## Lie Groups and Representation Theory

Seminar information archive ～11/01｜Next seminar｜Future seminars 11/02～

Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
---|

### 2022/06/28

17:00-18:00 Room #online (Graduate School of Math. Sci. Bldg.)

Computation of weighted Bergman inner products on bounded symmetric domains and Plancherel-type formulas for $(Sp(2r,\mathbb{R}),Sp(r,\mathbb{R})\times Sp(r,\mathbb{R}))$ (Japanese)

**Ryosuke Nakahama**(Kyushu University)Computation of weighted Bergman inner products on bounded symmetric domains and Plancherel-type formulas for $(Sp(2r,\mathbb{R}),Sp(r,\mathbb{R})\times Sp(r,\mathbb{R}))$ (Japanese)

[ Abstract ]

Let $(G,G_1)=(G,(G^\sigma)_0)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1:=(\mathfrak{p}^+)^\sigma\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)=\mathcal{O}_\lambda(D)$ on $D$ for sufficiently large $\lambda$. Its restriction to the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua--Kostant--Schmid--Kobayashi's formula in terms of the $\widetilde{K}_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on $\mathfrak{p}^+_2:=(\mathfrak{p}^+)^{-\sigma}\subset\mathfrak{p}^+$. Our goal is to understand the decomposition of the restriction $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}$ by studying the weighted Bergman inner product on each $\widetilde{K}_1$-type in $\mathcal{P}(\mathfrak{p}^+_2)\subset\mathcal{H}_\lambda(D)$.

Today we mainly deal with the symmetric pair $(G,G_1)=(Sp(2r,\mathbb{R}),Sp(r,\mathbb{R})\times Sp(r,\mathbb{R}))$.

Let $(G,G_1)=(G,(G^\sigma)_0)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1:=(\mathfrak{p}^+)^\sigma\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)=\mathcal{O}_\lambda(D)$ on $D$ for sufficiently large $\lambda$. Its restriction to the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua--Kostant--Schmid--Kobayashi's formula in terms of the $\widetilde{K}_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on $\mathfrak{p}^+_2:=(\mathfrak{p}^+)^{-\sigma}\subset\mathfrak{p}^+$. Our goal is to understand the decomposition of the restriction $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}$ by studying the weighted Bergman inner product on each $\widetilde{K}_1$-type in $\mathcal{P}(\mathfrak{p}^+_2)\subset\mathcal{H}_\lambda(D)$.

Today we mainly deal with the symmetric pair $(G,G_1)=(Sp(2r,\mathbb{R}),Sp(r,\mathbb{R})\times Sp(r,\mathbb{R}))$.