On the local constancy of characters

TL;DR

This paper provides an explicit description of neighborhoods around certain elements in p-adic groups where the characters of irreducible admissible representations remain constant, extending known local constancy results.

Contribution

It offers an explicit characterization of neighborhoods ensuring local constancy of characters for regular semisimple elements under mild conditions.

Findings

Explicit neighborhoods where characters are constant

Extension of local constancy to broader classes of elements

Provides tools for analyzing characters in p-adic representation theory

Abstract

The character of an irreducible admissible representation of a $p$-adic reductive group is known to be a constant function in some neighborhood of any regular semisimple element $\gamma$ in the group. Under certain mild restrictions on $\gamma$, we give an explicit description of a neighborhood of $\gamma$ on which the character is constant.