  We define the algebraic Dirac induction map $\Ind_D$ for graded affine Hecke
algebras. The map $\Ind_D$ is a Hecke algebra analog of the explicit
realization of the Baum-Connes assembly map in the $K$-theory of the reduced
$C^*$-algebra of a real reductive group using Dirac operators. The definition
of $\Ind_D$ is uniform over the parameter space of the graded affine Hecke
algebra. We show that the map $\Ind_D$ defines an isometric isomorphism from
the space of elliptic characters of the Weyl group (relative to its reflection
representation) to the space of elliptic characters of the graded affine Hecke
algebra. We also study a related analytically defined global elliptic Dirac
operator between unitary representations of the graded affine Hecke algebra
which are realized in the spaces of sections of vector bundles associated to
certain representations of the pin cover of the Weyl group. In this way we
realize all irreducible discrete series modules of the Hecke algebra in the
kernels (and indices) of such analytic Dirac operators. This can be viewed as a
graded Hecke algebra analogue of the construction of discrete series
representations for semisimple Lie groups due to Parthasarathy and
Atiyah-Schmid.
