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On two definitions of wave-front sets for $p$-adic groups

Cheng-Chiang Tsai

TL;DR

The paper investigates two definitions of wave-front sets for irreducible admissible representations of p-adic groups, contrasting WF^rat (analytic closure) with WF^Zar (Zariski closure). Using a concrete Sp_4 construction, it builds an epipelagic representation via compact induction from a depth -1/2 Moy-Prasad datum and analyzes Shalika germs to show WF^rat contains a subregular nilpotent orbit in addition to two regular orbits, whereas WF^Zar contains only the two regular orbits, providing a counterexample to their equivalence. This highlights subtle distinctions between analytic and geometric closures in wave-front theory and informs broader conjectures on geometric wave-front sets. The paper also details the Langlands parameters for the representation components, giving a explicit decomposition rho = rho1 ⊕ rho2 ⊕ triv with rho_j = Ind_{W_E}^{W_F} chi_j for E = F(√π), enumerating four candidate parameters.

Abstract

The wave-front set for an irreducible admissible representation of a $p$-adic reductive group is the set of maximal nilpotent orbits which appear in the local character expansion. By Mœglin-Waldspurger, they are also the maximal nilpotent orbits whose associated degenerate Whittaker models are non-zero. However, in the literature there are two versions commonly used, one defining maximality using analytic closure and the other using Zariski closure. We show that these two definitions are non-equivalent for $G=Sp_4$.

On two definitions of wave-front sets for $p$-adic groups

TL;DR

The paper investigates two definitions of wave-front sets for irreducible admissible representations of p-adic groups, contrasting WF^rat (analytic closure) with WF^Zar (Zariski closure). Using a concrete Sp_4 construction, it builds an epipelagic representation via compact induction from a depth -1/2 Moy-Prasad datum and analyzes Shalika germs to show WF^rat contains a subregular nilpotent orbit in addition to two regular orbits, whereas WF^Zar contains only the two regular orbits, providing a counterexample to their equivalence. This highlights subtle distinctions between analytic and geometric closures in wave-front theory and informs broader conjectures on geometric wave-front sets. The paper also details the Langlands parameters for the representation components, giving a explicit decomposition rho = rho1 ⊕ rho2 ⊕ triv with rho_j = Ind_{W_E}^{W_F} chi_j for E = F(√π), enumerating four candidate parameters.

Abstract

The wave-front set for an irreducible admissible representation of a -adic reductive group is the set of maximal nilpotent orbits which appear in the local character expansion. By Mœglin-Waldspurger, they are also the maximal nilpotent orbits whose associated degenerate Whittaker models are non-zero. However, in the literature there are two versions commonly used, one defining maximality using analytic closure and the other using Zariski closure. We show that these two definitions are non-equivalent for .
Paper Structure (4 sections, 6 theorems, 29 equations)

This paper contains 4 sections, 6 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Nilpotent orbits
  3. Shalika germs and proof of Theorem \ref{['main']}
  4. Langlands parameters

Key Result

Theorem 1.4

For any irreducible component $\pi$ of the compact induction we have that $\operatorname{WF}^{rat}(\pi)$ contains two regular nilpotent orbits and also a subregular nilpotent orbit. Consequently $\operatorname{WF}^{Zar}(\pi)$ contains only the two regular nilpotent orbits.

Theorems & Definitions (18)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3
  • proof : Proof of Proposition \ref{['prop:reg']}
  • ...and 8 more