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

TL;DR

This paper establishes a canonical local Langlands correspondence for principal series representations of quasi-split reductive p-adic groups, linking irreducible smooth representations to explicit enhanced L-parameters.

Contribution

It provides a complete, injective parametrization of principal series representations via enhanced L-parameters, determined by a Whittaker datum, and verifies key expected properties.

Findings

Injective parametrization of principal series representations.

Explicit description of the set of enhanced L-parameters.

Characterization of genericity through affine Hecke algebra representations.

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. Along the way we characterize genericity in terms of representations of an affine Hecke algebra.