The unicity of types for depth-zero supercuspidal representations

TL;DR

This paper proves the uniqueness of certain types within depth-zero supercuspidal representations of p-adic groups and derives an inertial Langlands correspondence for them, advancing understanding of their structure and classification.

Contribution

It establishes the unicity of types for depth-zero supercuspidal representations and connects this to an inertial Langlands correspondence, extending prior results to arbitrary p-adic groups.

Findings

Uniqueness of types for all depth-zero supercuspidal representations.

Establishment of an inertial Langlands correspondence for these representations.

Connection to the Langlands correspondence of DeBacker and Reeder.

Abstract

We establish the unicity of types for depth-zero supercuspidal representations of an arbitrary $p$-adic group $G$, showing that each depth-zero supercuspidal representation of $G$ contains a unique conjugacy class of typical representations of maximal compact subgroups of $G$. As a corollary, we obtain an inertial Langlands correspondence for these representations, via the Langlands correspondence of DeBacker and Reeder.