We define an analogue of the Casimir element for a graded affine Hecke algebra H, and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology HD(X) of an,-module X, and show that HD(X) carries a representation of a canonical double cover of the Weyl group,. Our main result shows that the, -structure on the Dirac cohomology of an irreducible, -module X determines the central character of X in a precise way. This can be interpreted as p-adic analogue of a conjecture of Vogan for Harish-Chandra modules. We also apply our results to the study of unitary representations of,. © 2012 Institut Mittag-Leffler.