Uniform bounds and uncertainty for asymptotics of representations of $p$-adic ${\rm GL}_N$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918202020200448 |
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| author | Dalal, Rahul Gerbelli-Gauthier, Mathilde Marshall, Simon |
| author_facet | Dalal, Rahul Gerbelli-Gauthier, Mathilde Marshall, Simon |
| contents | We prove two results on the growth of dimensions of fixed vectors of representations $π$ of $p$-adic ${\rm GL}_N$ under principal congruence subgroups: First, a uniform bound on the growth of fixed vectors in terms of the GK-dimension $π$, which we extend to a uniform bound on the Harish-Chandra--Howe coefficients. Second, for $π$ unitary, a quantitative relationship between the GK-dimension of $π$ and the rate of decay of its matrix coefficients. These results are independent of one another and proved in the framework of the Langlands and Zelevinsky classifications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11063 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform bounds and uncertainty for asymptotics of representations of $p$-adic ${\rm GL}_N$ Dalal, Rahul Gerbelli-Gauthier, Mathilde Marshall, Simon Representation Theory Number Theory 20G05, 11F70 We prove two results on the growth of dimensions of fixed vectors of representations $π$ of $p$-adic ${\rm GL}_N$ under principal congruence subgroups: First, a uniform bound on the growth of fixed vectors in terms of the GK-dimension $π$, which we extend to a uniform bound on the Harish-Chandra--Howe coefficients. Second, for $π$ unitary, a quantitative relationship between the GK-dimension of $π$ and the rate of decay of its matrix coefficients. These results are independent of one another and proved in the framework of the Langlands and Zelevinsky classifications. |
| title | Uniform bounds and uncertainty for asymptotics of representations of $p$-adic ${\rm GL}_N$ |
| topic | Representation Theory Number Theory 20G05, 11F70 |
| url | https://arxiv.org/abs/2511.11063 |