可約概均質ベクトル空間の$b$-関数と一般Verma加群

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1 $b$- Verma (Akihito Wachi) Faculty of Education, Hokkaido University of Education Capelli Capelli [4] $(1\leq i,j\leq n)$ $\det(a)= A =\sum_{\sigma}$ sgn $(\sigma)a_{\sigma(1)1}\cdots A_{\sigma(n)n}$ Capelli Capelli $\det(tt)\det(\frac{\partial}{\partial T})=\det(tT\frac{\partial}{\partial T}+[Matrix])$ (1) $T$ $\partial/\partial T$ $n$ $T=(T_{ij})_{1\leq i,j\leq n}, \frac{\partial}{\partial T}=(\frac{\partial}{\partial T_{ij}})_{1\leq i,j\leq n}$ Capelli (1) $ZU(\mathfrak{g}\mathfrak{l}_{n})$ Capelli $ \frac{\partial}{\partial T}(u_{1}) ^{t}t \frac{\partial}{\partial T}(u_{2}) ^{t}t \cdots tt \frac{\partial}{\partial T}(u_{l}) = \frac{\partial}{\partialt}(u_{1})^{t}t\frac{\partial}{\partial T}(u_{2})^{t}T\cdots tt\frac{\partial}{\partial T}(u_{l}) $ (2)

2 36 [4] $u_{i}$ $(\partial/\partial T)(u)$ $n$ $\frac{\partial}{\partial T}(u);=\frac{\partial}{\partial T}+u^{t}T^{-1}=\frac{\partial}{\partial T}+u\frac{\partial g}{\partial T}$ $=( \frac{\partial}{\partial T_{ij}}+u\frac{\partial g}{\partial T_{ij}})_{1\leqi,j\leq n}=(f^{-u}\frac{\partial}{\partial T_{ij}}f^{u})_{1\leq i,j\leq n}$ $(f=\det(tt), g=\log f)$. 1 2 $\partial g/\partial T=tT^{-1}$ $\mathbb{c}[t_{ij}, \partial/\partial T_{ij}, f^{-1}]$ (2) 2 Capelli Capelli (1) Capelli (2) [4] $b$- $\det(t)$ $\det(x^{(1)}x^{(2)}\cdots X^{(l)})$ [4] $ZU(\mathfrak{g}\mathfrak{l}_{n})$ $($ $5$ $)$ Howe-Umeda [1] ( 2 ) $\det(x^{(1)}x^{(2)}\cdots X^{(l)})$ $b$- Capelli $ \frac{\partial}{\partial T} (f^{s+1})=(s+1)(s+2)\cdots(s+n)f^{s}$ $b(s)=(s+1)(s+2)\cdots(s+n)$ $(f^{s+1})$ $f^{8+1}$ $ \frac{\partial}{\partial T}(u) (f^{s+1})= f^{-u}\frac{\partial}{\partial T}f^{u} (f^{s+1})=(f^{-u} \frac{\partial}{\partial T} f^{u})(f^{s+1})$ $=f^{-u} \frac{\partial}{\partial T} (f^{s+u+1})=(s+u+1)(s+u+2)\cdots(s+u+n)f^{s}.$ $b(s)$ $s$ 2 Capelli 3 4 $k$ $b$-

3 $\frac{\overline{\partial}}{\overline{\partial}s}(u):=\frac{\overline{\partial}}{\overline{\partial}s}+u^{t}s^{-1}=\frac{\overline{\partial}}{\overline{\partial}s}+u\frac{\overline{\partial}g}{\overline{\partial}s}$ 37 $\backslash Sato$-Sugiyama [2] [2] 5 Capelli Capelli Capelli (1) $ZU(\mathfrak{g}\mathfrak{l}_{n})$ Capelli 2 CapeIli Capelli (2) $T$ $S_{ij}(1\leq i,j\leq n)$ $S_{ij}=S_{ji}$ Capelli $\det(ts)\det(\frac{\overline{\partial}}{\overline{\partial}s})=\det(ts\frac{\overline{\partial}}{\overline{\partial}s}+(\begin{array}{llll}(n-1)/2 O (n-2)/2 \ddots O 0\end{array}))$ (3) $\overline{\partial}/\overline{\partial}s$ (Tumbull [3]) $S$ $n$ $S=(S_{ij})_{1\leq i,j\leq n}, \frac{\overline{\partial}}{\overline{\partial}s}=(\frac{1+\delta_{ij}}{2}\frac{\partial}{\partial S_{ij}})_{1\leq i,j\leq n}$ Capelli Capelli (2) Capelli Capelli 1 ( ). $f=\det(ts)$ $\mathbb{c}[s_{ij}, \frac{\partial}{\partial S_{ij}}, f^{-1}]$ $ \frac{\overline{\partial}}{\overline{\partial}s}(u_{1}) ts \frac{\overline{\partial}}{\overline{\partial}s}(u_{2})_{1} ts \cdots ts \frac{\overline{\partial}}{\overline{\partial}s}(u_{l}) $ $= \frac{\overline{\partial}}{\overline{\partial}s}(u_{1})^{t}s\frac{\overline{\partial}}{\overline{\partial}s}(u_{2})^{t}s\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(u_{l}) $. (4) $(\overline{\partial}/\overline{\partial}s)(u)$ $n$ $=( \frac{\overline{\partial}}{\overline{\partial}s_{ij}}+u\frac{\overline{\partial}g}{\overline{\partial}s_{ij}})_{1\leq i,j\leq n}=(f^{-u}\frac{\overline{\partial}}{\overline{\partial}s_{ij}}$ $)$ $(f=\det(ts), g=\log f)$. $1\leq i,j\leq n$

4 $\frac{\overline{\partial}g}{\overline{\partial}s}=ts^{-1}.$ Proof. $\frac{\overline{\partial}}{\overline{\partial}s_{ij}}=\frac{1+\delta_{ij}}{2}\frac{\partial}{\partial S_{ij}}$ $\partial $\overline{\partial}/\overline{\partial}s_{ij}$ $\partial$/ S$ $S_{ii}$ 1 $S_{ij}(i\neq j)$ $f=\det(ts)$ 1 $S_{ii}$ $S_{ij}(i\neq j)$ $i$ 1 1/2 ) $S$ 1/2 ( $\det(ts)$ $S_{ii}$ $S$ $i$ $i$ $S_{ii}$ $i$ $i$ 1 $i\neq j$ $S$ $i$ ( $j$ ) ( $i$ $i$ ) 2 $0$ $\sum_{a=1}^{n}s_{ai}\frac{\overline{\partial}f}{\overline{\partial}s_{aj}}=\delta_{ij}f$ $f$ $\sum_{a=1}^{n}s_{ai}\frac{\overline{\partial}g}{\overline{\partial}s_{aj}}=\delta_{ij}$ $\overline{\partial}g/\overline{\partial}s$ $ts$ Capelli (4) 3. $\mathcal{w}=\mathbb{c}[s_{ij}, \frac{\partial}{\partial S_{ij}}, f^{-1}]$ $(\mathcal{w}$ $\mathcal{r}$ 3 ) $\underline{u}=(u_{1}, u_{2}, \ldots, u_{l})$ $v\in \mathbb{c}$ $A^{(l)}(\underline{u}),$ $n$ $B(v)\in Mat(n;\mathcal{W})$ $A^{(l)}( \underline{u})=\frac{\overline{\partial}}{\overline{\partial}s}(u_{1})^{t}s\frac{\overline{\partial}}{\overline{\partial}s}(u_{2})^{t}s\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(u_{l}), B(v)=tS\frac{\overline{\partial}}{\overline{\partial}S}(v)$

5 $S_{bs} \frac{\overline{\partial}}{\overline{\partial}s_{bt}}]$ $\mathbb{c}$ $\underline{u}$ 39 $A^{(l)}(\underline{u})$ Capelli (4) $B(v)=B+v1_{n}$ $1_{n}$ ( $n$ $0$ ) $B(0)=B$ $\mathbb{c}^{n}$ $\wedge(\mathbb{c}^{n})$ $\mathcal{w}$ $\mathcal{r}=\wedge(\mathbb{c}^{n})\otimes_{\mathbb{c}}\mathcal{w}$ $\eta j(\underline{u}),$ $\zeta_{j}(\underline{u}, v)\in \mathcal{r}$ $\eta_{j}(\underline{u})=\sum_{i=1}^{n}e_{i}a_{ij}^{(l)}(\underline{u})$, $\zeta_{k}(\underline{u}, v)=\sum_{i=1}^{n}e_{i}a_{ik}^{(l+1)}(\underline{u}, v)=\sum_{j=1}^{n}\eta_{j}(\underline{u})b_{jk}(v)$ $e_{1},$ $e_{2},$ $\ldots,$ $e_{n}$ $\mathbb{c}^{n}$ $(\underline{u}, v)$ $\underline{u}$ $v$ $l+1$ $\eta j(\underline{u}),$ $\zeta_{k}(\underline{u}, v)$ 7 ). $A^{(l)}(\underline{u})$ 4( Proof. $u,$ $v\in \mathbb{c}$ $\frac{\overline{\partial}}{\overline{\partial}s}(u)^{t}s\frac{\overline{\partial}}{\overline{\partial}s}(v)=(\frac{\overline{\partial}}{\overline{\partial}s}+u^{t}s^{-1})ts(\frac{\overline{\partial}}{\overline{\partial}s}+v^{t}s^{-1})$ $= \frac{\overline{\partial}}{\overline{\partial}s}ts\frac{\overline{\partial}}{\overline{\partial}s}+(u+v)\frac{\overline{\partial}}{\overline{\partial}s}+uv^{t}s^{-1}$ 5. $[x, y]=xy-yx$ $[B_{ij}, B_{st}]= \frac{1}{2}(\delta_{js}b_{it}-\delta_{ti}b_{sj})$. $A^{(l)}(\underline{u})$ $\underline{u}$ Proof. (LHS) $= \sum_{a,b=1}^{n}[s_{ai}\frac{\overline{\partial}}{\overline{\partial}s_{aj}},$ $= \sum_{a,b}s_{ai}\cdot\frac{\delta_{ab}\delta_{js}+\delta_{as}\delta_{jb}}{2}\cdot\frac{\overline{\partial}}{\overline{\partial}s_{bt}}+\sum_{a,b}s_{bs}\cdot\frac{-\delta_{ab}\delta_{it}-\delta_{at}\delta_{ib}}{2}\cdot\frac{\overline{\partial}}{\overline{\partial}s_{aj}}$ $= \frac{1}{2}(\sum_{a}s_{ai}\frac{\overline{\partial}}{\overline{\partial}s_{at}}\delta_{js}+s_{si}\frac{\overline{\partial}}{\overline{\partial}s_{jt}}-\sum_{a}s_{as}\frac{\overline{\partial}}{\overline{\partial}s_{aj}}\delta_{it}-s_{is}\frac{\overline{\partial}}{\overline{\partial}s_{tj}})$ $= \frac{1}{2}(\delta_{js}b_{it}-\delta_{it}b_{sj})$ $=$ (RHS).

6 (3) 40 6( Capelli ). (1) $[A_{ij}^{(l)}( \underline{u}), B_{8}t(v)]=\frac{1}{2}(\delta_{j_{s}}A_{it}^{(l)}(\underline{u})+\delta_{i\epsilon}A_{tj}^{(l)}(\underline{u}))$ (2) $A^{(l)}(\underline{u})$ (3) $A^{(l)}(\underline{u})$ $l$ Proof. (1), (2), (3) $((1),$ (2) $,$ ) $B$ $v=0$ $l=1$ (2) (3) (1) $t(v)=b_{st}+v1_{n}$ $A^{(1)}(u_{1})= \frac{\overline{\partial}}{\overline{\partial}s}(u_{1})=f^{-u_{1}}\frac{\overline{\partial}}{\overline{\partial}s}f^{u_{1}}$ $l=1$ (1) $\underline{u}=(u)$ ((1) LHS) $=[ \frac{\overline{\partial}}{\overline{\partial}s_{ij}}(u),$ $\sum_{a}s_{as}\frac{\overline{\partial}}{\overline{\partial}s_{at}}]$ $= \sum_{a}\frac{1}{2}(\delta_{ia}\delta_{js}+\delta_{is}\delta_{ja})\frac{\overline{\partial}}{\overline{\partial}s_{at}}+\sum_{a}[u\frac{\overline{\partial}g}{\overline{\partial}s_{ij}}, S_{as}\frac{\overline{\partial}}{\overline{\partial}S_{at}}]$ $= \frac{1}{2}(\delta_{js}\frac{\overline{\partial}}{\overline{\partial}s_{it}}+\delta_{is}\frac{\overline{\partial}}{\overline{\partial}s_{jt}})-\sum_{a}(s_{as}\frac{\overline{\partial}}{\overline{\partial}s_{ij}}\frac{\overline{\partial}}{\overline{\partial}s_{at}})(ug)$ (5) $(ug)$ $ug$ $\sum_{a}(s_{as}\frac{\overline{\partial}}{\overline{\partial}s_{ij}}\frac{\overline{\partial}}{\overline{\partial}s_{at}})(ug)=\sum_{a}((\frac{\overline{\partial}}{\overline{\partial}s_{ij}}s_{as}\frac{\overline{\partial}}{\overline{\partial}s_{at}})(ug)-\frac{1}{2}(\delta_{ai}\delta_{sj}+\delta_{aj}\delta_{si})\frac{\overline{\partial}}{\overline{\partial}s_{at}}(ug))$ $= \frac{\overline{\partial}}{\overline{\partial}s_{ij}}(\sum_{a}s_{as}u\frac{\overline{\partial}g}{\overline{\partial}s_{at}})-\frac{1}{2}(\delta_{sj}u\frac{\overline{\partial}g}{\overline{\partial}s_{it}}+\delta_{si}u\frac{\overline{\partial}g}{\overline{\partial}s_{jt}})$ $u\delta_{st}$ (5) $\frac{1}{2}(\delta_{js}\frac{\overline{\partial}}{\overline{\partial}s_{it}}(u)+\delta_{is}\frac{\overline{\partial}}{\overline{\partial}s_{jt}}(u))=$ ( (1) RHS) $l=1$ (1)

7 41 $l$ (1), (2), (3) $l+1$ (1), (2), (3) (1) $[A_{ij}^{(\iota+1)}(\underline{u}, v), B_{st}]$ $= \sum_{a}[a_{ia}^{(l)}(\underline{u})b_{aj}(v), B_{st}]$ $= \sum_{a}\frac{1}{2}(\delta_{as}a_{it}^{(l)}(\underline{u})+\delta_{is}a_{ta}^{(l)}(\underline{u}))b_{aj}(v)+\sum_{a}a_{ia}^{(l)}(\underline{u})\cdot\frac{1}{2}(\delta_{js}b_{at}-\delta_{ta}b_{sj})$. 1 (1) 2 5 $= \frac{1}{2}(a_{it}^{(l)}(\underline{u})b_{sj}(v)+\delta_{is}a_{tj}^{(\iota+1)}(\underline{u}, v)+\delta_{js}a_{it}^{(\iota+1)}(\underline{u}, 0)-A_{it}^{(l)}(\underline{u})B_{sj})$ $= \frac{1}{2}(a_{it}^{(l)}(\underline{u})\cdot v\delta_{sj}+\delta_{is}a_{tj}^{(l+1)}(\underline{u}, v)+\delta_{js}a_{it}^{(l+1)}(\underline{u}, 0))$ $= \frac{1}{2}(\delta_{js}a_{it}^{(l+1)}(\underline{u}, v)+\delta_{is}a_{tj}^{(l+1)}(\underline{u}, v))$ (1) $l+1$ (2) $A_{ij}^{(l+1)}( \underline{u}, v)=\sum_{a=1}^{n}a_{ia}^{(l)}(\underline{u})b_{aj}(v)$ $= \sum_{a}(b_{aj}(v)a_{ia}^{(l)}(\underline{u})+\frac{1}{2}\delta_{aa}a_{ij}^{(l)}(\underline{u})+\frac{1}{2}\delta_{ia}a_{ja}^{(l)}(\underline{u}))$. (1) (2) 2 $= \sum_{a}b_{aj}(v)a_{ia}^{(l)}(\underline{u})+\frac{1}{2}(n+1)a_{ij}^{(l)}(\underline{u})$ (6) $B_{aj}(v)= \sum_{b=1}^{n}s_{ba}\frac{\overline{\partial}}{\overline{\partial}s_{bj}}(v)$ $= \sum_{b}(\frac{\overline{\partial}}{\overline{\partial}s_{bj}}(v)s_{ba}-\frac{1}{2}\delta_{aj}-\frac{1}{2}\delta_{bj}\delta_{ab})$ $= \sum_{b}\frac{\overline{\partial}}{\overline{\partial}s_{bj}}(v)s_{ba}-\frac{1}{2}(n+1)\delta_{a}$

8 42 (6) ( (6) ) $= \sum_{a}(\sum_{b}\frac{\overline{\partial}}{\overline{\partial}s_{bj}}(v)s_{ba}-\frac{n+1}{2}\delta_{aj})a_{ia}^{(l)}($ $)+ \frac{n+1}{2}a_{ij}^{(\iota)}(\underline{u})$ $= \sum_{a,b}\frac{\overline{\partial}}{\overline{\partial}s_{jb}}(v)s_{ab}a_{ai}^{(l)}(\underline{u})$. (2) $=A_{ji}^{(l+1)}(v,\underline{u})=A_{ji}^{(\iota+1)}(\underline{u}, v)$ (2) ( 4) $l+1$ (3) $A^{(\iota)}(\underline{u})$ $[A_{ij}^{(\iota+1)}(\underline{u}, v), A_{st}^{(l+1)}(\underline{u}, v)]$ $= \sum_{a,b=1}^{n}[a_{ia}^{(l)}(\underline{u})b_{aj(v),a_{sb}^{(l)}(\underline{u})b_{bt}(v)]}$ $\alpha\not\in$ (3) $\sum_{a,b}(a_{ia}^{(l)}(\underline{u})[b_{aj}(v), A_{sb}^{(l)}(\underline{u})]B_{bt}(v)$ $+A_{sb}^{(l)}(\underline{u})[A_{ia}^{(l)}(\underline{u}),$ $B_{bt}(v)]B_{aj}(v)+A_{sb}^{(l)}(\underline{u})A_{ia}^{(l)}$ $($ $)$ $[B_{aj}(v),$ $B_{bt}(v)])$ $= \frac{1}{2}\sum_{a,b}(-a_{ia}^{(l)}(\underline{u})(\delta_{ba}a_{sj}^{(l)}(\underline{u})+\delta_{sa}a_{jb}^{(l)}(\underline{u}))b_{bt}(v)$ $+A_{sb}^{(l)}(\underline{u})(\delta_{ab}A_{it}^{(l)}(\underline{u})+\delta_{ib}A_{ta}^{(l)}(\underline{u}))B_{aj(v)}+A_{sb}^{(l)}(\underline{u})A_{ia}^{(l)}(\underline{u})(\delta_{jb}B_{at}(v)-\delta_{ta}B_{bj(v)))}.$ (3) $= \frac{1}{2}()(\underline{u}, v)+a_{it}^{(l)}(\underline{u})a_{sj}^{(l+1)}(\underline{u}, v)$ $+A_{si}^{(l)}(\underline{u})A_{tj}^{(l+1)}(\underline{u}, v)+a_{sj}^{(l)}(\underline{u})a_{it}^{(l+1)}(\underline{u}, v)-a_{it}^{(l)}(\underline{u})a_{sj}^{(l+1)}(\underline{u}, v))$ $=0.$ 2 4 ( ) (2) $l+1$ (3) 7( $\eta$ $\zeta$ ). $\eta_{j}(\underline{u})\zeta_{k}(\underline{u}, v)=-\zeta_{k}(\underline{u}, v-\frac{1}{2})\eta_{j}(\underline{u})$

9 43 Proof. (LHS) $= \sum_{i=1}^{n}e_{i}a_{ij}^{(l)}(\underline{u})\sum_{a,b=1}^{n}e_{a}a_{ab}^{(l)}(\underline{u})b_{bk}(v)$ $= \sum_{i,a,b}e_{i}e_{a}a_{ab}^{(l)}(\underline{u})(b_{bk}(v)a_{ij}^{(l)}(\underline{u})+\frac{\delta_{jb}}{2}a_{ik}^{(l)}(\underline{u})+\frac{\delta_{ib}}{2}a_{kj}^{(l)}(\underline{u}))$ $=- \zeta_{k}(\underline{u}, v)\eta_{j}(\underline{u})+\frac{1}{2}(\sum_{i,a}e_{i}e_{a}a_{aj}^{(l)}(\underline{u})a_{ik}^{(l)}(\underline{u})+\sum_{i,a}e_{i}e_{a}a_{ai}^{(l)}(\underline{u})a_{kj}^{(l)}(\underline{u}))$ $( 2 e_{i}e_{a} i, a A_{ai}^{(l)}(\underline{u})A_{kj}^{(l)}(\underline{u})$ $i,$ $a$ 0 ) $=- \zeta_{k}(\underline{u}, v)\eta_{j}(\underline{u})+\frac{1}{2}\eta_{k}(\underline{u})\eta_{j}(\underline{u})$ $=- \zeta_{k}(\underline{u}, v-\frac{1}{2})\eta_{j}(\underline{u})$ $=$ (RHS). 8. $ A^{(l+1)}( \underline{u}, v) = A^{(l)}(\underline{u}) ts \frac{\overline{\partial}}{\overline{\partial}s}(v).$ Proof. $\zeta_{1}(\underline{u}, v)\cdots\zeta_{n}(\underline{u}, v)=(\sum_{i=1}^{n}e_{i}a_{i1}^{(\iota+1)}(\underline{u}, v))\cdots(\sum_{i=1}^{n}e_{i}a_{in}^{(l+1)}(\underline{u}, v))$ $=e_{1}\cdots e_{n}\det(a^{(l+1)}(\underline{u}, v))$ $\zeta_{1}(\underline{u}, v)\cdots\zeta_{n}(\underline{u}, v)=\zeta_{1}(\underline{u}, v)\cdots\zeta_{n-1}(\underline{u}, v)\sum_{j_{n}=1}^{n}\eta_{j_{n}}(\underline{u})b_{j_{n},n}(v)$ $g=7(-1)^{n-1}\sum\eta_{j_{n}}(\underline{u})\cdot\zeta_{1}(\underline{u}, v+\frac{1}{2})\cdots\zeta_{n-1}(\underline{u}, v+\frac{1}{2})\cdot B_{j_{n},n}(v)$ $j_{n}$

10 44 $=(-1)^{n(n-1)} \sum_{j_{1},\ldots,j_{n}}\eta_{j_{1}}(\underline{u})\eta_{j_{2}}(\underline{u})\cdots\eta_{j_{n}}(\underline{u})$ $\cross B_{j_{1},1}(v+\frac{n-1}{2})B_{j_{2},2}(v+\frac{n-2}{2})\cdots B_{j_{n},n}(v+\frac{n-n}{2})$ $=e_{1}\cdots e_{n}\det(a^{(l)}(\underline{u}))\det(b+(\begin{array}{llll}v+(n-1)/2 v+(n-2)/2 \ddots v\end{array}))$ Capelli (3) $f^{-v}$ $v$ $=e_{1} \cdots e_{n}\det(a^{(l)}(\underline{u}))\det(ts)\det(\frac{\overline{\partial}}{\overline{\partial}s}(v))$ 8 $ \frac{\overline{\partial}}{\overline{\partial}s}(u_{1}) ts \frac{\overline{\partial}}{\overline{\partial}s}(u_{2}) ts \cdots ts \frac{\overline{\partial}}{\overline{\partial}s}(u_{l}) $ $= \frac{\overline{\partial}}{\overline{\partial}s}(u_{1})^{t}s\frac{\overline{\partial}}{\overline{\partial}s}(u_{2})^{t}s\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(u_{l}) $ Capelli (4) 3 $\det((x^{(1)}x^{(2)}\cdots X^{(l)})t(X^{(1)}X^{(2)}\cdots X^{(l)}))$ $b$ Capelli (4) 2 b- 1 - Sato-Sugiyama [2] $b$ $\mathbb{c}$ $m_{1},$ $m_{2},$ $\ldots$, $(G, V)$ $G=GL(m_{0})\cross GL(m_{1})\cross\cdots\cross GL(m_{l-1})\cross SO(m_{l})$, $V=$ Mat $(m_{0}, m_{1})\oplus$ Mat $(m_{1}, m_{2})\oplus\cdots\oplus$ Mat $(m_{l-1}, m\iota)$, $(g_{0}, \ldots, g_{l}).(x^{(1)}, \ldots, X^{(l)})=(g_{0}X^{(1)}g_{1}^{-1}, \ldots, g_{l-1}x^{(l)}g_{l}^{-1})$. $i$ $m_{0}\leq m_{i}$ $f=\det((x^{(1)}x^{(2)}\cdots X^{(l)})t(X^{(1)}X^{(2)}\cdots X^{(l)}))$ $(G, V)$ $f$ b. $b_{f}(s)$ $f$ $f(\partial)$ $f(\partial)(f^{s+1})=b_{f}(s)f^{s}$

11 45 ( monic $b_{f}(s)$ ) $b_{f}(s)= \prod_{r=1}^{\iota}(s+m_{r}/2)^{((m_{0}))}\prod_{r=0}^{l-1}(s+(m_{r}+1)/2)^{((m_{0}))},$ $a^{((k))}=a(a-1/2)\cdots(a-(k-1)/2)$ Sato-Sugiyama [2, Proposition 4. 1] Capelli (4) $b$- 3.1 $m_{0},$ $m_{1},$ $\ldots$, $X^{(i)}$ $m_{l}$ $i$ $m_{i-1}\cross m_{i}$ $x^{(i,j)}=x^{(i)x(i+1)\ldots X^{(j)}}$ $m_{0}$ $S$ $m_{i}\geq m_{0}$ $(1\leq i\leq l)$ $i\leq i$ $S:=X^{(1)}X^{(2)}\cdots X^{(l)}t(X^{(1)}X^{(2)}\cdots X^{(l)})=X^{(1,l)}tX^{(1,l)}$ $\frac{\partial}{\partial X(r)}=(\frac{\partial}{\partial X_{ij}^{(r)}})_{1\leqi\leq m_{r-1},1\leq j\leq m_{r}} (1\leq r\leq l)$ $\frac{\overline{\partial}}{\overline{\partial}s}=(\frac{\overline{\partial}}{\overline{\partial}s_{ij}})_{1\leq i,j\leq m_{0}}=(\frac{1+\delta_{ij}}{2}\frac{\partial}{\partial S_{ij}})_{1\leq i,j\leq m_{0}}$ $f=\det(ts)$, $g=$ log det $(ts)$ $\frac{\overline{\partial}}{\partial}s^{=ts^{-1}}a$ 2 $\frac{\overline{\partial}}{\overline{\partial}s}(u)=\frac{\overline{\partial}}{\overline{\partial}s}+u^{t}s^{-1}=\frac{\overline{\partial}}{\overline{\partial}s}+u\frac{\overline{\partial}g}{\overline{\partial}s}=f^{-u}\frac{\overline{\partial}}{\overline{\partial}s}f^{u}$ $\frac{\partial}{\partial X(r)}(u):=\frac{\partial}{\partial X(r)}+u\frac{\partial g}{\partial X(r)}=f^{-u}\frac{\partial}{\partial X(r)}f^{u} (1\leq r\leq l)$ $X^{(r)}$ $T$ $S$

12 $\frac{\partial\phi}{\partial Y_{ij}}=\sum_{a\leq b}\frac{\partial\phi}{\partial S_{ab}}\frac{\partial S_{ab}}{\partial Y_{ij}}=\sum_{a,b=1}^{n}\frac{\overline{\partial}\phi}{\overline{\partial}S_{ab}}\frac{\partial S_{ab}}{\partial Y_{ij}}$ ( ). $\phi=\phi(s)=\phi(s_{11}, \ldots, S_{m_{0},m_{0}})$ $1\leq r\leq l$ (1) $\frac{\partial\phi}{\partial X(r)}(u)=2\cdot tx^{(1,r-1)}\frac{\overline{\partial}\phi}{\overline{\partial}s}(u)t(x^{(r+1,l)}tx^{(1,l)})$ (2) $t( \frac{\partial\phi}{\partial X(r)}(u))=2\cdott(X^{(1,l)}tX^{(r+1,l)})\frac{\overline{\partial}\phi}{\overline{\partial}S}(u)tX^{(1,r-1)}$ Proof. (2) (1) (1) $X=X^{(1,r-1)}, Y=X^{(r)}, Z=X^{(r+1,l)}tX^{(r+1,l)}$ $S=XYZ^{t}Y^{t}X$ $Z$ $S_{ab}= \sum_{p,q,s,t}x_{ap}y_{pq}z_{q }Y_{s}X_{bt}$ $= \sum \frac{\overline{\partial}\phi}{\overline{\partial}s_{ab}}(x_{ap}\delta_{ip}\delta_{jq}z_{qs}y_{ts}x_{bt}+x_{ap}y_{pq}z_{qs}\delta_{it}\delta_{js}x_{bt})$ $a,b,p,q,s,t$ $=(tx \frac{\overline{\partial}\phi}{\overline{\partial}s}xytz)(i,j)$ $+(tx^{t}( \frac{\overline{\partial}\phi}{\overline{\partial}s})xyz)(i,j)$ $S$ $Z$ $\frac{\partial\phi}{\partial Y}=2^{t}X\frac{\overline{\partial}\phi}{\overline{\partial}S}XYZ$ (1) $u=0$ $u$ $\phi=ug$ 10. (1) $\frac{\partial}{\partial X(r)}(u-\frac{m_{r}}{2})tX^{(1,r)}=2\cdot tx^{(1,r-1)}\frac{\overline{\partial}}{\overline{\partial}s}(u-\frac{m_{0}+1}{2})ts$ (2) $t( \frac{\partial}{\partial X(r)}(u-\frac{m_{r-1}}{2}))X^{(r,l)}tX^{(1,l)}=2\cdott(X^{(1,l)}tX^{(r+1,\iota)})\frac{\overline{\partial}}{\overline{\partial}S}(u-\frac{m_{0}}{2})^{t}S$ (3) $t( \frac{\partial}{\partial X(1)}(u))=2\cdott(X^{(1,l)}tX^{(2,l)})\frac{\overline{\partial}}{\overline{\partial}S}(u)$

13 47 Proof. (2) (3) (1) (1) 9 $\frac{\partial\phi}{\partial X(r)}=2tX^{(1,r-1)}\frac{\overline{\partial}\phi}{\overline{\partial}S}t(X^{(r+1,l)}tX^{(1,l)})$ $\phi$ $tx^{(1,r)}$ $\frac{\partial\phi}{\partial X(r)}tX^{(1,r)}=2tX^{(1,r-1)}\frac{\overline{\partial}\phi}{\overline{\partial}S}tS,$ $t(x^{(r)^{t}}( \frac{\partial\phi}{\partial X(r)}))tX^{(1,r-1)}=2tX^{(1,r-1)^{t}}(S^{t}(\frac{\overline{\partial}\phi}{\overline{\partial}S}))$ (7) $\phi$ $t(x^{(r)^{t}}( \frac{\partial}{\partial X(r)}))=\frac{\partial}{\partial X(r)}tX^{(r)}-m_{r}1_{m_{r-1}},$ $t(s^{t}( \frac{\overline{\partial}}{\overline{\partial}s}))=\frac{\overline{\partial}}{\overline{\partial}s}ts-\frac{m_{0}+1}{2}1_{m_{0}}$ ( ) (7) $1_{m_{r-1}}$ $\phi$ $\frac{\partial}{\partial X(r)}tX^{(1,r)}=2tX^{(1,r-1)}\frac{\overline{\partial}}{\overline{\partial}S}(\frac{m_{r}-m_{0}-1}{2})^{t}S.$ $f^{-u+m_{r}/2}$ 11 ( ). $u_{i},$ $v_{i}\in \mathbb{c}$ $2^{2l} \cdot\frac{\overline{\partial}}{\overline{\partial}s}(u_{1}-\frac{m_{0}}{2})^{t}s\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(u_{l}-\frac{m_{0}}{2})\cdot ts\cdot\frac{\overline{\partial}}{\overline{\partial}s}(v_{l}-\frac{m_{0}}{2})^{t}s\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(v_{1}-\frac{m_{0}}{2})$ $= \frac{\partial}{\partial X(1)}(u_{1}-\frac{m_{1}-1}{2})\cdots\frac{\partial}{\partial X(l)}(u_{l}-\frac{m_{l}-1}{2})$ $\cross t(\frac{\partial}{\partial X(\iota)}(v_{l}-\frac{m_{l-1}}{2}))\cdots t(\frac{\partial}{\partial X(1)}(v_{1}-\frac{m_{0}}{2}))$. Proof. 10 (1) $r=1$ $2^{2l-1} \cdot\frac{\partial}{\partial X(1)}(u_{1}-\frac{m_{1}-1}{2})tX^{(1)}\cdot\frac{\overline{\partial}}{\overline{\partial}S}(u_{2}-\frac{m_{0}}{2})ts\cdots\cdot$ 10 (1) $r=2$ $2^{2l-2} \cdot\frac{\partial}{\partial X(1)}(u_{1}-\frac{m_{1}-1}{2})\cdot\frac{\partial}{\partial X(2)}(u_{2}-\frac{m_{2}-1}{2})tX^{(1,2)}\cdot\frac{\overline{\partial}}{\overline{\partial}S}(u_{3}-\frac{m_{0}}{2})^{t}S\cdots\cdot$ $2^{l} \cdot\frac{\partial}{\partial X(1)}(u_{1}-\frac{m_{1}-1}{2})\cdots\frac{\partial}{\partial X(\iota)}(u_{l}-\frac{m_{l}-1}{2})$ $\cross tx^{(1,l)}\cdot\frac{\overline{\partial}}{\overline{\partial}s}(v_{l}-\frac{m_{0}}{2})^{tt}t\cdot-\cdot\cdot\frac{\overline{\partial}}{\overline{\partial}s}(v_{1}-\frac{m_{0}}{2})$

14 (2) $\frac{\partial}{\partial X(1)}(u_{1}-\frac{m_{1}-1}{2})\cdots\frac{\partial}{\partial X(\iota)}(u_{l}-\frac{m_{l}-1}{2})\cdot\frac{\partial}{\partial X(\iota)}(v_{l}-\frac{m_{l-1}}{2})\cdots\frac{\partial}{\partial X(1)}(v_{1}-\frac{m_{0}}{2})$ 3.3 $b$- $f= (X^{(1)}X^{(2)}\cdots X^{(l)})t(X^{(1)}X^{(2)}\cdots X^{(l)}) $ - $b$ $( S ^{s+1} )$ $ S ^{s+1}$ $ \frac{\partial}{\partial X(1)}\cdots\frac{\partial}{\partial X(\iota)}.$ $t( \frac{\partial}{\partial X(\iota)})\ldots t(\frac{\partial}{\partial X(1)}) ( S ^{s+1})$ $11_{2^{2lm_{0}}} \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{1}-m_{0}-1}{2})^{t}s\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{2}-m_{0}-1}{2})\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l}-m_{0}-1}{2})$. $ts \cdot\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l-1}-m_{0}}{2})^{t}s\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l-2}-m_{0}}{2})\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{0}-m_{0}}{2})( S ^{s+1})$ $1 2^{2lm_{0}} \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{1}-m_{0}-1}{2}) ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{2}-m_{0}-1}{2}) \cdots ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l}-m_{0}-1}{2}) $. $ ts \cdot \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l-1}-m_{0}}{2}) ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l-2}-m_{0}}{2}) \cdots ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{0}-m_{0}}{2}) ( S ^{s+1})$ $ S ^{s+1}$ Capelli (3) $2^{2lm_{0}} \cdot b(s+\frac{m_{1}-m_{0}-1}{2})b(s+\frac{m_{2}-m_{0}-1}{2})\cdots b(s+\frac{m_{l}-m_{0}-1}{2})$. $b(s+ \frac{m\iota-1^{-m_{0}}}{2})\cdots b(s+\frac{m_{1}-m_{0}}{2})\cdot b(s+\frac{m_{0}-m_{0}}{2}) S ^{S}$ $b(s)$ $b$- $b(s)=(s+1)(s+ \frac{3}{2})\cdots(s+\frac{m_{0}+1}{2})=(s+\frac{m_{0}+1}{2})^{((m_{0}))}$ (8)

15 49 $a^{((n))}=a(a-1/2)\cdots(a-(n-1)/2)$ 1/2 $f$ - $b$ $b_{f}(s)=(s+ \frac{m_{1}}{2})^{((m_{0}))}(s+\frac{m_{2}}{2})^{((m_{0}))}\cdots(s+\frac{m_{l}}{2})^{((m_{0}))}$ $\cross(s+\frac{m_{l-1}+1}{2})^{((m_{0}))}(s+\frac{m_{l-2}+1}{2})^{((m_{0}))}\cdots(s+\frac{m_{0}+1}{2})^{((m_{0}))}$ 4 $\det((x^{(1)}x^{(2)} \cdot X^{(l)})Yt(X^{(1)}X^{(2)} \cdot X^{(l)}))$ $b$ - $b$ Capelli (4) Sato-Sugiyama [2] [2] $m_{1},$ $m_{2},$ $m_{l}$ $\ldots,$ $(G, V)$ $G=GL(m_{0})\cross GL(m_{1})\cross\cdots\cross GL(m_{l})$, $V=$ Mat $(m_{0}, m_{1})\oplus\cdots\oplus$ Mat $(m_{l-1}, m_{l})\oplus$ Sym $(m\iota)$, $(go, \ldots, g\iota).(x^{(1)}, \ldots, X^{(l)}, Y)=(g_{0}X^{(1)}g_{1}^{-1}, \ldots, g_{l-1}x^{(l)}g_{l}^{-1}, g_{l}ytg_{l})$ $i$ $m_{0}\leq m_{i}$ $f=\det((x^{(1)}x^{(2)}\cdots X^{(l)})Yt(X^{(1)}X^{(2)}\cdots X^{(l)}))$ $(G, V)$ $b_{f}(s)$ $b_{f}(s)= \prod_{r=1}^{\iota}(s+m_{r}/2)^{((m_{0}))}\prod_{r=0}^{\iota}(\mathcal{s}+(m_{r}+1)/2)^{((m_{0}))}$ Capelli (4) 4.1 $i$ $m_{0},$ $m_{1},$ $\ldots$, $m_{l}$ $X^{(i)}$ $m_{i-1}\cross m_{i}$ $m_{l}$ $i\leq j$ $m_{0}$ $S$ $m_{i}\geq m_{0}$ $(1\leq i\leq l)$ $Y$ $Y_{j}=Y_{ji}$ $X^{(i,j)}=X^{(i)}X^{(i+1)}\cdots X$ ( $S:=X^{(1)}X^{(2)}\cdots X^{(l)}Yt(X^{(1)}X^{(2)}\cdots X^{(l)})=X^{(1,l)}YtX^{(1,l)}$

16 $\frac{\overline{\partial}}{\overline{\partial}s}(u)=\frac{\overline{\partial}}{\overline{\partial}s}+u\frac{\overline{\partial}g}{\overline{\partial}s}=f^{-u}\frac{\overline{\partial}}{\overline{\partial}s}f^{u},$ $\frac{\overline{\partial}}{\overline{\partial}y}(u)=\frac{\overline{\partial}}{\overline{\partial}y}+u\frac{\overline{\partial}g}{\overline{\partial}y}=f^{-u}\frac{\overline{\partial}}{\overline{\partial}y}f^{u}$ 50 $\frac{\partial}{\partial X(r)}=(\frac{\partial}{\partial X_{ij}^{(r)}})_{1\leqi\leq m_{r-1},1\leq J\leq m_{r}}$ $\frac{\overline{\partial}}{\overline{\partial}s}=(\frac{\overline{\partial}}{\overline{\partial}s_{ij}})_{1\leq i,j\leq m_{0}}=(\frac{1+\delta_{ij}}{2}\frac{\partial}{\partial S_{ij}})_{1\leq i,j\leq m_{0}}$ $\frac{\overline{\partial}}{\overline{\partial}y}=(\frac{\overline{\partial}}{\overline{\partial}y_{ij}})_{1\leq i,j\leq m_{l}}=(\frac{1+\delta_{ij}}{2}\frac{\partial}{\partial Y_{ij}})_{1\leq i,j\leq m_{l}}$ $f=\det(ts)$, $g=$ log det $(ts)$ $\frac{\partial}{\partial X^{(r)}}(u)=\frac{\partial}{\partial X(r)}+\cdot u\frac{\partial g}{\partial X(r)}=f^{-u}\frac{\partial}{\partial X(r)}f^{u},$ ( ). $\phi=\phi(s)=\phi(s_{11}, \ldots, S_{m_{O},m_{O}})$ $1\leq r\leq l$ (1) $\frac{\partial\phi}{\partial X(r)}(u)=2\cdot tx^{(1,r-1)}\frac{\overline{\partial}\phi}{\overline{\partial}s}(u)t(x^{(r+1,l)}y^{t}x^{(1,l)})$ (2) $\frac{\overline{\partial}\phi}{\overline{\partial}y}(u)=tx^{(1,l)}\frac{\overline{\partial}\phi}{\overline{\partial}s}(u)x^{(1.l)}$ (3) $t( \frac{\partial\phi}{\partial X(r)}(u))=2\cdott(X^{(1,l)}YtX^{(r+1,l)})\frac{\overline{\partial}\phi}{\overline{\partial}S}(u)tX^{(1,r-1)}$ Proof (1) $\frac{\partial}{\partial X^{(r)}}(u-\frac{m_{r}}{2})tX^{(1,r)}=2\cdot tx^{(1,r-1)}\frac{\overline{\partial}}{\overline{\partial}s}(u-\frac{m_{0}+1}{2})^{t}s$

17 51 (2) $\frac{\overline{\partial}}{\overline{\partial}y}(u-\frac{m_{l}}{2})t(x^{(1.l)}y)=tx^{(1,l)}\frac{\overline{\partial}}{\overline{\partial}s}(u-\frac{m_{0}}{2})^{t}s$ (3) $t( \frac{\partial}{\partial X(r)}(u-\frac{m_{r-1}}{2}))t(X^{(1,l)}Y^{t}X^{(r,l)})=2\cdott(X^{(1,\iota)}Y^{t}X^{(r+1,l)})\frac{\overline{\partial}}{\overline{\partial}S}(u-\frac{m_{0}}{2})tS$ Proof ( ). $2^{2l} \cdot\frac{\overline{\partial}}{\overline{\partial}s}(u_{1}-\frac{m_{0}}{2})^{t}s\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(u\iota-\frac{m_{0}}{2})\cdot ts\frac{\overline{\partial}}{\overline{\partial}s}(v)ts\cdot\frac{\overline{\partial}}{\overline{\partial}s}(w_{l}-\frac{m_{0}}{2})^{t}s\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(w_{1}-\frac{m_{0}}{2})$ $= \frac{\partial}{\partial X(1)}(u_{1}-\frac{m_{1}-1}{2})\cdots\frac{\partial}{\partial X(l)}(u_{l}-\frac{m_{l}-1}{2})\cdot\frac{\overline{\partial}}{\overline{\partial}Y}(v-\frac{m_{l}}{2})$ $\cross t(\frac{\partial}{\partial X(\iota)}(w_{l}-\frac{m_{l-1}}{2}))\cdots t(\frac{\partial}{\partial X(1)}(w_{1}-\frac{m_{0}}{2}))$ Proof $f= (X^{(1)}X^{(2)}\cdots X^{(l)})Yt(X^{(1)}X^{(2)}\cdots X^{(l)}) $ $( S ^{s+1} )$ $ S ^{s+1}$ $ \frac{\partial}{\partial X(1)}\cdots\frac{\partial}{\partial X(l)}\cdot\frac{\overline{\partial}}{\overline{\partial}Y}.$ $tt ( S ^{s+1})$ $14_{2^{2lm_{0}}} \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{1}-m_{0}-1}{2})^{t}s\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{2}-m_{0}-1}{2})\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l}-m_{0}-1}{2})$. $ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l}-m_{0}}{2})^{t}s$. $\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l-1}-m_{0}}{2})^{t}s\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l-2}-m_{0}}{2})\cdots ts\frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{0}-m_{0}}{2})( S ^{s+1})$ $1_{2^{2lm_{0}}} \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{1}-m_{0}-1}{2}) ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{2}-m_{0}-1}{2}) \cdots ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l}-m_{0}-1}{2}) $. $ ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l}-m_{0}}{2}) ts $. $ \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l-1}-m_{0}}{2}) ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{l-2}-m_{0}}{2}) \cdots ts \frac{\overline{\partial}}{\overline{\partial}s}(\frac{m_{0}-m_{0}}{2}) ( S ^{s+1})$

18 $\mathfrak{g}$ $\lambda$ : 52 $ S ^{s+1}$ $2^{2lm_{0}} \cdot b(s+\frac{m_{1}-m_{0}-1}{2})b(s+\frac{m_{2}-m_{0}-1}{2})\cdots b(s+\frac{m_{l}-m_{0}-1}{2\prime})$. $b(s+ \frac{m_{l}-m_{0}}{2})\cdot b(s+\frac{m_{l-1}-m_{0}}{2})\cdots b(s+\frac{m_{1}-m_{0}}{2})\cdot b(s+\frac{m_{0}-m_{0}}{2}) S ^{s}$ $b(s)$ (8) $f$ $k$ $b_{f}(s)=(s+ \frac{m_{1}}{2})^{((m_{0}))}(s+\frac{m_{2}}{2})^{((m_{0}))}\cdots(s+\frac{m_{l}}{2})^{((m_{0}))}$ $\cross(s+\frac{m_{l}+1}{2})^{((mo))}(s+\frac{m_{l-1}+1}{2})^{((m_{0}))}\cdots(s+\frac{m_{0}+1}{2})^{((m_{o}))}$ 5 $\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{2n}$ $\mathfrak{g}=\mathfrak{s}\mathfrak{p}_{2n}$ $\mathfrak{g}=\mathfrak{k}\oplus(\mathfrak{n}^{+}+\mathfrak{n}^{-})$ $\mathfrak{p}arrow $\mathfrak{p}=\mathfrak{k}\oplus \mathfrak{n}^{+}$ \mathbb{c}$ $\mathfrak{g}$ $\mathfrak{p}$ 1 $\lambda$ $M(\lambda):=U(\mathfrak{g})\otimes_{U(\mathfrak{p})}\mathbb{C}_{\lambda}$ $\mathbb{c}_{\lambda}$ $\lambda$ $M(\lambda)\simeq U(\mathfrak{n}^{-})\simeq$ $\mathbb{c}[\mathfrak{n}^{+}]$ $U(\mathfrak{g})arrow \mathbb{c}[\mathfrak{n}^{+}]\pi_{\lambda}$ 5.1 Capelli $)$ Capelli (2) $\mathfrak{g}=\mathfrak{g}$ 12n $(\mathbb{c}$ $\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{2n},$ $\mathfrak{k}=\{(\begin{array}{ll}a 00 D\end{array}) A,$ $D\in$ Mat $(n)\}\simeq \mathfrak{g}\mathfrak{l}_{n}\oplus \mathfrak{g}\mathfrak{l}_{n},$ $\mathfrak{n}^{+}=\{(\begin{array}{ll}0 B0 0\end{array}) B\in$ Mat $(n)\},$ $\mathfrak{p}=\{(\begin{array}{ll}a B0 D\end{array}) A,$ $B,$ $D\in$ Mat $(n)\}=\mathfrak{k}\oplus \mathfrak{n}^{+},$ $\mathfrak{n}^{-}=\{(\begin{array}{ll}0 0C 0\end{array}) C\in$ Mat $(n)\}.$ $e_{ij}\in \mathfrak{g} _{}2n$ $\{e_{ii}\}$ $\mathfrak{g}b_{n}$ $\mathbb{c}e_{11}+\mathbb{c}e_{22}+\cdots+\mathbb{c}e_{2n,2n}$ $\{\epsilon_{i}\}$ $\mathfrak{p}$ 1 $\lambda_{1},$ $\lambda_{2}\in \mathbb{c}$ $\lambda=\lambda_{1}(\epsilon_{1}+\cdots+\epsilon_{n})+\lambda_{2}(\epsilon_{n+1}+\cdots+\epsilon_{2n})$

19 $(\mathfrak{g}\mathfrak{l}_{2n}, \mathfrak{g}\mathfrak{l}_{n}\oplus \mathfrak{g}\mathfrak{l}_{n})$ $\mathfrak{n}^{+}$ 53 $\mathfrak{g}\mathfrak{l}_{2n}$ $\mathbb{c}[\mathfrak{n}^{+}]$ $\pi_{\lambda}(e_{ij})$ 15. $T_{ij}(1\leq i,j\leq n)$ $\mathfrak{n}^{+}$ $\pi_{\lambda}(e_{ij})=-\sum_{k=1}^{n}t_{jk}\frac{\partial}{\partial T_{ik}}+\lambda_{1}\delta_{ij},$ $\pi_{\lambda}(e_{n+i,n+j})=\sum_{k=1}^{n}t_{ki}\frac{\partial}{\partial T_{kj}}+\lambda_{2}\delta_{ij},$ $\pi_{\lambda}(e_{n+j,i})=t_{ij},$ $\pi_{\lambda}(e_{i,n+j})=-\sum_{k,l=1}^{n}t_{lk}\frac{\partial}{\partial T_{ik}}\frac{\partial}{\partial T_{lj}}+(\lambda_{1}-\lambda_{2})\frac{\partial}{\partial T_{ij}}$ $=-( \frac{\partial}{\partial T}(\lambda_{2}-\lambda_{1}-h)^{t}T\frac{\partial}{\partial T})_{(i,j)}$ 15 $\pi_{\lambda}(\det(e_{i,n+j}))=\det(\pi_{\lambda}(e_{i,n+j})))$ $= - \frac{\partial}{\partial T}(\lambda_{2}-\lambda_{1}-n)^{t}T\frac{\partial}{\partial T} = -\frac{\partial}{\partial T}(\lambda_{2}-\lambda_{1}-n) tt \frac{\partial}{\partial T} $ $[\pi_{\lambda}(\det(e_{i,n+j}))](\det(t)^{s+1})=(-1)^{n}b(s+\lambda_{2}-\lambda_{1}-n)b(s)\det(t)^{s}$ $(\det(t)^{s+1})$ $\det(t)^{s+1}$ $b(s)=(s+1)(s+2)\cdots(s+n)$ $M(\lambda)$ ( $7])$ $b(s+\lambda_{2}-\lambda_{1}-n)$ $[$5, 6, Capelli (2) $l=2$ ( )

20 $\mathfrak{s}\mathfrak{p}_{2n}$ $\pi_{\lambda}$ Capelli $\mathfrak{g}=\mathfrak{s}\mathfrak{p}_{2n}(\mathbb{c})$ Capelli (4) $\mathfrak{g}=\{(\begin{array}{ll}a BC -ta\end{array}) A\in$ Mat $(n),$ $B,$ $C\in$ Sym $(n)\}=\mathfrak{s}\mathfrak{p}_{2n},$ $\mathfrak{k}=\{(\begin{array}{ll}a 00 -ta\end{array}) A\in$ Mat $(n)\}\simeq \mathfrak{g}\mathfrak{l}_{n},$ $\mathfrak{n}^{+}=\{(\begin{array}{ll}0 B0 0\end{array}) B\in Sym(n)\},$ $\mathfrak{n}^{-}=\{(\begin{array}{ll}0 0C 0\end{array}) C\in$ Sym $(n)\},$ $\mathfrak{p}=\{(\begin{array}{ll}a B0 -ta\end{array}) A\in$ Mat $(n),$ $B\in$ Sym $(n)\}=\mathfrak{k}\oplus \mathfrak{n}^{+}.$ $e_{ij}\in \mathfrak{g}\mathfrak{l}_{2n}$ $\mathfrak{s}\mathfrak{p}_{2n}$ $\mathbb{c}(e_{11}-e_{n+1,n+1})+\mathbb{c}(e_{22}-$ $e_{n+2,n+2})+\cdots+\mathbb{c}(e_{nn}-e_{2n,2n})$ $\lambda_{0}\in \mathbb{c}$ 1 $\{\epsilon_{i}\}$ $\{e_{ii}-e_{n+i,n+i}\}$ $\mathfrak{p}$ $\lambda=\lambda_{0}(\epsilon_{1}+\cdots+\epsilon_{n})$ $\mathfrak{s}\mathfrak{p}_{2n}$ $\mathbb{c}[\mathfrak{n}^{+}]$ $n^{+}$ 16. $S_{ij}(1\leq i\leq j\leq n)$ $\mathfrak{n}^{+}$ $S_{ji}=S_{ij}$ $\pi_{\lambda}(e_{ij}-e_{n+j,n+i})=-2\sum_{k=1}^{n}s_{jk}\frac{\overline{\partial}}{\overline{\partial}s_{ik}}+\lambda_{0}\delta_{ij},$ $\pi_{\lambda}(e_{n+i,j}+e_{n+j,i})=s_{ij},$ $\pi_{\lambda}(e_{i,n+j}+e_{j,n+i})=-4\sum_{k,l=1}^{n}s_{kl}\frac{\overline{\partial}}{\overline{\partial}s_{il}}\frac{\overline{\partial}}{\overline{\partial}s_{jk}}+4\lambda_{0}\frac{\overline{\partial}}{\overline{\partial}s_{ij}}$ $=-4( \frac{\overline{\partial}}{\overline{\partial}s}(-\lambda_{0}-\frac{n+1}{2})^{t}s\frac{\partial}{\partial S})_{(i,j)}$ $\Re$ 16 $\pi_{\lambda}(\det(e_{i,n+j}+e_{j,n+i}))=\det(\pi_{\lambda}(e_{i,n+j}+e_{j,n+i})))$ $= -4 \frac{\overline{\partial}}{\overline{\partial}s}(-\lambda_{0}-\frac{n+1}{2})^{t}s\frac{\partial}{\partial S} = -4\frac{\overline{\partial}}{\overline{\partial}S}(-\lambda_{0}-\frac{n+1}{2}) ts \frac{\partial}{\partials} $ $\det(ts)^{8+1}$ $[ \pi_{\lambda}(\det(e_{i,n+j}+e_{j,n+i}))](\det(ts)^{s+1})=(-4)^{n}b(s-\lambda_{0}-\frac{n+1}{2})b(s)\det(ts)^{s}$

21 55 $b(s)=(s+1)(s+3/2)\cdots(s+(n-1)/2)$ (8) $b(s- \lambda_{0}-\frac{n+1}{2})$ ( [5,6,7]) $M(\lambda)$ Capelli (4) $l=2$ ( ) $(\mathfrak{s}\mathfrak{p}_{2n}, \mathfrak{g}\mathfrak{l}_{n})$ [1] Roger Howe and Toru Umeda. The Capelli identity, the double commutant theorem, and multiplicity-free actions. Math. Ann., $290(3): $, [2] Fumihiro Sato and Kazunari Sugiyama. Multiplicity one property and the decomposition of -functions. Internat. J. Math., $17(2): $, [3] H. W. Turnbull. Symmetric determinants and the Cayley and Capelli operators. Proc. Edinburgh Math. Soc. (2), 8:76-86, [4] Akihito Wachi. Logarithmic derivative and Capelli identities. [5] Akihito Wachi. Contravariant forms on generalized Verma modules and $b$-functions. Hiroshima Math. $J$., $29(1): $, [6] Akihito Wachi. Capelli type identities on certain scalar generalized Verma modules. J. Math. Kyoto Univ., $40(4): $, [7] Akihito Wachi. Capelli type identities on certain scalar generalized Verma modules. II. J. Math. Soc. Japan, $56(2): $, 2004.