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A nonabelian Fourier transform for tempered unipotent representations

Anne-Marie Aubert, Dan Ciubotaru, Beth Romano

Abstract

We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple $p$-adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence of the conjecture, including proofs for $\mathsf{SL}_n$ and $\mathsf{PGL}_n$.

A nonabelian Fourier transform for tempered unipotent representations

Abstract

We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple -adic group and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence of the conjecture, including proofs for and .

Paper Structure

This paper contains 38 sections, 32 theorems, 270 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Notation and conventions
  5. Recollection on inner twists
  6. Inner twists
  7. The L-group
  8. Generalized Springer correspondence for disconnected groups
  9. The Langlands parametrization
  10. Langlands parameters
  11. Inertial classes
  12. Unipotent representations
  13. Lusztig's nonabelian Fourier transform for finite groups
  14. Fourier transforms
  15. Families of Weyl group representations
  16. ...and 23 more sections

Key Result

Theorem 1.1

Suppose that $G$ is split and adjoint. The local Langlands correspondence induces an isometric isomorphism where the left-hand side has a natural elliptic inner product while the right-hand side is endowed with the Euler--Poincaré product. The element $u$ ranges over representatives of unipotent conjugacy classes in $G^\vee$ and $\Pi(u,s,h)$ is defined in (eqn:pi_ush).

Theorems & Definitions (99)

  • Theorem 1.1: (Theorem \ref{['t:main-elliptic']})
  • Conjecture 1.2: (Conjecture \ref{['c:elliptic']})
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 3.1
  • Definition 3.2
  • Example 3.3
  • Remark 4.1
  • ...and 89 more