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On Harish-Chandra's integrability theorem in positive characteristic

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

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.

On Harish-Chandra's integrability theorem in positive characteristic

Abstract

The celebrated Harish-Chandra's integrability theorem states that the distributional character of an irreducible smooth representation of a p-adic group is integrable, that is represented by an function. Here is a non-Archimedean local field of characteristic and is a reductive algebraic group defined over . In this paper we focus on cuspidal representations of for a field 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 we establish the regularity of such characters unconditionally.
Paper Structure (58 sections, 79 theorems, 110 equations)

This paper contains 58 sections, 79 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Previous results
  5. The original argument of Harish-Chandra
  6. Difficulties with Harish-Chandra's argument in positive characteristic
  7. The approach of JL
  8. Results of AGKS2
  9. The approach of rod
  10. Our approach
  11. The role of the assumption $\mathbf{G}=\operatorname{GL}_n$
  12. Statements for the orbital integrals
  13. Unconditional results
  14. Summary of the logic of the paper
  15. Ideas of the proofs
  16. ...and 43 more sections

Key Result

Theorem 3

conj:2 implies conj:1.

Theorems & Definitions (155)

  • Conjecture 1
  • Conjecture 2
  • Theorem 3: S\ref{['Sec: proof of Theorem C and D']}
  • Proposition 4: S\ref{['Sec: proof of Theorem C and D']}
  • Remark 1.1.1
  • Remark 1.1.2
  • Theorem 5: \ref{['rem:intro.Omega1']}
  • Theorem \ref{thm:intro.Omega}'
  • Theorem 6: \ref{['rem:intro.Omega1']}
  • Theorem \ref{thm:Lie}'
  • ...and 145 more