Cartan motion groups: regularity of K-finite matrix coefficients
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910069484945408 |
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| author | Dumas, Guillaume |
| author_facet | Dumas, Guillaume |
| contents | If $G$ is a connected semisimple Lie group with finite center and $K$ is a maximal compact subgroup of G, then the Lie algebra of $G$ admits a Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$. This allows us to define the Cartan motion group $H=\mathfrak{p}\rtimes K$. In this paper, we study the regularity of $K$-finite matrix coefficients of unitary representations of $H$. We prove that the optimal exponent $κ(G)$ for which all such coefficients are $κ(G)$-Hölder continuous coincides with the optimal regularity of all $K$-finite coefficients of the group $G$ itself. Our approach relies on stationary phase techniques that were previously employed by the author to study regularity in the setting of $(G,K)$. Furthermore, we provide a general framework to reduce the question of regularity from $K$-finite coefficients to $K$-bi-invariant coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04368 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cartan motion groups: regularity of K-finite matrix coefficients Dumas, Guillaume Group Theory Functional Analysis Representation Theory 22E46 (Primary) 43A85, 43A90 (Secondary) If $G$ is a connected semisimple Lie group with finite center and $K$ is a maximal compact subgroup of G, then the Lie algebra of $G$ admits a Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$. This allows us to define the Cartan motion group $H=\mathfrak{p}\rtimes K$. In this paper, we study the regularity of $K$-finite matrix coefficients of unitary representations of $H$. We prove that the optimal exponent $κ(G)$ for which all such coefficients are $κ(G)$-Hölder continuous coincides with the optimal regularity of all $K$-finite coefficients of the group $G$ itself. Our approach relies on stationary phase techniques that were previously employed by the author to study regularity in the setting of $(G,K)$. Furthermore, we provide a general framework to reduce the question of regularity from $K$-finite coefficients to $K$-bi-invariant coefficients. |
| title | Cartan motion groups: regularity of K-finite matrix coefficients |
| topic | Group Theory Functional Analysis Representation Theory 22E46 (Primary) 43A85, 43A90 (Secondary) |
| url | https://arxiv.org/abs/2502.04368 |