Asymptotics and Local Constancy of Characters of p-adic Groups

Abstract

In this paper we study quantitative aspects of trace characters Θπ of reductive p-adic groups when the representation π 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 Θπ relative to the formal degree of π, which we are able to prove in the case where π is a tame supercuspidal. The proof builds on J.-K. Yu’s construction and the structure of Moy–Prasad subgroups. Keywords: Maximal Torus; Regular Element; Discrete Series; Formal Degree; Open Compact Subgroup