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

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 compute explicitly the inner product $⟨f(x_2),{\mathrm{e}}^{(x|\overline{z}{)}_{{\mathfrak{p}}^{+}}}{⟩}_{\lambda }$ 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 ${\mathfrak{p}}^{+}$, ${\mathfrak{p}}_2^{+}$ are of tube type and $f(x_2)=\det (x_2{)}^k$, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials ${}_2{F}_1$. Also, as an application, we construct explicitly ${\tilde{G}}_1$-intertwining operators (symmetry breaking operators) ${\mathcal{H}}_{\lambda }(D){|}_{{\tilde{G}}_1}\to {\mathcal{H}}_{\mu }(D_1)$ from holomorphic discrete series representations of $\tilde{G}$ to those of ${\tilde{G}}_1$, which are unique up to constant multiple for sufficiently large $\lambda$.

References in corpus (2)

• Exceptional Lie groups
• Norm computation and analytic continuation of vector valued holomorphic discrete series representations

Cited by in corpus (1)

• Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups