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

TL;DR

This paper compares two different definitions of wave-front sets for representations of p-adic groups and demonstrates their non-equivalence in the case of Sp_4, highlighting a subtlety in the theory.

Contribution

It proves that the analytic and Zariski closure-based definitions of wave-front sets are not equivalent for the group Sp_4, clarifying a key aspect of the theory.

Findings

The two definitions are non-equivalent for Sp_4.

The discrepancy arises from the specific structure of Sp_4.

This result impacts the understanding of wave-front sets in p-adic representation theory.

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 M\oe glin-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 $G=Sp_4$.