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

Abstract

Let (𝐺, 𝐺₁) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces 𝐷₁ = 𝐺₁/𝛫₁ ⊂ 𝐷 = 𝐺/𝛫, realized as bounded symmetric domains in complex vector spaces p⁺₁ ⊂ p⁺ respectively. Then the universal covering group 𝐺˜ of 𝐺 acts unitarily on the weighted Bergman space ℋλ(𝐷) ⊂ 𝒪(𝐷) on 𝐷. Its restriction to the subgroup 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 𝛫₁-decomposition of the space 𝒫(𝔭⁺₂) of polynomials on the orthogonal complement 𝔭⁺₂ of 𝔭⁺₁ in 𝔭⁺. The object of this article is to compute the inner product ⟨𝑓(𝑥₂),e⁽ˣ|ᶻ¯⁾𝔭⁺⟩λ explicitly for 𝑓(𝑥₂) ∈ 𝒫(𝔭⁺₂), 𝑥 = (𝑥₁, 𝑥₂), 𝑧 ∈ 𝔭⁺ = 𝔭⁺₁ ⊕ 𝔭⁺₂ explicitly. For example, when 𝔭⁺, 𝔭⁺₂ are of tube type and 𝑓(𝑥₂) = det(𝑥₂)ᵏ, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials ₂𝐹₁. Also, as an application, we construct 𝐺˜₁-intertwining operators (symmetry-breaking operators) ℋλ(𝐷)|𝐺˜₁ → ℋμ(𝐷₁) explicitly from holomorphic discrete series representations of 𝐺˜ to those of 𝐺˜₁, which are unique up to a constant multiple for sufficiently large λ.