  Graded Hecke algebras can be constructed geometrically, with constructible
sheaves and equivariant cohomology. The input consists of a complex reductive
group G (possibly disconnected) and a cuspidal local system on a nilpotent
orbit for a Levi subgroup of G. We prove that every such "geometric" graded
Hecke algebra is naturally isomorphic to the endomorphism algebra of a certain
G x C*-equivariant semisimple complex of sheaves on the nilpotent cone $g_N$.
  From there we provide an algebraic description of the G x C*-equivariant
bounded derived category of constructible sheaves on $g_N$. Namely, it is
equivalent with the bounded derived category of finitely generated differential
graded modules of a suitable direct sum of graded Hecke algebras. This can be
regarded as a categorification of graded Hecke algebras.
