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On certain identities between Fourier transforms of weighted orbital integrals on infinitesimal symmetric spaces of Guo-Jacquet

Huajie Li

Abstract

In an infinitesimal variant of Guo-Jacquet trace formulae, the regular semi-simple terms are expressed as noninvariant weighted orbital integrals on two global infinitesimal symmetric spaces. We prove some relations between the Fourier transforms of invariant weighted orbital integrals on the corresponding local infinitesimal symmetric spaces. These relations should be useful in the noninvariant comparison of the infinitesimal variant of Guo-Jacquet trace formulae.

On certain identities between Fourier transforms of weighted orbital integrals on infinitesimal symmetric spaces of Guo-Jacquet

Abstract

In an infinitesimal variant of Guo-Jacquet trace formulae, the regular semi-simple terms are expressed as noninvariant weighted orbital integrals on two global infinitesimal symmetric spaces. We prove some relations between the Fourier transforms of invariant weighted orbital integrals on the corresponding local infinitesimal symmetric spaces. These relations should be useful in the noninvariant comparison of the infinitesimal variant of Guo-Jacquet trace formulae.

Paper Structure

This paper contains 45 sections, 56 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Groups
  4. Symmetric pairs
  5. The case of $(G,H)$
  6. The case of $(G',H')$
  7. Labesse's lemma
  8. Matching of orbits
  9. Definition by invariants
  10. Matching orbits
  11. Matching Levi subgroups
  12. $M$-matching orbits
  13. Criterion via the Kottwitz sign
  14. Centralisers of semi-simple elements
  15. Classification of $E\hookrightarrow{\mathfrak{g}}'(F)$
  16. ...and 30 more sections

Key Result

Theorem 1.1

Theorems & Definitions (116)

  • Theorem 1.1: see Corollary \ref{['parcommutecor1']} and Proposition \ref{['vancommute']}
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • ...and 106 more