Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups - Nakahama, Ryosuke (Symmetry, Integrability and Geometry: Methods and Applications, 2022)
Let (, ₁) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces ₁ = ₁/₁ ⊂ = /, realized as bounded symmetric domains in complex vector spaces p⁺₁ ⊂ p⁺ respectively. Then the universal covering group ˜ of acts unitarily on the weighted Bergman space ℋλ() ⊂ () on . Its restriction to the subgroup ˜₁ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the ₁-decomposition of the space (⁺₂) of polynomials on the orthogonal complement ⁺₂ of ⁺₁ in ⁺. The object of this article is to compute the inner product ⟨(₂),e⁽ˣ|ᶻ¯⁾⁺⟩λ explicitly for (₂) ∈ (⁺₂), = (₁, ₂), ∈ ⁺ = ⁺₁ ⊕ ⁺₂ explicitly. For example, when ⁺, ⁺₂ are of tube type and (₂) = det(₂)ᵏ, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials ₂₁. Also, as an application, we construct ˜₁-intertwining operators (symmetry-breaking operators) ℋλ()|˜₁ → ℋμ(₁) explicitly from holomorphic discrete series representations of ˜ to those of ˜₁, which are unique up to a constant multiple for sufficiently large λ.