d}.\cdot\eta’\dot{.}\omega(E_{S(:)T(j)}+(u-k/2)\delta_{S(:),T(j)})$$\mu_{1}(u_{1})\cdots\mu_{d}(u_{d})$$=\mu_{1}(u_{1})$.. .

$\mu_{d-1}(u_{d-1})\sum_{i=1}^{d}\eta_{i}’\omega(E_{S(:)T(d)}+ (\mathrm{J}d - \mathrm{C}/2)\delta_{S(\cdot)T(d)}.)$$=$$(-1)^{d-1}$$=$$((-1)^{d-1})^{d} \sum_{i_{1},\ldots,i_{d}=1}^{d}\eta_{\dot{1}1}’$$\sum_{\dot{|}}\eta’\dot{.}\cdot\mu_{1}(u_{1})\cdots\mu_{d-1}(u_{d-1})\cdot\omega(E_{S(:)T(d)}+(u_{d}-k/2)\delta_{S(:)T(d)})$.

.

$\cdot$...

$\eta_{d}’.\cdot\cdot\omega(E_{S(:_{1})T(1)}+(u_{1}-k/2)\delta_{S(:_{1})T(1)})\cdots$...

$\omega(E_{S(:_{d})T(d)}+(u_{d}-k/2)\delta_{S(_{d})T(d)}.\cdot)$.

Since ’s are anti-commutative, ’s are distinct in the above expression. Similarly
to the proof of Lemma 2.6, the expression above is thus equal to$i_{j}$$\eta_{\dot{\iota}_{j}}’$$e_{1}\cdots e_{d}\det X_{SI}\cdot\det(\omega(E_{S(:)T(j)}+(u_{j}-k/2)\delta s(:),T(j)))$Second we compute(2) th”’s to the right in the product$\mu_{1}(u_{1})\cdots$ $\mu_{d}(u_{d})$.

in another way. It follows from Lemma 2.8. So this time we move, and we have$\mu_{j}(u)=\sum_{i=1}^{d}\omega(E_{S(:)T(\mathrm{j})}+(u-1-k/2)\delta_{S(i),T(j)})\eta_{\dot{1}}’$$\eta’$$\mu_{1}(u_{1})\cdot\cdot$.

$\mu_{d}(u_{d})=e_{1}\cdot$$\cdot$.

$\mu_{1}(u_{1})$ $\ldots\mu_{d}(u_{d})$$e_{d}\det(\omega(E_{S(*)T(j)}.+(u_{j}-1-k/2)\delta_{S(\cdot),T(j)}.))\cdot$ $\det XSI$Comparing these two computations we have the lemma.
2,5ClProof of Theorem 2.1

To begin with the first equality of (2.3), using Lemma 2.6 and Lemma 2.9, we
can immediately prove Theorem 2.1. We ffist have,$\sum_{I,J\in \mathrm{I}_{d}^{k}}\omega(\det \mathrm{G}_{IJ}\cdot\det \mathrm{F}_{JI})=(-1)^{d}\sum_{I,J}\sum_{S,T\in \mathrm{I}_{d}^{2n}}\det X_{SI})\det X_{SJ}\det \mathrm{t}b_{J})\det\partial_{TI}$164

from the first equality of (2.3). It follows from Lemma 2.6 that this is equal to$(-1)^{d}E$ $\det X_{SI})\det(\omega(E_{S(i)T(j)}+(d-j-k/2)\delta_{S(:),T(j)}))_{1\leq:j\leq d}\det\theta_{TI}$.

$I,S,T$

By Lemma 2.9, it turns out that the expression above equals$(-1)^{d} \sum_{I,S,T}\det(\omega(E_{S(:)T(j)}+(d-j-1-k/2)\delta_{S(:),T(j)}))_{1\leq:\dot{o}\leq d}$.

$\det\phi X_{SI}$) $\det b_{I}$ .

By using Lemma 2.6 again, this is equal to$(-1)^{d} \sum_{s,\tau}\det(\omega(E_{S(:)T(j)}+(d-j-1-k/2)\delta_{S(\cdot),T0)}..))_{1\leq:}|j\leq d$$\mathrm{x}\det$$(\omega(E_{S(:)T(\mathrm{j})}+(d-j-k/2)\delta_{S(\dot{\cdot}),T(i)}))_{1\leq:\dot{o}\leq d}$.

We thus have proved Theorem 2.1.

2.6

Invariance of the Capelli elements

the Capelli element for the symmetric pair,
We call the following element
which is the element appearing on the right-hand side of the formula of Theorem
2.1:$C_{d}^{\mathrm{R}}$$C_{d}^{\mathrm{R}}=(-1)^{d} \sum_{S,T\in \mathrm{I}_{d}^{n}}\det(E_{S(\dot{*})T(j)}+ (d-j- 1 -k/2)\delta_{S(\cdot),T(j)}.):\dot{o}$$\mathrm{x}$det(Es(i)Tcy) $+(d-j-k/2)\delta_{S(\dot{\cdot}),T(j)}):,j$ ,under the Weilfor $d=1,2$ , , . Note that the image of the Capelli element
representation is zero when $k<d\leq n,$ since the left-hand side of the Capell
for $1\leq d\leq n.$ The
identity in Theorem 2.1 becomes an empty sum, while, but an $H$-invariant element.
is not a central element ofCapell element$\ldots$$C_{d}^{\mathrm{R}}$$n$$\omega$$C_{d}^{\mathrm{R}}\neq 0$$U(\mathrm{m})$$C_{d}^{\mathrm{R}}$Proposition 2.10. The Capelli element is $H$ -invariant, that is,.
where$C_{d}^{\mathrm{R}}\in U(\mathrm{m})^{H}$,$H=O_{n}(\mathrm{C})$Proof. We just give an outline of the proof. Consider the tensor product algebra$W=\wedge \mathrm{C}^{n}\otimes \mathrm{c}\wedge \mathrm{C}^{n}\otimes \mathrm{c}U(\mathrm{m})$the second factor by$e_{t}$

and$e_{t}’$. Denote the standard basis ofrespectively. Define$\eta_{t}(u)=\sum_{\epsilon=1}^{n}e_{s}(E_{st}+u\delta_{st})$,$\eta_{t}(u)$and$\eta_{t}’(u)=\sum_{s=1}^{n}e_{s}’$($\mathrm{C}^{n}$$\eta:(1\#)$$E_{t}.$$+$

in the first andin $W$ byill), ,$t)$

185

Note that we have for ,$u$$v\in \mathrm{C}$$\eta_{T}(u)=\sum_{S\in \mathrm{I}_{d}^{n}}e_{S}\det \mathrm{E}_{ST}(u)$,$(T\in \mathrm{I}_{d}^{n})$$\eta_{T}’(v)=\sum_{S\in \mathrm{I}_{d}^{n}}e_{S}’\det \mathrm{E}_{ST}(v)$,where $\eta_{T}(u)=$ $77(1)(u-1)\eta T(2)(u-2)\cdots m(d)(u-d)$ , $e_{S}=$ e5(1)$eS(d)$ and-entry is Es(:)T(j)+(u-j)\mbox{\boldmath $\delta$}s( .),T(j).
denotes thematrix whose$M$-module structure to $W$ .
Now we give an -module structure, that is, aFirst,
is a-module through the adjoint action. Second,, we give the module structure dual to the natural representation offor bothbe. The tensor product $W$ thus has a-module structure. Let, with ,. Then it is known that thethe submodule of $W$ spanned by
mapping$\mathrm{e}5(2)\cdots$$d\cross d$$\mathrm{E}_{ST}(u)$$(\mathrm{z}, \mathrm{y})$$\cdot$$GL_{n}(\mathrm{C})$$U(\mathrm{m})=U(\mathfrak{g}\mathfrak{l}_{n})$$GL_{n}(\mathrm{C})$$\mathrm{C}^{n}$$W_{d}$$GL_{n}(\mathrm{C})$$GL_{n}(\mathrm{C})$$T$$e_{T}e_{T}’$$\Delta$:$W_{d}$$arrow$$W$$e_{T}e_{T’}’$”$\eta_{T}(u)\eta_{T’}’(v)$$T’\in \mathrm{I}_{d}^{n}$$(T,T’\in T_{d})$-homomorphism foris aparticular. The-module-homomorphism in-module through the-invariant. We canrestricted action, and the element $\sum_{T}e_{T}e_{T}’\in W_{d}$ is, which is naturallybydefine the contraction mapping on-homomorphism.
, and it is anextended to-homomorphisms and .
We can prove the assertion by using the$GL_{n}(\mathrm{C})$$GL_{n}(\mathrm{C})$$u$, $v\in$ C, and it isis also a self-dual$\mathrm{C}^{n}$$O_{n}(\mathrm{C})$$O_{n}(\mathrm{C})$$O_{n}(\mathrm{C})$$\epsilon$$\epsilon(e_{T}e_{T’}’)=\delta_{T,T’}$$W_{d}$$O_{n}(\mathrm{C})$$W_{d}\otimes \mathrm{c}U(\mathrm{m})$$\Delta$$O_{n}(\mathrm{C})$$\epsilon$$\epsilon(\Delta(\sum_{T}e_{T}e_{T}’))=\epsilon(\sum_{T}r\pi(u)\eta_{T}’(v))$$=\epsilon$$( \sum_{\tau,s,s},$$e_{S}\det \mathrm{E}_{ST}(u)e_{S’}’\det \mathrm{E}_{S’T}(v))$$= \sum_{s,\tau}\det \mathrm{E}_{ST}(u)\det \mathrm{E}_{ST}(v)$-invariant, since
The last expression is also
thus have proved the assertion.
$O_{n}(\mathrm{C})$

3

Case

$\sum e_{T}e_{T}’$

.
is

. We

$O_{n}(\mathrm{C})- \mathrm{i}\mathrm{n}1’\mathfrak{N}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{t}$$\square$$\mathrm{C}$In this section, we give the Capeli identity for the symmetric pair of Case
, and thenin Table 1. We first fix the notation, describe the generators ofprove the main theorem for Case C.
$\mathrm{C}$$S(\mathfrak{p})^{K}$

166

3.1

Preliminary

Define complex Lie algebras ,$\mathrm{f}$$\mathfrak{g}$$\mathfrak{g}$$\mathrm{t}$

$=$

g

and

$\mathfrak{p}^{\pm}$and elements of these algebras by,$\mathfrak{l}_{p+q}=\{$$=\{$$\mathfrak{p}^{+}=\{$$(\begin{array}{ll}H^{(x)} GF H^{(y)}\end{array})$$|H^{(y)}\in \mathfrak{g}\mathfrak{l}_{q}H^{(x)}\in \mathfrak{g}1_{p},’ F\in \mathrm{M}\mathrm{a}\mathrm{t}(q,p\cdot., \mathrm{C})G\in \mathrm{M}\mathrm{a}\mathrm{t}(p,q,\mathrm{C})$$(\begin{array}{ll}H^{(x)} 00 H^{(y)}\end{array})\in \mathfrak{g}\}\simeq \mathfrak{g}\mathfrak{l}_{p}\oplus \mathfrak{g}\mathfrak{l}_{q}$$H_{\dot{\iota}j}^{(x)}=E_{\dot{|}j}\in t$$H_{\dot{|}j}^{(y)}=E_{\mathrm{p}+,p+j}\dot{|}\in$t$\mathrm{m}$,$G_{ij}=E_{,p+j}\in \mathfrak{p}^{+}$$(1\leq i,j\leq q)$,$F_{\dot{*}i}=E_{p+\dot{1}_{1}j}\in \mathfrak{p}^{-}$$=\mathfrak{g}\mathfrak{l}_{n}\oplus$$\mathrm{m}$$p;\mathrm{C}$$(1\leq i\leq p, 1\leq j\leq q)$,$(1\leq i\leq q, 1\leq j\leq p)$.

, its subalgebra , and elements of$\mathfrak{h}$g ,$\mathrm{m}$$\mathfrak{h}=\{(X, -{}^{t}X)\in \mathrm{m}\}\simeq \mathfrak{g}\mathfrak{l}_{n}$$\mathfrak{l}_{n}$Mat(n,$V$ by$\oplus$$q;\mathrm{C}$,.

) and denote the linear coordinate functions$(1\leq s\leq n, 1\leq i\leq p, 1\leq j\leq q)$$x_{s}:,y_{sj}$by$(1\leq s, t\leq n)$$E_{st}^{(y)}=(0, E_{st})\in \mathrm{m}$$E_{st}^{(x)}=(E_{st}, 0)\in \mathfrak{m},$)Set $V=$ Mat(yz,on each component of$(_{F}^{0}00)\in \mathfrak{g}\}$$(1\leq i,j\leq p)$

Define a complex Lie algebra,,$\mathfrak{p}^{-}=\{$$(\begin{array}{ll}0 G0 0\end{array})\in g\},$

$\}$

,

respectively.
form dual pairs. We have theandSet) in which both, and its explicit forms on 9 and are as follows:
Weil representation of on$\mathrm{B}$$=$$(t, \mathrm{m})$$(\mathfrak{g}, \mathfrak{h})$$\mathrm{s}\mathfrak{p}_{2(\mathrm{p}}+$$\mathrm{r}$$n$$\omega$$\mathrm{B}$$\mathrm{C}[V]$$\mathrm{m}$,$\omega(H_{j}^{(x)}\dot{.})=\sum_{s=1}^{n}X_{s}:\frac{\partial}{\partial x_{sj}}+\frac{n}{2}\delta_{ij}$,$\omega(H_{\dot{\iota}j}^{(y)})=-\sum_{s=1}^{n}y_{\epsilon j}\frac{\partial}{\partial y_{*}}.\cdot-\frac{n}{2}\delta_{j}\dot{.}$.ya ,$\omega(G_{\dot{\iota}j})=\sqrt{-1}\sum_{s=1}^{n}x_{\mathrm{f}1}$$\omega(F_{j:})=\sqrt{-1}\sum_{s=1}^{n}\frac{\partial}{\partial x_{l\dot{1}}}\frac{\partial}{\partial y_{sj}}$$j$,$\omega(E_{\epsilon t}^{(x)})=\sum_{\dot{|}=1}^{p}x_{\dot{n}}\frac{\partial}{\partial x_{t\cdot}}.+\frac{p}{2}\delta_{st}$We now recall the structure of,
position of$S(\mathfrak{p})^{K}$$\omega(E_{st}^{(y)})=\sum_{\mathrm{j}=1}^{q}y_{\epsilon j}\frac{\partial}{\partial y_{tj}}+\frac{q}{2}\delta_{\epsilon t}$,.

. Similarly to Case , we have the decom$\mathrm{R}$$S(\mathfrak{p})^{K}$$S(\mathfrak{p})^{K}=\oplus_{\mu}(W_{\mu}$Sc$W_{\mu}^{*})^{K}$

,

187

respectively,andare the simple submodules of
where
and
they are dual to each other, and runs over the set of all the partitions withis isomorphic to a polynomial ring withlength at most $\min(p, q)$ . In fact,$\min(p, q)$ algebraically independent generators, and their explicit forms are$W_{\mu}$$S(\mathfrak{p}^{+})$$W_{\mu}^{*}$$S(\mathfrak{p}^{-})$$\mu$$S(\mathfrak{p})^{K}$$X_{d}= \sum_{I\in \mathrm{I}_{d}^{p},J\in \mathrm{I}_{d}^{q}}\det \mathrm{G}_{IJ}\cdot\det \mathrm{F}_{JI}$$(d=1,2 \ldots, r; r=\min(p, q))$

.

(3.1)

, and that
Note that the generators above belong to the symmetric algebra
appearing in the generators commute with each other in this context.
and$S(\mathfrak{p})$$G_{\dot{\iota}j}$3.2$F_{i’j’}$Capelli identity for Case$\mathrm{C}$

Theorem 3.1. For $1 \leq d\leq\min(p,$q, n), we have the Capelli identities for thesymmetric pair of Case C in Table 1:rw$( \sum_{I\in \mathrm{I}_{d}^{p},J\in \mathrm{I}_{d}^{q}}\det \mathrm{G}_{IJ}\cdot\det \mathrm{F}_{JI})$$= \omega((-1)^{d}\sum_{S,T\in \mathrm{I}_{d}^{n}}\det(E_{S(!)T(j)}^{(x}.+(d-j-p/2)\delta_{S(i),T(j)}):,j$$\cross$

det $(E\mathrm{g}(:_{)T(j)}+(d-j-q/2)\delta_{s(\cdot),T(j)}.)\dot{.}|j)$ .

The expression on the right-hand side is the image under of a surn of products. There are $\min(p, q)$ generators ofminors with entries inof twoas (3.1), however the equation above is trivial when $n<d \leq\min(p, q)$ since theright-hand side becomes an empty sum.
$\omega$$d\mathrm{x}d$$S(\mathfrak{p})^{K}$$U(\mathrm{m})$Proof As in \S \S 2.2 of Case$\mathrm{R}$, we define the matrices$X=(x_{s}:)_{1\leq s\leq n,1\leq i\leq p}$,$\partial^{X}=(\frac{\partial}{\partial x_{s}}.\cdot)_{1\leq s\leq n,1\leq}$$\mathrm{Y}=(y_{sj})_{1\leq\iota\leq n,1\leq j\leq q}$,.

$\partial^{\mathrm{Y}}=(\frac{\partial}{\partial y_{\iota j}})_{1\leq s\leq n,1<\leq q}\lrcorner$

i9,

and we thus have,$\omega(\det(\mathrm{G}_{IJ}))=(\sqrt{-1})^{d}\sum_{S\in \mathrm{I}_{\text{\’{e}}}^{n}}\det \mathrm{t}X_{SI})\det \mathrm{Y}_{SJ}$$\det \mathrm{a}\mathrm{e}_{I}$$\omega(\det(\mathrm{F}_{JI}))=(\sqrt{-1})^{d}\sum_{T\in \mathrm{I}_{d}^{\mathfrak{n}}}\det{}^{t}(\partial(_{J})$

,

,

1 es

for

$I\in \mathrm{I}_{d}^{p}$,$J\in \mathrm{I}_{d}^{q}$$\sum$. Using these formulas, we can prove the theorem as follows:$\omega(\det \mathrm{G}_{IJ}\cdot\det \mathrm{F}_{JI})$$I\in\Pi_{d},J\in \mathrm{I}_{d}^{q}$$=(-1)^{d} \sum_{I,J}\sum_{S,T\in \mathrm{I}_{d}^{n}}\det{}^{t}(X_{SI})\det \mathrm{Y}_{SJ}\det{}^{t}(\partial_{TJ}^{\mathrm{Y}})\det\#_{I}$$=(-1)^{d} \sum_{I,J,S,T}\det X_{SI}\det\#_{I}\cdot\det \mathrm{Y}_{SJ}\det\partial_{TJ}^{\mathrm{Y}}$$(\det(E_{S(!:\dot{o}}^{(oe}.+(d-7-)T(l)p/2)\delta_{S(\dot{1}),T(j)})$$=(-1)^{d} \sum_{s,\tau\epsilon p_{d^{\iota}}})$$\mathrm{x}$$\det(E\mathrm{s}(:)T\mathrm{C}\mathrm{y})$$+(d-j-q/2)\delta s(:),T(j)):i)$The last equality above folows from Lemma 2.6 of Case$\mathrm{R}$by replacing

or .

$k$

.
with

$p$

$\square$

$q$

3.3

Invariance of the Capelli elements

The Capell element is$C_{d}^{\mathrm{C}}=(-1)^{d} \sum$$\det(E_{s(!:\dot{s}}^{(x}.)T(j)+(d-j-p/2)\delta_{S(\cdot),T\circ)}..)$$S,T\in \mathrm{I}_{d}^{n}$$\cross\det(E_{S(}^{(y}i_{)T(;)} +(d-j-q/2)\delta_{S(:),T(f)})\dot{.}i$,for $d=1,$ 2, . .., , which appears on the right-hand side of the formula of Theorem
for $1\leq d\leq n.$
is zero when $\min(p, q)<d\leq n,$ while
3.1. Note that
, but anInvarianceis not a central element of
The Capelli element
element. The following proposition is proved similarly to Proposition 2.10 of CaseR.
$n$$C_{d}^{\mathrm{C}}\neq 0$$i(C_{d}^{\mathrm{C}})$$C_{d}^{\mathrm{C}}$$U(\mathrm{m})$Proposition 3.2. The Capelli element is $H$ -invariant, that is,
:
.
where
$H\simeq$4$C_{d}^{\mathrm{C}}\in U(\mathrm{m})^{H}$,$\square$$GL_{n}(\mathrm{C})$Case$\mathrm{H}$In this section, we give the Capell identity for the symmetric pair of Case
, and thenin Table 1. We ffist fix the notation, describe the generator of
prove the main theorem for Case H. The proof needs a lemma due to IshikawaWakayama [IWOO].
$\mathrm{H}$$S(\mathfrak{p})^{K}$

169

4.1

Preliminary

Define complex Lie algebras ,$\mathrm{P}$$\mathfrak{g}$$9=0_{2k}=\{$$\mathrm{f}$$(\begin{array}{ll}H GF -{}^{t}H\end{array})$$|H\in \mathfrak{g}\mathfrak{l}_{k}$

and

,

$G$

$\mathfrak{p}^{\pm}$and elements of these algebras by, $F\in$ Alt$(k;\mathrm{C})\}$

,

,

$=\{$

$(\begin{array}{ll}H 00 -{}^{t}H\end{array})\in \mathfrak{g}\}\simeq \mathfrak{g}\mathfrak{l}_{k}$$\mathfrak{p}^{+}=\{$$(\begin{array}{ll}0 G0 0\end{array})\in \mathfrak{g}\}$$\mathfrak{p}^{-}=\{$$(\begin{array}{ll}0 0F 0\end{array})\in \mathfrak{g}\}$,,$H_{\mathrm{j}}.\cdot=E_{j}.\cdot-E_{k+j,k+i}\in t,$$G_{\dot{|}j}=E_{\dot{1},k+j}-E_{j,k+:}\in \mathfrak{p}^{+}$$F_{ij}=E_{k+\cdot i}.-E_{k+j,*}$.

,

where Alt(fc; C) denotes the set of the alternating
Lie algebra and its subalgebra [ bymatrices. Define a complex$k\cross k$$)$$\mathrm{m}$$\mathrm{m}$,$(1\leq i,j\leq k)$$\in \mathfrak{p}^{-}$$=\mathfrak{g}\mathfrak{l}_{2n}$,$\mathfrak{h}=\{$$(\begin{array}{ll}H GF -{}^{t}H\end{array})$

$|HG$

,

,( ; C)$\}\simeq z\mathfrak{p}_{2n}$$F\in \mathrm{S}_{\mathfrak{M}}\in \mathfrak{g}\mathfrak{l}_{n}$) and denote the linear coordinate functions on
Set $V=$ Mat(2n,corresponding differential operators by$k;\mathrm{C}$$x_{ai}$,$(1\leq s\leq 2n, 1\leq i\leq k)$$\partial_{si}$

.

$n$

$V$

and the,respectively.
in which bothform dual pairs. We have the WeilLetand
, and its explicit forms on 9 and are as follows:
representation of on$\mathrm{B}$$(t, \mathrm{m})$$(\mathfrak{g}, \mathfrak{h})$$=z\mathfrak{p}_{4kn}$$\omega$$\mathrm{C}[V]$$\mathrm{B}$$u(H_{ij})= \sum_{s=1}^{2n}x_{S\dot{1}}\partial_{sj}+$$\omega(F_{j1}.)=\sqrt{-1}\sum_{s=1}^{n}$($\mathrm{m}$,n6|.j,$\omega(G_{j}\dot{.})=\sqrt{-1}\sum_{s=1}^{n}(x_{\dot{n}}x_{\overline{s}j}-x_{\overline{l}\dot{1}}x_{sj})$.h$\partial_{\delta 1}$$\mathrm{j}-\ .\cdot\partial_{sj}$where $\overline{s}=s+n.$
We now recal the structure of,
position of),$S(\mathfrak{p})^{K}$$\omega(E_{st})$$= \sum_{\dot{\iota}=1}^{k}x_{s}:\partial_{t\dot{*}}+\frac{k}{2}\delta_{at}$. Similarly to Case$\mathrm{R}$,, we have the decom-$S(\mathfrak{p})^{K}$$5(\mathfrak{p})^{K}=\oplus_{\mu}(W_{\mu}\otimes_{\mathrm{C}}W_{\mu}^{*})^{K}$,

respectively,andare the simple submodules of
and
where
they are dual to each other, and runs over the set of all the partitions of theis isomorphic to awith length at most . In fact,form$W_{\mu}$$S(\mathfrak{p}^{+})$$W_{\mu}^{*}$$\mu$$(\mu_{1},\mu_{1}, \mu_{2},\mu_{2}, \ldots)$$k$$S(\mathfrak{p})^{K}$$S(\mathfrak{p}^{-})$

70

polynomial ring withforms are$\mathrm{P}\mathrm{f}$$\lfloor k/2\rfloor$$\mathrm{G}_{II}$$X_{d}= \sum_{I\in \mathrm{I}_{2d}^{k}}$algebraically independent generators, and their explicit. Pf$\mathrm{F}_{II}$$(d= 1, 2, \ldots, r; r=\lfloor k/2\rfloor)$

.

(4.1)

where Pf denotes the Pfaffian of a alternating matrix. Note that the generators
appearing inand, and thatabove belong to the symmetric algebra
the generators commute with each other in this context.
$S(\mathfrak{p})$4.2Capelli identity for Case’

$F_{j’}\dot{.}$$G_{\dot{\iota}j}$$\mathrm{H}$Theorem 4.1. For $1 \leq d\leq\min(\lfloor k/2\rfloor, n)$ , we have the Capelli identities for thein Table 1:
symmetric pair of Case$\mathrm{H}$$\omega(\sum_{I\in \mathrm{I}_{2d}^{k}}\mathrm{P}\mathrm{f}\mathrm{G}_{II}\cdot \mathrm{P}\mathrm{f}\mathrm{F}_{II})$$= \omega(\sum_{S_{\mathrm{O}},T_{0}\in \mathcal{D}_{d}}\det(E_{S(:)T(j)}+(2d-j-k/2)\delta_{S(),T(j)})_{1\leq:\dot{o}\leq 2d)}$.

by $S(i)=S_{0}(i)$ ,is defined using
similarly.
$S(d+i)=n+$ $S_{0}(i)(1\leq i\leq d)$ , and is defined ffom
The expression on the right-hand side is the image under of a sum of$2d$ minors with entrries inas (4.1),
generators of. There are
however the equation above is trivial when $n<d\leq\lfloor k/2\rfloor$ since the right-hand side

On the right-hand side above,$S_{0}\in \mathrm{I}_{d}^{n}$$S\in \mathfrak{B}_{d}^{n}$$T$$7_{0}$$2d\cross$$\omega$$U(\mathrm{m})$$S(\mathfrak{p})^{K}$$\lfloor k/2\rfloor$

becomes an empty sum.

For proving the theorem, we use the folowing lemma to compute Pfaffans:Lemma 4.2 (Ishikawa-Wakayama [IWOO]). Let be a commutative ring and$d\leq n.$ For, define $P={}^{t}AXB-$ $tBXA\in$,,Alt(2d; R). We then have
$R$

$A$

$B\in \mathrm{M}\mathrm{a}\mathrm{t}(n, 2d;R)$

Pf(P)$X\in \mathrm{S}\mathrm{y}\mathrm{m}(n;R)$$(\begin{array}{ll}0 X-X 0\end{array})$$(\begin{array}{l}AB\end{array})$$ss\det$$= \sum_{s\in \mathrm{B}_{d}^{n}}\mathrm{P}\mathrm{f}$

$s,$

.

In particular, when $X=I_{n}$ we have
Pf $({}^{t}AB-{}^{t}BA)=$$\det$,$(\begin{array}{l}AB\end{array})$$(-1)^{d(d-1)/2} \sum_{S_{0}\in \mathrm{I}_{d}^{\mathfrak{n}}}$$S,.$

where $S\in l$$n$

is defined using So by $S(i)=S_{0}(i)$ , 5$(\mathrm{c}\mathrm{i}+\mathrm{i})$$=n+S_{0}(i)(1\leq i\leq d)$

.

171

Proof. The first formula is due to Ishikawa-Wakayama [IWOO, Corollary 2.1]. For
implies thethe second formula we use two facts. The condition Pf$(_{-I_{n}0}^{0I_{n}})_{SS}\neq 0$

condition$\mathrm{S}(d+\mathrm{i})$$(-1)^{d(d-1)/2}$$=n+S(i)(1\leq i\leq d)$ , and we have the formula Pf$(\begin{array}{ll}0 I_{d}-I_{d} 0\end{array})=$. Thege facts give the second formula.

Proof of Theorem 4.1. Similarly to Case
$X=(x_{s}:)_{1\leq s\leq n,1\leq i\leq k}$$\partial$$=(\partial_{s}.\cdot)_{1\leq s\leq n,1\leq i\leq k}$$\mathrm{R}$Cl

and Case$\mathrm{C}$, define matrices by,$\overline{X}=(x_{\overline{s}\dot{\iota}})_{1\leq s\leq n,1\leq:\leq k}$,$=(\ _{\mathrm{S}1}\cdot)_{1\leq s\leq n,1\leq i\leq k}$a,,and we have the equations of matrices,$\omega(\mathrm{G}_{II})=\sqrt{-1}(\mathrm{t}X.,I)\overline{X}.J-(\overline{X}.,I)X.\mathit{4})$$\omega(\mathrm{F}_{II})=-\sqrt{-1}((\mathrm{C}\}.,I)\overline{\partial}.,I-\lambda\overline{\partial}).,I\partial.J)$

for$I\in \mathrm{I}_{2d}^{k}$,. Using these equations we can prove the theorem as follows:($\mathrm{P}\mathrm{f}\mathrm{G}_{II}$$\sum_{I\in \mathrm{I}_{2d}^{k}}\omega$. Pf$\mathrm{F}_{II}$

)

.

.,’

$\mathrm{P}\mathrm{f}(\mathrm{t}\partial.,I)\overline{\partial}.J-{}^{t}(\overline{\partial}$$= \sum_{I\in \mathrm{I}_{2d}^{k}}\mathrm{P}\mathrm{f}(qX.,I)\overline{X}.,I-\mathrm{t}\overline{X}.J)X.,I)$$)\mathrm{C}?.,’$$= \sum_{I}\sum_{S_{0}\in \mathrm{I}_{d}^{n}}(-1)^{d(d-1)/2}\det(_{\frac{X}{X}}\cdot.,J)_{S}I,.\sum_{T_{0}\in \mathrm{I}_{d}^{n}}(-1)^{d(d-1)\int 2}\det$)$(\begin{array}{ll}\partial ,I\overline{\partial} ,I\end{array})$

$T,$

.

(by Lemma 4.2)$= \sum_{I,S_{0},T_{0}}\det(\frac{X}{X})_{SI}\det(\frac{\partial}{\partial})_{TI}$$= \sum_{S_{\mathrm{O}},T_{0}\in \mathrm{I}_{d}^{n}}\omega$().

$\det(E_{S(\dot{\iota})T(j)}+(2d-j-k/2)\delta_{S(i),T(j)})_{1\leq:,j\leq 2d}$(by Lemma 2.6)
口

4.3

Invariance of the Capelli elements

The Capelli element is$C_{d}^{\mathrm{H}}=$$\mathrm{E}$:$\det(E_{S(\cdot)T(j)}.+(2d-j-k/2)\delta_{S(:),T(j)})_{1\leq:_{\dot{\theta}}\leq 2d}$,

$C_{d}^{\mathrm{H}}= \sum_{s_{0},\tau_{\mathrm{O}}\in z_{d}^{n}}\det(E_{S(\cdot)T(j)}.+(2d-j-k/2)\delta_{S(:),T(j)})_{1\leq\leq 2d}$$s_{0},\tau_{\mathrm{O}}\in z_{d}^{n}$for $d=1,2$ , . . . , , which appears on the right-hand side of the formula of Theorem
for $1\leq d\leq n$ . As
while
is zero when
4.1. Note that$\ldots$$n$$\omega(C_{d}^{\mathrm{H}})$$\lfloor k/2\rfloor<d\leq n,$$C_{d}^{\mathrm{H}}\neq 0$

172$U(\mathrm{m})$,$C_{d}^{\mathrm{H}}\in U(\mathrm{m})^{H}$,

is not a central element ofin Case and Case , the Capelli element
but an $H$-invariant element. We omit the proof of the following proposition.
$\mathrm{R}$$\mathrm{C}$$C_{d}^{\mathrm{H}}$

Proposition 4.3. The Capelli element is $H$ -invariant, that is,where $H=Sp_{2n}(\mathrm{C})$ .

$\square$

参考文献[HOw89] Roger Howe, Transcending classical invariant theory, J. Amer. Math. Soc.
2 (1989), no. 3, 535-552.
[Hua02] Jing-Song Huang, Admissible square quadruplets and semisimple symrnetric spaces, Adv. Math. 165 (2002), no. 1, 101-123.

[It003]$[\mathrm{I}\mathrm{t}\mathrm{o}04\mathrm{a}]$$[\mathrm{I}\mathrm{t}\mathrm{o}04\mathrm{b}]$Minoru Itoh, Correspondences of the Gelfand invariants in reductive dual
pairs, J. Aust. Math. Soc. 75 (2003), no. 2, 263-277.
Minoru Itoh, Capelli identities for the dual pair(2004), no. 1-2, 125-154.

$(\mathrm{O}_{M}, \mathrm{S}\mathrm{p}_{N})$, Math. Z. 246

Minoru Itoh, Central elements of permanent type in the universal enveloping algebras of the symplectic Lie algebras, preprint (2004).

[IUOI]

M. Itoh and T. Umeda, On central elements in the universal enveloping
algebras of the orthogonal Lie algebras, Compositio Math. 127 (2001),
no. 3, 333-359.

[IWOO]

Masao Ishikawa and Masato Wakayama, Minor summation formulas ofPfaffians, survey and new identity, Combinatorial methods in representation theory (Kyoto, 1998), Adv. Stud. Pure Math., vol. 28, Kinokuniya,
Tokyo, 2000, pp. 133-142.
$a$[Lee04] Soo Teck Lee, Highest weight vectors in the space ofnornials, preprint (2004).
[Ume]$\mathrm{G}\mathrm{L}_{n}$harmonic poly-

Toru Umeda, On the proof of the Capelli identities, preprint.

[Wac03] Akihito Wachi, Central elements in the universal enveloping algebras forthe split realization of the orthogonal Lie algebras, preprint (2003).
[Wac04] Akihito Wachi, Capelli type identities on certain scalar generalized Vermamodules. $II$, J. Math. Soc. Japan 56 (2004), no. 2, 447-473.