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 ${\mathfrak{p}}_1^{+}\subset {\mathfrak{p}}^{+}$ respectively. Then the universal covering group $\tilde{G}$ of $G$ acts unitarily on the weighted Bergman space ${\mathcal{H}}_{\lambda }(D)\subset \mathcal{O}(D)$ on $D$. Its restriction to the subgroup ${\tilde{G}}_1$ 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 $\mathcal{P}({\mathfrak{p}}_2^{+})$ 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 ${\tilde{G}}_1$-intertwining operators (symmetry breaking operators) ${\mathcal{H}}_{\lambda }(D){|}_{{\tilde{G}}_1}\to {\mathcal{H}}_{{\epsilon }_1\lambda }(D_1,{\mathcal{P}}_{\mathbf{k}}({\mathfrak{p}}_2^{+}))$ from holomorphic discrete series representations of $\tilde{G}$ to those of ${\tilde{G}}_1$, 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 $\mathrm{rank}{\mathfrak{p}}^{+}=\mathrm{rank}{\mathfrak{p}}_2^{+}$. When $\mathrm{rank}{\mathfrak{p}}^{+}=3$, we treat all partitions $\mathbf{k}$, and when $\mathrm{rank}{\mathfrak{p}}^{+}$ is general, we treat partitions of the form $\mathbf{k}=(k,\dots ,k,k-l)$.

78 pages. arXiv admin note: text overlap with arXiv:2207.11663