We define an analogue of the Casimir element for a graded affine Hecke algebra $$ \mathbb{H} $$ , and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology H D (X) of an $$ \mathbb{H} $$ -module X, and show that H D (X) carries a representation of a canonical double cover of the Weyl group $$ \widetilde{W} $$ . Our main result shows that the $$ \widetilde{W} $$ -structure on the Dirac cohomology of an irreducible $$ \mathbb{H} $$ -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 $$ \mathbb{H} $$ .