Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups

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 ${\mathfrak p}^+_2:=({\mathfrak p}^+)^{-\sigma}\subset{\mathfrak p}^+$. The object of this article 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 ${\mathcal P}({\mathfrak p}^+_2)\subset{\mathcal H}_\lambda(D)$. For example, by computing explicitly the norm $\Vert f\Vert_\lambda$ for $f=f(x_2)\in{\mathcal P}({\mathfrak p}^+_2)$, we can determine the Parseval-Plancherel-type formula for the decomposition of ${\mathcal H}_\lambda(D)|_{\widetilde{G}_1}$. Also, by computing the poles of $\langle f(x_2),{\rm e}^{(x|\overline{z})_{{\mathfrak p}^+}}\rangle_{\lambda,x}$ for $f(x_2)\in{\mathcal P}({\mathfrak p}^+_2)$, $x=(x_1,x_2)$, $z\in{\mathfrak p}^+={\mathfrak p}^+_1\oplus{\mathfrak p}^+_2$, we can get some information on branching of ${\mathcal O}_\lambda(D)|_{\widetilde{G}_1}$ also for $\lambda$ in non-unitary range. In this article we consider these problems for all $\widetilde{K}_1$-types in ${\mathcal P}({\mathfrak p}^+_2)$.