  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.
