Dirac cohomology of one K-type representations

TL;DR

This paper proves that certain one K-type representations of split reductive p-adic groups have nonzero Dirac cohomology and uses this to classify their Langlands parameters.

Contribution

It establishes nonzero Dirac cohomology for these representations and determines their Langlands parameters using Dirac operator techniques.

Findings

All such one K-type modules have nonzero Dirac cohomology.

The semisimple part of their Langlands parameters is explicitly determined.

The classification of these modules is completed using Dirac cohomology methods.

Abstract

The smooth hermitian representations of a split reductive p-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single K-type with Iwahori fixed vectors have been studied in [D. Barbasch, A. Moy, Classification of one K-type representations, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4245-4261] in the more general setting of modules for graded affine Hecke algebras with parameters. We show that every such one K-type module has nonzero Dirac cohomology (in the sense of [D. Barbasch, D. Ciubotaru, P. Trapa, The Dirac operator for graded affine Hecke algebras, arXiv:1006.3822]), and use Dirac operator techniques to determine the semisimple part of the Langlands parameter for these modules, thus completing their classification.