  We introduce graded Hecke algebras H based on a (possibly disconnected)
complex reductive group G and a cuspidal local system L on a unipotent orbit of
a Levi subgroup M of G. These generalize the graded Hecke algebras defined and
investigated by Lusztig for connected G.
  We develop the representation theory of the algebras H. obtaining complete
and canonical parametrizations of the irreducible, the irreducible tempered and
the discrete series representations. All the modules are constructed in terms
of perverse sheaves and equivariant homology, relying on work of Lusztig. The
parameters come directly from the data (G,M,L) and they are closely related to
Langlands parameters.
  Our main motivation for considering these graded Hecke algebras is that the
space of irreducible H-representations is canonically in bijection with a
certain set of "logarithms" of enhanced L-parameters. Therefore we expect these
algebras to play a role in the local Langlands program. We will make their
relation with the local Langlands correspondence, which goes via affine Hecke
algebras, precise in a sequel to this paper.
