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Relative Fundamental Lemmas for Spherical Hecke Algebras and Multiplicative Hitchin Fibrations: the Jacquet--Rallis Case

X. Griffin Wang, Zhiyu Zhang

Abstract

We prove the Jacquet--Rallis fundamental lemma for spherical Hecke algebras over local function fields using multiplicative Hitchin fibrations. Our work is inspired by the proof of [Yun11] in the Lie algebra case and builds upon the general framework of multiplicative Hitchin fibrations in [Wang26].

Relative Fundamental Lemmas for Spherical Hecke Algebras and Multiplicative Hitchin Fibrations: the Jacquet--Rallis Case

Abstract

We prove the Jacquet--Rallis fundamental lemma for spherical Hecke algebras over local function fields using multiplicative Hitchin fibrations. Our work is inspired by the proof of [Yun11] in the Lie algebra case and builds upon the general framework of multiplicative Hitchin fibrations in [Wang26].
Paper Structure (37 sections, 63 theorems, 423 equations)

This paper contains 37 sections, 63 theorems, 423 equations.

Table of Contents

  1. Introduction
  2. Jacquet--Rallis fundamental lemmas for spherical Hecke algebras
  3. Applications
  4. Sketch of the proof
  5. Structure of the paper
  6. Acknowledgement
  7. Notations
  8. Monoids and Relative Invariant Theory
  9. The setup
  10. Review of the usual adjoint action
  11. The H-action and an explicit section from the GIT quotient
  12. Some examples
  13. The discriminant divisor
  14. The involutions and twisted forms
  15. Deformed quotient stack
  16. ...and 22 more sections

Key Result

Theorem 1.1.1

Let $n\ge 1$ be an integer and $F=k\lauser{\pi}$ be a local function field with $\mathop{\mathrm{char}}\nolimits(k)>2n$. For any strongly regular semisimple $A\in \mathrm{S}_n(F)$ matching $A'\in G'(F)$ and any pair of spherical functions $(f,f')\in {\mathcal{H}}_0\times{\mathcal{H}}_0'$ matched und

Theorems & Definitions (135)

  • Theorem 1.1.1: Jacquet--Rallis fundamental lemma for spherical Hecke algebras, \ref{['thm:main_theorem']}
  • Theorem 2.3.3
  • proof
  • Example 2.4.1
  • Example 2.4.2
  • Remark 2.4.3
  • Lemma 2.5.3
  • proof
  • Corollary 2.5.4
  • proof
  • ...and 125 more