On depth-zero characters of p-adic groups

TL;DR

This paper investigates depth-zero characters of reductive p-adic groups, establishing their properties and equivalences, which aids in advancing the local Langlands correspondence for depth-zero representations.

Contribution

It introduces new properties and equivalences of depth-zero characters for p-adic groups, including wildly ramified cases, facilitating progress in local Langlands correspondence.

Findings

Depth-zero characters have multiple equivalent definitions.

New properties of depth-zero characters are established.

Results support extending local Langlands correspondence to more cases.

Abstract

We show new properties of the Langlands correspondence for arbitrary tori over local fields. Furthermore, we give a detailed analysis of depth-zero characters of reductive p-adic groups, for groups that may be wildly ramified. We present several different definitions of ``depth-zero'' for characters, and show that these notions are in fact equivalent. These results are useful for proving new cases of local Langlands correspondences, in particular for depth zero representations.