Typical representations via fixed point sets in Bruhat--Tits buildings

TL;DR

This paper explores the geometric structure of Bruhat-Tits buildings to establish conditions for the occurrence of certain irreducible components in the restriction of supercuspidal representations, aiding understanding of their branching rules.

Contribution

It introduces two new geometric criteria based on fixed point sets in Bruhat-Tits buildings for analyzing supercuspidal representations of p-adic groups.

Findings

Provides sufficient conditions for irreducible components in restrictions of supercuspidal representations.

Links geometric fixed point data in buildings to representation-theoretic properties.

Offers tools for studying branching rules and type unicity in p-adic groups.

Abstract

For an essentially tame supercuspidal representation $\pi$ of a connected reductive $p$-adic group $G$, we establish two distinct and complementary sufficient conditions for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of $G$ which is not inertially equivalent to $\pi$. These two results are further formulated in terms of the geometry of the Bruhat-Tits building of $G$ and its fixed points under the action of certain tori. The consequence is a set of broadly applicable tools for addressing the branching rules of $\pi$ and the unicity of $[G,\pi]_G$-types.