Asymptotics and local constancy of characters of p-adic groups

TL;DR

This paper investigates the behavior of trace characters of p-adic group representations, establishing new results on their local constancy and formulating a conjecture relating characters to formal degrees, proven for tame supercuspidal cases.

Contribution

It introduces a conjecture on the relation between characters and formal degrees, proven for tame supercuspidal representations using Yu's construction and Moy-Prasad subgroups.

Findings

Proved the conjecture for tame supercuspidal representations.

Established local constancy properties of characters.

Surveyed related results on p-adic group characters.

Abstract

In this paper we study quantitative aspects of trace characters $\Theta_\pi$ of reductive $p$-adic groups when the representation $\pi$ varies. Our approach is based on the local constancy of characters and we survey some other related results. We formulate a conjecture on the behavior of $\Theta_\pi$ relative to the formal degree of $\pi$, which we are able to prove in the case where $\pi$ is a tame supercuspidal. The proof builds on J.-K.~Yu's construction and the structure of Moy-Prasad subgroups.