Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups
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() ⊂ ()...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2022
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211635 |
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| 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 Restriction to Subgroups. Ryosuke Nakahama. SIGMA 18 (2022), 033, 105 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862545153335492608 |
|---|---|
| author | Nakahama, Ryosuke |
| author_facet | Nakahama, Ryosuke |
| citation_txt | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups. Ryosuke Nakahama. SIGMA 18 (2022), 033, 105 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2026-03-12T23:35:05Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211635 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T23:35:05Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nakahama, Ryosuke 2026-01-07T13:42:36Z 2022 Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups. Ryosuke Nakahama. SIGMA 18 (2022), 033, 105 pages 1815-0659 2020 Mathematics Subject Classification: 22E45; 43A85; 17C30; 33C67 arXiv:2105.13976 https://nasplib.isofts.kiev.ua/handle/123456789/211635 https://doi.org/10.3842/SIGMA.2022.033 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. The author would like to thank Professor T. Kobayashi for a lot of helpful advice on this research, and also thank Professor H. Ochiai for a lot of helpful comments on this paper. This work was supported by the Grant-in-Aid 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 Restriction to Subgroups Article published earlier |
| spellingShingle | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups Nakahama, Ryosuke |
| title | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups |
| title_full | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups |
| title_fullStr | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups |
| title_full_unstemmed | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups |
| title_short | Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups |
| title_sort | computation of weighted bergman inner products on bounded symmetric domains and restriction to subgroups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211635 |
| work_keys_str_mv | AT nakahamaryosuke computationofweightedbergmaninnerproductsonboundedsymmetricdomainsandrestrictiontosubgroups |