Regularity of K-finite matrix coefficients of semisimple Lie groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912024667095040 |
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| author | Dumas, Guillaume |
| author_facet | Dumas, Guillaume |
| contents | We consider $G$ a semisimple Lie group with finite center and $K$ a maximal compact subgroup of $G$. We study the regularity of $K$-finite matrix coefficients of unitary representations of $G$. More precisely, we find the optimal value $κ(G)$ such that all such coefficients are $κ(G)$-Hölder continuous. The proof relies on analysis of spherical functions of the symmetric Gelfand pair $(G,K)$, using stationary phase estimates from Duistermaat, Kolk and Varadarajan. If $U$ is a compact form of $G$, then $(U,K)$ is a compact symmetric pair. Using the same tools, we study the regularity of $K$-finite coefficients of unitary representations of $U$, improving on previous results obtained by the author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07944 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity of K-finite matrix coefficients of semisimple Lie groups Dumas, Guillaume Group Theory Functional Analysis 22E46 (Primary) 43A85, 43A90 (Secondary) We consider $G$ a semisimple Lie group with finite center and $K$ a maximal compact subgroup of $G$. We study the regularity of $K$-finite matrix coefficients of unitary representations of $G$. More precisely, we find the optimal value $κ(G)$ such that all such coefficients are $κ(G)$-Hölder continuous. The proof relies on analysis of spherical functions of the symmetric Gelfand pair $(G,K)$, using stationary phase estimates from Duistermaat, Kolk and Varadarajan. If $U$ is a compact form of $G$, then $(U,K)$ is a compact symmetric pair. Using the same tools, we study the regularity of $K$-finite coefficients of unitary representations of $U$, improving on previous results obtained by the author. |
| title | Regularity of K-finite matrix coefficients of semisimple Lie groups |
| topic | Group Theory Functional Analysis 22E46 (Primary) 43A85, 43A90 (Secondary) |
| url | https://arxiv.org/abs/2409.07944 |