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

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 $\mathcal{P}\bigl(\mathfrak{p}^+_2\bigr)$ 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 $\mathcal{P}\bigl(\mathfrak{p}^+_2\bigr)\subset\mathcal{H}_\lambda(D)$. For example, by computing explicitly the norm $\Vert f\Vert_\lambda$ for $f=f(x_2)\in\mathcal{P}\bigl(\mathfrak{p}^+_2\bigr)$, 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 $\bigl\langle f(x_2),{\rm e}^{(x|\overline{z})_{\mathfrak{p}^+}}\bigr\rangle_{\lambda,x}$ for $f(x_2)\in\mathcal{P}\bigl(\mathfrak{p}^+_2\bigr)$, $x=(x_1,x_2)$, $z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$, 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 $\mathcal{P}\bigl(\mathfrak{p}^+_2\bigr)$.