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

Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider Hermitian spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded domains in complex vector $\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ $G$ acts unitarily on weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ $D$. Its restriction to subgroup $\widetilde{G}_1$ decomposes discretely multiplicity-freely, its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula terms $K_1$-decomposition $\mathcal{P}(\mathfrak{p}^+_2)$ polynomials orthogonal complement $\mathfrak{p}^+_2$ $\mathfrak{p}^+_1$ $\mathfrak{p}^+$. The object this article compute 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 $\mathfrak{p}^+$, are tube type $f(x_2)=\det(x_2)^k$, introducing multivariate generalization Gauss' hypergeometric ${}_2F_1$. Also, an application, construct $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_\mu(D_1)$ from discrete series representations those $\widetilde{G}_1$, which unique up constant multiple sufficiently large $\lambda$.