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arXiv · 2208.05139

Gelfand-Kirillov dimension of representations of $\mathrm{GL}_n$ over a non-archimedean local field

Kenta Suzuki
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Abstract

We calculate the asymptotic behavior of the dimension of the fixed vectors of $\pi$ with respect to compact open subgroups $1+ M_n(\mathfrak{p}^N)\subset\mathrm{GL}_n(F)$ for $\pi$ an admissible representation of $\mathrm{GL}_n(F)$, and $F$ a nonarchimedean local field. Such dimensions can be calculated by germs of the character of $\pi$. We also make some observations on how those dimensions behave under instances of Langlands functoriality, such as the Jacquet-Langlands correspondence and cyclic base change, where relations between characters are known.

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BibTeXRIS

Kenta Suzuki. 2022-08-10. Gelfand-Kirillov dimension of representations of $\mathrm{GL}_n$ over a non-archimedean local field. https://arxiv.org/abs/2208.05139

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