Uniform bounds on the Harish-Chandra characters

2023/03/03 by Anna Szumowicz, Szumowicz, Anna
Mathematics · #11F55 #20C15 #20G25 #22E35 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2303.01752

openalex publication_date 2023/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a connected reductive algebraic group over a p-adic local field F. In this paper we study the asymptotic behaviour of the trace characters θ_π evaluated at a regular element γ of G(F) as π varies among supercuspidal representations of G(F). Kim, Shin and Templier conjectured that \fracθ_π(γ)\rm deg(π) tends to 0 when π runs over irreducible supercuspidal representations of G(F) with unitary central character and the formal degree of π tends to infinity. For G semisimple we prove that the trace character is uniformly bounded on γ under the assumption, which is expected to hold true for every G (F), that all irreducible supercuspidal representations of G(F) are compactly induced from an open compact modulo center subgroup. Moreover, we give an explicit upper bound in the case of γ ellitpic.

Related

• Asymptotics and local constancy of characters of p-adic groups
• Local Summability of Characters on p-adic Reductive Groups
• Integrability and singularities of Harish-Chandra characters
• Characterization of supercuspidal representations and very regular elements
• Progrès récents sur les représentations supercuspidales