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

TL;DR

This paper explicitly constructs symmetry breaking operators between holomorphic discrete series representations on bounded symmetric domains, using differential operators and polynomial inner products, focusing on tube type domains with equal ranks.

Contribution

It provides explicit formulas for intertwining operators in the case of simple tube type domains with equal rank, extending previous branching law results.

Findings

Constructed explicit symmetry breaking differential operators

Operators are unique up to scalar for large parameters

Handled cases with rank 3 and specific partitions in general rank

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 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 $\lambda$. These operators are given by differential operators whose symbols are computed as the inner products of polynomials on $\mathfrak{p}^+_2$. 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 $\mathbf{k}=(k,\ldots,k,k-l)$.