Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups II (2406.01905v2)

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 $\bigl\langle f(x_2),e^{(x|\overline{z}){\mathfrak{p}^+}}\bigr\rangle_\lambda$ for $f(x_2)\in\mathcal{P}{\mathbf{k}}(\mathfrak{p}^+_2)\subset\mathcal{P}(\mathfrak{p}^+_2)$, $x=(x_1,x_2),z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$, and to construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}\lambda(D)|{\widetilde{G}_1}\to\mathcal{H}{\varepsilon_1\lambda}(D_1,\mathcal{P}_{\mathbf{k}}(\mathfrak{p}^+_2))$ from holomorphic discrete series representations of $\widetilde{G}$ to those of $\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large $\lambda$. In this article, we treat the case $\mathfrak{p}^+,\mathfrak{p}^+_2$ are both simple of tube type and $\operatorname{rank}\mathfrak{p}^+=\operatorname{rank}\mathfrak{p}^+_2$. When $\operatorname{rank}\mathfrak{p}^+=3$, we treat all partitions $\mathbf{k}$, and when $\operatorname{rank}\mathfrak{p}^+$ is general, we treat partitions of the form $\mathbf{k}=(k,\ldots,k,k-l)$.