Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups II
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| Formato: | Preprint |
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2024
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| _version_ | 1866918135869734912 |
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| author | Nakahama, Ryosuke |
| author_facet | Nakahama, Ryosuke |
| contents | Let $(G,G_1)$ 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 $\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_λ(D)\subset\mathcal{O}(D)$ on $D$. 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 $K_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement $\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object of this article is to construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_λ(D)|_{\widetilde{G}_1}\to\mathcal{H}_{\varepsilon_1λ}(D_1,\mathcal{P}_{\mathbf{k}}(\mathfrak{p}^+_2))$ from holomorphic discrete series representations of $\widetilde{G}$ to those of $\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large $λ$. These operators are given by differential operators whose symbols are computed as the inner products of polynomials on $\mathfrak{p}^+_2$. In this article, we treat the case $\mathfrak{p}^+,\mathfrak{p}^+_2$ are both simple of tube type and $\operatorname{rank}\mathfrak{p}^+=\operatorname{rank}\mathfrak{p}^+_2$. When $\operatorname{rank}\mathfrak{p}^+=3$, we treat all partitions $\mathbf{k}$, and when $\operatorname{rank}\mathfrak{p}^+$ is general, we treat partitions of the form $\mathbf{k}=(k,\ldots,k,k-l)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01905 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups II Nakahama, Ryosuke Representation Theory 22E45, 43A85, 17C30, 33C67 Let $(G,G_1)$ 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 $\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_λ(D)\subset\mathcal{O}(D)$ on $D$. 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 $K_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement $\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object of this article is to construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_λ(D)|_{\widetilde{G}_1}\to\mathcal{H}_{\varepsilon_1λ}(D_1,\mathcal{P}_{\mathbf{k}}(\mathfrak{p}^+_2))$ from holomorphic discrete series representations of $\widetilde{G}$ to those of $\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large $λ$. These operators are given by differential operators whose symbols are computed as the inner products of polynomials on $\mathfrak{p}^+_2$. In this article, we treat the case $\mathfrak{p}^+,\mathfrak{p}^+_2$ are both simple of tube type and $\operatorname{rank}\mathfrak{p}^+=\operatorname{rank}\mathfrak{p}^+_2$. When $\operatorname{rank}\mathfrak{p}^+=3$, we treat all partitions $\mathbf{k}$, and when $\operatorname{rank}\mathfrak{p}^+$ is general, we treat partitions of the form $\mathbf{k}=(k,\ldots,k,k-l)$. |
| title | Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups II |
| topic | Representation Theory 22E45, 43A85, 17C30, 33C67 |
| url | https://arxiv.org/abs/2406.01905 |