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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• Main Author: Nakahama, Ryosuke
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211982
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups. Ryosuke Nakahama. SIGMA 19 (2023), 049, 74 pages

Let (, ₁) = (, (σ)₀) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces D₁ = ₁/₁ ⊂ D = /, realized as bounded symmetric domains in complex vector spaces p⁺₁ := (p⁺)σ ⊂ p⁺ respectively. Then the universal covering group ˜ of acts unitarily on the weighted Bergman space ℋλ() ⊂ () = λ() on D for sufficiently large λ. 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 ⁺₂ := (⁺)⁻σ ⊂ ⁺. The object of this article is to understand the decomposition of the restriction ℋλ()|˜1 by studying the weighted Bergman inner product on each ˜₁-type in (⁺₂) ⊂ ℋλ(). For example, by computing the norm ∥∥λ for = (₂) ∈ (⁺₂) explicitly, we can determine the Parseval-Plancherel-type formula for the decomposition of ℋλ()|˜₁. Also, by computing the poles of ⟨(₂),e(ˣ|ᶻ¯)⁺⟩λ,ₓ for (₂) ∈ (⁺₂), = (₁, ₂), z ∈ ⁺ = ⁺₁ ⊕ p⁺₂, we can get some information on the branching of λ()|˜₁ also for λ in the non-unitary range. In this article, we consider these problems for all ˜₁-types in (⁺₂).