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arXiv · 2602.16389

On Harish-Chandra's integrability theorem in positive characteristic

Avraham Aizenbud·Dmitry Gourevitch·David Kazhdan·Eitan Sayag·Itay Glazer·Yotam Hendel
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Abstract

The celebrated Harish-Chandra's integrability theorem states that the distributional character of an irreducible smooth representation of a p-adic group $G(F)$ is integrable, that is represented by an $L^1_{loc}(G(F))$ function. Here $F$ is a non-Archimedean local field of characteristic $0$ and $G$ is a reductive algebraic group defined over $F$. In this paper we focus on cuspidal representations of $GL_n(F)$ for a field $F$ of positive characteristic. We show that in this case the integrability holds under the hypothesis of existence of desingularization of (certain) algebraic varieties in positive characteristics. Furthermore, in the case $char(F)>n/2$ we establish the regularity of such characters unconditionally.

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Avraham Aizenbud, Dmitry Gourevitch, David Kazhdan, Eitan Sayag, Itay Glazer, Yotam Hendel. 2026-02-18. On Harish-Chandra's integrability theorem in positive characteristic. https://arxiv.org/abs/2602.16389

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