Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups

TL;DR

This paper explicitly computes weighted Bergman inner products on bounded symmetric domains, derives branching laws for subgroup restrictions, and constructs intertwining operators for holomorphic discrete series representations.

Contribution

It provides explicit formulas for inner products and branching laws in the context of symmetric pairs and constructs explicit symmetry breaking operators.

Findings

Explicit computation of inner products involving polynomial functions and exponential kernels.

Derivation of branching laws for unitary representations restricted to subgroups.

Construction of explicit intertwining operators between holomorphic discrete series.

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 $\lambda$.