Construction of Intertwining Operators between Holomorphic Discrete Series Representations
In this paper, we explicitly construct G₁-intertwining operators between holomorphic discrete series representations H of a Lie group G and those H₁ of a subgroup G₁⊂G when (G, G₁) is a symmetric pair of holomorphic type. More precisely, we construct G₁-intertwining projection operators from H onto...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2019 |
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Інститут математики НАН України
2019
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210186 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Construction of Intertwining Operators between Holomorphic Discrete Series Representations / R. Nakahama // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 51 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862736768772603904 |
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| author | Nakahama, R. |
| author_facet | Nakahama, R. |
| citation_txt | Construction of Intertwining Operators between Holomorphic Discrete Series Representations / R. Nakahama // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 51 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we explicitly construct G₁-intertwining operators between holomorphic discrete series representations H of a Lie group G and those H₁ of a subgroup G₁⊂G when (G, G₁) is a symmetric pair of holomorphic type. More precisely, we construct G₁-intertwining projection operators from H onto H₁ as differential operators, in the case (G, G₁)=(G₀×G₀, ΔG₀) and both H, H₁ are of scalar type, and also construct G₁-intertwining embedding operators from H₁ into H as infinite-order differential operators, in the case G is simple, H is of scalar type, and H₁ is multiplicity-free under a maximal compact subgroup K₁⊂K. In the actual computation, we make use of series expansions of integral kernels and the result of Faraut-Korányi (1990) or the author's previous result (2016) on norm computation. As an application, we observe the behavior of residues of the intertwining operators, which define the maps from some subquotient modules, when the parameters are at poles.
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| first_indexed | 2025-12-07T21:24:40Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210186 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:40Z |
| publishDate | 2019 |
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| spelling | Nakahama, R. 2025-12-03T14:30:31Z 2019 Construction of Intertwining Operators between Holomorphic Discrete Series Representations / R. Nakahama // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 51 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E45; 43A85; 17C30 arXiv: 1804.07100 https://nasplib.isofts.kiev.ua/handle/123456789/210186 https://doi.org/10.3842/SIGMA.2019.036 In this paper, we explicitly construct G₁-intertwining operators between holomorphic discrete series representations H of a Lie group G and those H₁ of a subgroup G₁⊂G when (G, G₁) is a symmetric pair of holomorphic type. More precisely, we construct G₁-intertwining projection operators from H onto H₁ as differential operators, in the case (G, G₁)=(G₀×G₀, ΔG₀) and both H, H₁ are of scalar type, and also construct G₁-intertwining embedding operators from H₁ into H as infinite-order differential operators, in the case G is simple, H is of scalar type, and H₁ is multiplicity-free under a maximal compact subgroup K₁⊂K. In the actual computation, we make use of series expansions of integral kernels and the result of Faraut-Korányi (1990) or the author's previous result (2016) on norm computation. As an application, we observe the behavior of residues of the intertwining operators, which define the maps from some subquotient modules, when the parameters are at poles. The author would like to thank his supervisor, Professor T. Kobayashi, for a lot of helpful advice on this paper. He also thanks Professor H. Ochiai for helpful advice on the structure of this paper, and his colleagues, especially Dr. M. Kitagawa, for a lot of helpful discussion. In addition, he would like to thank anonymous referees for a lot of helpful suggestions to improve this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Construction of Intertwining Operators between Holomorphic Discrete Series Representations Article published earlier |
| spellingShingle | Construction of Intertwining Operators between Holomorphic Discrete Series Representations Nakahama, R. |
| title | Construction of Intertwining Operators between Holomorphic Discrete Series Representations |
| title_full | Construction of Intertwining Operators between Holomorphic Discrete Series Representations |
| title_fullStr | Construction of Intertwining Operators between Holomorphic Discrete Series Representations |
| title_full_unstemmed | Construction of Intertwining Operators between Holomorphic Discrete Series Representations |
| title_short | Construction of Intertwining Operators between Holomorphic Discrete Series Representations |
| title_sort | construction of intertwining operators between holomorphic discrete series representations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210186 |
| work_keys_str_mv | AT nakahamar constructionofintertwiningoperatorsbetweenholomorphicdiscreteseriesrepresentations |