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

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.