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

Abstract

Let $(G,G_1)=(G,(G^{\sigma }{)}_0)$ 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 ${\mathfrak{p}}_1^{+}:=({\mathfrak{p}}^{+}{)}^{\sigma }\subset {\mathfrak{p}}^{+}$ respectively. Then the universal covering group $\tilde{G}$ of $G$ acts unitarily on the weighted Bergman space ${\mathcal{H}}_{\lambda }(D)\subset \mathcal{O}(D)={\mathcal{O}}_{\lambda }(D)$ on $D$ for sufficiently large $\lambda$. 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 ${\tilde{K}}_1$-decomposition of the space $\mathcal{P}({\mathfrak{p}}_2^{+})$ of polynomials on ${\mathfrak{p}}_2^{+}:=({\mathfrak{p}}^{+}{)}^{-\sigma }\subset {\mathfrak{p}}^{+}$. The object of this article is to understand the decomposition of the restriction ${\mathcal{H}}_{\lambda }(D){|}_{{\tilde{G}}_1}$ by studying the weighted Bergman inner product on each ${\tilde{K}}_1$-type in $\mathcal{P}({\mathfrak{p}}_2^{+})\subset {\mathcal{H}}_{\lambda }(D)$. For example, by computing explicitly the norm $\Vert f{\Vert }_{\lambda }$ for $f=f(x_2)\in \mathcal{P}({\mathfrak{p}}_2^{+})$, we can determine the Parseval-Plancherel-type formula for the decomposition of ${\mathcal{H}}_{\lambda }(D){|}_{{\tilde{G}}_1}$. Also, by computing the poles of $⟨f(x_2),{\mathrm{e}}^{(x|\overline{z}{)}_{{\mathfrak{p}}^{+}}}{⟩}_{\lambda ,x}$ for $f(x_2)\in \mathcal{P}({\mathfrak{p}}_2^{+})$, $x=(x_1,x_2)$, $z\in {\mathfrak{p}}^{+}={\mathfrak{p}}_1^{+}\oplus {\mathfrak{p}}_2^{+}$, we can get some information on branching of ${\mathcal{O}}_{\lambda }(D){|}_{{\tilde{G}}_1}$ also for $\lambda$ in non-unitary range. In this article we consider these problems for all ${\tilde{K}}_1$-types in $\mathcal{P}({\mathfrak{p}}_2^{+})$.

References in corpus (2)

• Branching of some holomorphic representations of SO(2,N)
• Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups