On depth-zero characters of p-adic groups

TL;DR

The paper investigates depth-zero phenomena in the local Langlands program for tori and more general reductive $p$-adic groups. It establishes that the local Langlands correspondence for tori preserves depth-zero information across wildly ramified cases, via the isomorphism $Hom(T/T_{0+}, \mathbb{C}^\times) \cong H^1(\mathbf{W}_F/\mathbf{P}_F, T^{\vee,\mathbf{P}_F})$ and the bijection $H^1(\mathbf{W}_F, Z(G^\vee)) \to Hom(G/G_{sc}, \mathbb{C}^\times)$, extending norm compatibility over extensions. It then defines and characterizes depth-zero characters of arbitrary reductive $G$, introducing the group $\mathfrak{X}^0(G)$ and proving several equivalent descriptions, including a Bruhat–Tits/Moy–Prasad-based viewpoint. These results provide a robust framework for constructing depth-zero representations and for advancing depth-zero cases of the local Langlands correspondence, even in non-split or wildly ramified settings.

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.

Theorems & Definitions (18)

• Theorem 1.1: LLC for tori
• Proposition 1.2: see Proposition \ref{['prop:1.8']}
• Proposition 1.3: see Proposition \ref{['prop:1.7']}
• Theorem 1.4: see Theorem \ref{['thm:1.4']} and Lemma \ref{['lem:1.9']}
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 3.1
• proof