Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups II

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 $p_1^{+}\subset p^{+}$ respectively. Then the universal covering group $\tilde{G}$ of G acts unitarily on the weighted Bergman space $H_{\lambda }\left(D\right)\subset O\left(D\right)$ on D. Its restriction to the subgroup ${\tilde{G}}_1$ 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 $P\left(p_2^{+}\right)$ of polynomials on the orthogonal complement $p_2^{+}$ of $p_1^{+}$ in $p^{+}$. The object of this article is to construct explicitly ${\tilde{G}}_1$-intertwining operators (symmetry breaking operators) $H_{\lambda }\left(D\right){|}_{{\tilde{G}}_1}\to H_{{\epsilon }_1\lambda }(D_1,P_k(p_2^{+}\left)\right)$ from holomorphic discrete series representations of $\tilde{G}$ to those of ${\tilde{G}}_1$, which are unique up to constant multiple for sufficiently large λ. These operators are given by differential operators whose symbols are computed as the inner products of polynomials on $p_2^{+}$. In this article, we treat the case $p^{+},p_2^{+}$ are both simple of tube type and $\mathrm{rank}{p}^{+}=\mathrm{rank}{p}_2^{+}$. When $\mathrm{rank}{p}^{+}=3$, we treat all partitions k, and when $\mathrm{rank}{p}^{+}$ is general, we treat partitions of the form $k=(k,\dots ,k,k-l)$. This article is a continuation of the author's previous article [38].