Regularity of matrix coefficients of a compact symmetric pair of Lie groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913366014951424 |
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| author | Dumas, Guillaume |
| author_facet | Dumas, Guillaume |
| contents | We consider symmetric Gelfand pairs $(G,K)$ where $G$ is a compact Lie group and $K$ a subgroup of fixed point of an involutive automorphism. We study the regularity of $K$-bi-invariant matrix coefficients of $G$. The results rely on the analysis of the spherical functions of the Gelfand pair $(G,K)$. When the symmetric space $G/K$ is of rank $1$ or isomorphic to a Lie group, we find the optimal regularity of $K$-bi-invariant matrix coefficients. Furthermore, in rank $1$ we also show the optimal regularity of $K$-bi-invariant Herz-Schur multipliers of $S_p(L^2(G))$. We also give a lower bound for the optimal regularity in some families of higher rank symmetric spaces. From these results, we make a conjecture in the general case involving the root system of the symmetric space. Finally, we prove that if all $K$-bi-invariant matrix coefficients of $G$ have the same regularity, then so do all $K$-finite matrix coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_09291 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regularity of matrix coefficients of a compact symmetric pair of Lie groups Dumas, Guillaume Functional Analysis Group Theory Operator Algebras We consider symmetric Gelfand pairs $(G,K)$ where $G$ is a compact Lie group and $K$ a subgroup of fixed point of an involutive automorphism. We study the regularity of $K$-bi-invariant matrix coefficients of $G$. The results rely on the analysis of the spherical functions of the Gelfand pair $(G,K)$. When the symmetric space $G/K$ is of rank $1$ or isomorphic to a Lie group, we find the optimal regularity of $K$-bi-invariant matrix coefficients. Furthermore, in rank $1$ we also show the optimal regularity of $K$-bi-invariant Herz-Schur multipliers of $S_p(L^2(G))$. We also give a lower bound for the optimal regularity in some families of higher rank symmetric spaces. From these results, we make a conjecture in the general case involving the root system of the symmetric space. Finally, we prove that if all $K$-bi-invariant matrix coefficients of $G$ have the same regularity, then so do all $K$-finite matrix coefficients. |
| title | Regularity of matrix coefficients of a compact symmetric pair of Lie groups |
| topic | Functional Analysis Group Theory Operator Algebras |
| url | https://arxiv.org/abs/2305.09291 |