A Pinned Local Langlands Correspondence for Depth-Zero Supercuspidal Representations

TL;DR

This paper constructs a canonical, pinning-normalized local Langlands correspondence for depth-zero supercuspidal representations of reductive groups over non-archimedean fields, integrating toral, unipotent, and Clifford data.

Contribution

It introduces a new, canonical bijection between depth-zero supercuspidal representations and enhanced Langlands parameters, utilizing a detailed decomposition approach.

Findings

The correspondence is canonical and compatible with various invariants.

It matches toral parts via unramified elliptic tori and normalized L-embeddings.

Under certain hypotheses, the resulting packets are stable.

Abstract

We construct a pinning-normalized local Langlands correspondence for depth-zero supercuspidal representations of a connected reductive group over a non-archimedean local field. After fixing a pinned splitting of the quasi-split inner form, we obtain a canonical bijection between irreducible depth-zero supercuspidal representations and relevant cuspidal enhanced depth-zero Langlands parameters. The construction is organized around the two pieces naturally present in a depth-zero type: a tame toral part and a finite cuspidal representation of a parahoric quotient. The toral part is matched using the local Langlands correspondence for maximally unramified elliptic tori and normalized \(L\)-embeddings. The finite cuspidal part is compared with the parameter side by a pinned Jordan decomposition for the relevant finite reductive quotients. Since these quotients may be disconnected, the finite comparison must retain the Clifford-theoretic data that records the possible extension ambiguity. On the connected unipotent part we use the correspondence of Feng--Opdam--Solleveld for supercuspidal unipotent representations. Combining the toral, unipotent, and Clifford-theoretic pieces gives the enhanced parameter attached to a depth-zero supercuspidal representation, and the inverse map is obtained by reversing the same construction. The correspondence is canonical relative to the fixed pinned normalization. It is compatible with the tame inertial parameter attached to the depth-zero character, with weakly unramified twists, and with central characters via the torus correspondence. Under the DeBacker--Reeder logarithm hypothesis, the dimension-weighted packet distributions attached to the resulting packets are stable.