Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The stable wave front set of theta representations

Edmund Karasiewicz, Emile Okada, Runze Wang

Abstract

We compute the stable wave front set of theta representations for certain tame Brylinski-Deligne covers of a connected reductive $p$-adic group. The computation involves two main inputs. First we use a theorem of Okada, adapted to covering groups, to reduce the computation of the wave front set to computing the Kawanaka wave front set of certain representations of finite groups of Lie type. Second, to compute the Kawanaka wave front sets we use Lusztig's formula. This requires a careful analysis of the action of the pro-$p$ Iwahori-Hecke algebra on the theta representation, using the structural results about Hecke algebras developed by Gao-Gurevich-Karasiewicz and Wang.

The stable wave front set of theta representations

Abstract

We compute the stable wave front set of theta representations for certain tame Brylinski-Deligne covers of a connected reductive -adic group. The computation involves two main inputs. First we use a theorem of Okada, adapted to covering groups, to reduce the computation of the wave front set to computing the Kawanaka wave front set of certain representations of finite groups of Lie type. Second, to compute the Kawanaka wave front sets we use Lusztig's formula. This requires a careful analysis of the action of the pro- Iwahori-Hecke algebra on the theta representation, using the structural results about Hecke algebras developed by Gao-Gurevich-Karasiewicz and Wang.

Paper Structure

This paper contains 32 sections, 33 theorems, 97 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Covering Groups
  5. Modified Root Datum
  6. Tame Covers
  7. Parahoric subgroups
  8. Hecke Algebras
  9. Moy-Prasad filtration and the exponential map
  10. DeBacker's lifting map and stable orbits
  11. Character distributions and the local character expansion
  12. The wave front set and test functions
  13. Theta Representations
  14. Basic properties of theta representations
  15. The Kawanaka wave front set of semisimple principal series representations
  16. ...and 17 more sections

Key Result

Theorem 1.1

Suppose that hypotheses hyp:exp, hyp:lift and hyp:bilin hold. This holds for example if the residue characteristic is larger than some constant depending on the absolute root datum of $\mathbf G$ and the ramification index of $F/\mathbb Q_p$. Fix $\pi^{\dagger}$ a distinguished genuine irreducible $

Figures (12)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 7 more figures

Theorems & Definitions (68)

  • Theorem 1.1: Theorem \ref{['WFThetaThm']}
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 58 more