Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups
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 B...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2023 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211982 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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 |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862742725223251968 |
|---|---|
| author | Nakahama, Ryosuke |
| author_facet | Nakahama, Ryosuke |
| citation_txt | 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 |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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).
|
| first_indexed | 2026-04-17T18:18:15Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211982 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T18:18:15Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nakahama, Ryosuke 2026-01-20T16:14:17Z 2023 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 1815-0659 2020 Mathematics Subject Classification: 22E45;43A85;17C30 arXiv:2207.11663 https://nasplib.isofts.kiev.ua/handle/123456789/211982 https://doi.org/10.3842/SIGMA.2023.049 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). The author would like to thank Professor T. Kobayashi for much helpful advice on this research. Grant-in-Aid supported this work for JSPS Fellows Grant Number JP20J00114. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups Article published earlier |
| spellingShingle | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups Nakahama, Ryosuke |
| title | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups |
| title_full | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups |
| title_fullStr | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups |
| title_full_unstemmed | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups |
| title_short | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups |
| title_sort | computation of weighted bergman inner products on bounded symmetric domains and parseval-plancherel-type formulas under subgroups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211982 |
| work_keys_str_mv | AT nakahamaryosuke computationofweightedbergmaninnerproductsonboundedsymmetricdomainsandparsevalplanchereltypeformulasundersubgroups |