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Orbital integral bounds the character for cuspidal representations of $GL_n(\mathbb{F}_{\ell}((t)))$

Avraham Aizenbud, Dmitry Gourevitch, David Kazhdan, Eitan Sayag

Abstract

We prove that the character of an irreducible cuspidal representation of $GL_n(\mathbb{F}_{\ell}((t)))$ is locally bounded up to a logarithmic factor by the orbital integral of a matrix coefficient of this representation. The characteristic $0$ analog of this result is part of the proof of the celebrated Harish-Chandra's integrability theorem. In a sequel work [AGKS] we use this result in order to prove a positive characteristic analog of Harish-Chandra's integrability theorem under some additional assumptions.

Orbital integral bounds the character for cuspidal representations of $GL_n(\mathbb{F}_{\ell}((t)))$

Abstract

We prove that the character of an irreducible cuspidal representation of is locally bounded up to a logarithmic factor by the orbital integral of a matrix coefficient of this representation. The characteristic analog of this result is part of the proof of the celebrated Harish-Chandra's integrability theorem. In a sequel work [AGKS] we use this result in order to prove a positive characteristic analog of Harish-Chandra's integrability theorem under some additional assumptions.
Paper Structure (34 sections, 54 theorems, 94 equations)

This paper contains 34 sections, 54 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Orbital integrals
  3. Main results
  4. Background and motivation
  5. Idea of the proof
  6. Idea of the proof of \ref{['thm:Om.bnd.Av']}
  7. Comparison to the characteristic zero case
  8. The role of the assumption $\mathbf{G}=\operatorname{GL}_n$
  9. The Lie algebra case
  10. Structure of the paper
  11. Acknowledgments
  12. Conventions and Notation
  13. Conventions
  14. Notations
  15. Weak convergence of the averaging to the character - Proof of \ref{['prop:ChartAv']}
  16. ...and 19 more sections

Key Result

Theorem 1

Let $\rho$ be a cuspidal irreducible representation of $G$. Let $m$ be a matrix coefficient of $\rho$$m(1)\neq 0$. Then there exists a polynomial $\textcolor{DefColor}{\alpha^{\rho{{,m}}}} \IfNoValueTF{-NoValue-}{}{} \in \mathbb{N}[t]$ such that for every $\eta\in C^{\infty}_c(G)$ we have where $f \in C^{\infty}(G^{rss})$ is defined by $f(g)=\alpha^{\rho{{,m}}}(ov_{G^{rss}}(g)).$

Theorems & Definitions (94)

  • Theorem 1: S \ref{['subsec:Idea']}
  • Remark
  • Definition 1.4.1
  • Theorem 1.4.2: S\ref{['ssec:pf.ChartAv']}, cf. HC_VD
  • Definition 1.4.3
  • Theorem 2: S\ref{['subsec:PfStab']}
  • Theorem 3: S \ref{['subsec:PfStab']}
  • Theorem 4: S \ref{['subsec:Pf.bnd.A']}
  • Theorem 1.5.1: cf. HC_VD
  • Theorem 1.5.2: cf HC_VD
  • ...and 84 more