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2 papers
Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups
Ryosuke Nakahama Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku Fukuoka 819-0395, Japan
Abstract:
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
$\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ 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 compute explicitly the inner product $\big\langle f(x_2),{\rm e}^{(x|\overline{z})_{\mathfrak{p}^+}}\big\rangle_\lambda$ for
$f(x_2)\in\mathcal{P}(\mathfrak{p}^+_2)$,
$x=(x_1,x_2)$, $z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$. For example, when
$\mathfrak{p}^+$,
$\mathfrak{p}^+_2$ are of tube type and
$f(x_2)=\det(x_2)^k$, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials
${}_2F_1$. Also, as an application, we construct explicitly
$\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_\mu(D_1)$ from holomorphic discrete series representations of
$\widetilde{G}$ to those of
$\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large
$\lambda$.
Keywords:
weighted Bergman spaces, holomorphic discrete series representations, branching laws, intertwining operators, symmetry breaking operators, highest weight modules.
MSC: 22E45,
43A85,
17C30,
33C67 Received: June 14, 2021; in final form
April 6, 2022; Published online
May 3, 2022
Language: English
DOI:
10.3842/SIGMA.2022.033
Fulltext:
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