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1 CAPELLI (T\^o $\mathrm{r}\mathrm{u}$ UMEDA) MATHEMATICS KYOTO UNIVERSITY DEPARTMENT $\mathrm{o}\mathrm{p}$ $0$: Cape i Capelli 1991 ( ) (1994 ; 1998 ) 100 Capelli Capelli Capelli ( ) ( ) 2002 ( ) Capelli ($\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\dot{\mathrm{a}}\mathrm{c}\mathrm{a}\mathrm{l}$ groups)

2 2 1: Capelli $\mathfrak{g}$ Lie $U(\mathfrak{g})$ (A) (B) (C) ( ) ( ) Capelli (A) (C) (B) (A) (B) (C) (1) $-(3)$ (1) \searrow (2) (3) Capelli - [1] [3] [1] ( ) : ( ) $-$ ( ) U( ) ( ) ( fusion process) Harish-Chandra Schur unique (Gelfand ) [2] ( = ): (1) fusion process Capelli (l) :

$3 - ( ) [3] ( ) : Capelli $\rho$-shift Okounkov higher Capelli Young contents Howe-Umeda Appendix anisotropic Lie ( ) Lie $\rho-$ -shift $R$- Itoh-Umeda Lie \epsilon 2 dual$

3 3 - ( ) [3] ( ) : Capelli $\rho$-shift Okounkov higher Capelli Young contents Howe-Umeda Appendix anisotropic Lie ( ) Lie $\rho-$ -shift $R$- Itoh-Umeda Lie \epsilon 2 dual $\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}$ 2: Capelli $U(\mathfrak{g}\mathfrak{l}_{n})$ Capelli Howe-Umeda CapeUi Capelli $U(\mathfrak{g}\mathfrak{l}_{n})$ $GL_{n}$ ( 1 ) $(\pi V_{\pi})$ $GL_{n}$ $\pi_{\mu\nu}(g)\#\mathrm{h}g$ $g_{ij}$ $\mathcal{p}(\mathrm{m}\mathrm{a}\mathrm{t}(n))$ : $\mathcal{p}(\mathrm{m}\mathrm{a}\mathrm{t}(m\cross n))$ $t_{ij}$ $\frac{\partial}{\partial t_{1\mathrm{j}}}$ $\mathfrak{g}\mathfrak{l}_{n}$ Lie ( )

4 $\mathfrak{g}\mathfrak{l}_{m}$ 4 ( ) $\rho(e_{ij})=\sum_{a=1}^{m}t_{ai}\partial_{aj}$ $\lambda(e_{ij}^{\mathrm{o}})=\sum_{b=1}^{n}t_{jb}\partial_{ib}$ Lie $T=(t_{ij})_{1\leq i\leq m1\leq j\leq n}$ $D=(\partial_{ij})_{1\leq i\leq m1\leq j\leq n}$ $\Pi=(\rho(E_{ij}))_{1\leq ij\leq n}$ $\Pi^{0}=(\lambda(\mathrm{E}_{ij}^{\mathrm{o}}))_{1\leq ij\leq m}$ $\Pi={}^{t}TD$ $t_{\pi^{\mathrm{o}}=t{}^{t}d}$ $t$ $\mathcal{p}(\mathrm{m}\mathrm{a}\mathrm{t}(n))$ $\pi$ $\mathrm{r}(\pi(^{t}t)\pi(d))=\sum_{\mu\nu}\pi_{\nu\mu}(^{t}t)\pi_{\mu\nu}(d)$ $GL_{n}$ $\Pi={}^{t}TD$ $\pi$ ( ) $\pi$ $U(\mathfrak{g}\mathfrak{l}_{n})$ $C_{\pi}= \mathrm{h}(\pi^{\mathfrak{y}}(\mathrm{e}))=\sum_{\mu}\pi_{\mu\mu}^{\mathfrak{h}}(\mathrm{e})$ $\pi\#$ $C_{\pi}$ $\pi_{\mu\mu}^{\mathfrak{h}}$ \urcorner \beta A Okounkov (1996) [HU] higher order Capelli $C_{\pi}$ $\mathcal{p}(\mathrm{m}\mathrm{a}\mathrm{t}(n))$ $\pi$ Capelli ( multiplicity-foee

$5 action ) $C_{\pi}$ $\pi$ ( $P(\mathrm{M}\mathrm{a}\mathrm{t}(n))$ (Lie ) (?$

5 5 action ) $C_{\pi}$ $\pi$ ( $P(\mathrm{M}\mathrm{a}\mathrm{t}(n))$ (Lie ) (?) vanishing property Capelli ) 3: Capelli (column determinant) $\det(\phi)=\sum_{\sigma\in \mathfrak{s}_{n}}$ sign(a) $\Phi_{\sigma(1)1}\Phi_{\sigma(2)2}\cdots\Phi_{\sigma(n)n}$ (row determinant) Capelli ( highest weight ) Capelli (double deterninant; symmetrized determinant) (permanent) (Pfaffian) (Hafnian) Lie Lie Lie j(n

$$\mathfrak{g}$ 6 ( ) $U(\mathfrak{g})$ Lie $\mathfrak{g}$ $=\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}$ ( typical invariant) U( )$

6 $\mathfrak{g}$ 6 ( ) $U(\mathfrak{g})$ Lie $\mathfrak{g}$ $=\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}$ ( typical invariant) U( ) $\mathfrak{g}$ Okounkov (?) Capelli $(\mathrm{a})-(\mathrm{c})$ 1 Lie $\mathfrak{g}\mathfrak{l}_{n}$ multiplicity-free action Capelli split Lie dual pair Capelli Wronski $\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{n}$

7 $\mathfrak{g}$ 7 Lie anisotropic $R$ ( ) 4: Lie (B) Harish-Chandra ZU $(\mathfrak{g})\simeq U(\mathfrak{h})^{W}=S(\mathfrak{h})^{W}$ h Cartan W Weyl $\mathfrak{h}$ $\mathfrak{s}_{n}$ Weyl $\{\pm 1\}^{n}$ $\{\pm 1\}^{n-1}$ $S(\mathfrak{h})^{W}$ modify Schur Schur-Weyl duality Weyl Han $\mathrm{s}\mathrm{h}$-chandra ( ) (Schur ) as $\text{ }$ $\Sigma \mathrm{j}$ Hanish-Chandra IEI ZU $(\mathfrak{g})arrow S(\mathfrak{h})^{W}$

8 8 $(\mathfrak{g})$ (1) ZU (2) Harish-Chandra (1) (2) Hari $\mathrm{s}\mathrm{h}$-chandra Harish-Chandra Hari $\mathrm{s}\mathrm{h}$-chandra 5: $\mathfrak{g}\mathrm{t}_{n}$ $\mathfrak{g}\mathfrak{l}_{n}$ ( ) 2 1 ( Lie ) ( ;Lie ) $GL$ ( ) oscillator spin dual pair ( ) \sim [2 Lie $0_{n}$ s[2

9 $i$ 9 - fcapelli Lie Howe-Umeda multiplicity-free actions Capelli 6: (1) [Capelli ] Capelli Capelli $-$ typical $\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\bm{\mathrm{r}}\mathrm{t}\mathrm{s}$ - Capeui ( ) ( ) (doubling the variables) R Howe dual pair (dual) Capelli dual $\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}$ dual \searrow Ge and ( ) Newton Koszul cyclic cohomology

10 10 (2) [ ] R Lie $\mathrm{a}\mathrm{d}\mathrm{j}(\dot{\mathrm{n}}\mathrm{n}\mathrm{t}$ Capelli pshift $\rho-$-shift - Capelli ( ) $\mathrm{a}\mathrm{d}\mathrm{j}\dot{\alpha}\mathrm{n}\mathrm{t}$ - Euler Lie Capelli [1] 100 Capelli Identity Identities a century after in Selected Papers on Harmonic Analysi $\mathrm{s}$ $46(1994)$ ( : The Capelli Groups and Invariants (Ed by K Nomizu) AMS Translations Series 2 vol 183 (1998) pp [2] R Howe and T Umeda The Capelli identity the double commutant theorem and multiplicity-free actions Math Ann 290 (1991) [3] T Umeda Newton s $fo$ rmula for $\mathfrak{g}\mathfrak{l}_{n}$ Proc Amer Math Soc 126 (1998) [4] T Umeda On the proofs of the Capdli identities preprint 1997 [5] T Umeda On TUmbull identity for $skew- s\psi nmet\dot{n}c$ matrices Proc Edinburgh Math Soc 43 (2000) [6] T Umeda Application of Koszul complex to Wronski relations for $U(\mathfrak{g}\mathfrak{l}_{n})$ Commentanii Math Helv 78 (2003) [7] M Itoh and T Umeda On central elements in the universal enveloping algebras of

11 11 the orthogonal Lie algebras Compositio Math 127(2001) [8] Capelli No 429 ( ) [2] M Noumi T Umeda and M Wakayama A quantum analogue fo the Capelli identity and an elementary differential calculus on $GL_{q}(n)$ Duke Math J 76(1994) [3] M Noumi T Umeda and M Wakayama Dual pairs spherical harmonics and a Capelli identity in quantum group theory Compositio Math 104(1996) [5] M Itoh: Capelli identities for the dual pair $(O_{M} Sp_{N})$ Math Zeit 246(2004) [6] A Wachi: Central elements in the universal enveloping algebras for the split realization of the orthogonal Lie algebras to appear in Lett Math Phys

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