On the types for supercuspidal representations of inner forms of $\mathrm{GL}_{N}$

TL;DR

This paper proves the uniqueness of simple types over maximal compact subgroups for supercuspidal representations of inner forms of GL_N, under certain unramifiedness conditions, advancing the understanding of their classification.

Contribution

It establishes the uniqueness (up to conjugation) of simple types over maximal compact subgroups for supercuspidal representations under unramifiedness assumptions.

Findings

Uniqueness of simple types over maximal compact subgroups

Conditions under which types are unique up to conjugation

Advancement in classification of supercuspidal representations

Abstract

Let $F$ be a non-Archimedean local field, $A$ be a central simple $F$-algebra, and $G$ be the multiplicative group of $A$. It is known that for every irreducible supercuspidal representation $\pi$, there exists a $[G, \pi]_{G}$-type $(J, \lambda)$, called a (maximal) simple type. We will show that $[G, \pi]_{G}$-types defined over some maximal compact subgroup are unique up to $G$-conjugations under some unramifiedness assumption on a simple stratum.