On principal series representations of quasi-split reductive $p$-adic groups

Maarten Solleveld

Vol. 334 (2025), No. 2, 269–327

DOI: 10.2140/pjm.2025.334.269

Abstract

Let $G$ be a quasi-split reductive group over a non-archimedean local field. We establish a local Langlands correspondence for all irreducible smooth complex $G$-representations in the principal series. The parametrization map is injective, and its image is an explicitly described set of enhanced $L$-parameters. Our correspondence is determined by the choice of a Whittaker datum for $G$, and it is canonical given that choice.

We show that our parametrization satisfies many expected properties, among others with respect to the enhanced $L$-parameters of generic representations, temperedness, cuspidal supports and central characters. Our correspondence lifts to a categorical level, where it makes the appropriate Bernstein blocks of $G$-representations naturally equivalent to module categories of Hecke algebras coming from Langlands parameters. Along the way we characterize genericity of $G$-representations in terms of representations of an affine Hecke algebra.

Keywords

$p$-adic groups, principal series, local Langlands correspondence

Mathematical Subject Classification

Primary: 20C08, 22E50

Secondary: 20G25

Milestones

Received: 19 January 2024

Revised: 13 August 2024

Accepted: 25 January 2025

Published: 8 February 2025