On Harish-Chandra's integrability theorem in positive characteristic
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914336749912064 |
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| author | Aizenbud, Avraham Gourevitch, Dmitry Kazhdan, David Sayag, Eitan Glazer, Itay Hendel, Yotam |
| author_facet | Aizenbud, Avraham Gourevitch, Dmitry Kazhdan, David Sayag, Eitan Glazer, Itay Hendel, Yotam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16389 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Harish-Chandra's integrability theorem in positive characteristic Aizenbud, Avraham Gourevitch, Dmitry Kazhdan, David Sayag, Eitan Glazer, Itay Hendel, Yotam Representation Theory 14L30, 20G25, 46F10, 14B05, 14B10, 14E15 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. |
| title | On Harish-Chandra's integrability theorem in positive characteristic |
| topic | Representation Theory 14L30, 20G25, 46F10, 14B05, 14B10, 14E15 |
| url | https://arxiv.org/abs/2602.16389 |