  Graded Hecke algebras can be constructed in terms of equivariant cohomology
and constructible sheaves on nilpotent cones. In earlier work, their standard
modules and their irreducible modules where realized with such geometric
methods.
  We pursue this setup to study properties of module categories of (twisted)
graded Hecke algebras, in particular what happens geometrically upon formal
completion with respect to a central character. We prove a version of the
Kazhdan--Lusztig conjecture for (twisted) graded Hecke algebras. It expresses
the multiplicity of an irreducible module in a standard module as the
multiplicity of an equivariant local system in an equivariant perverse sheaf.
  This is applied to smooth representations of reductive p-adic groups. Under
some conditions, we verify the p-adic Kazhdan--Lusztig conjecture from [Vogan].
Here the equivariant constructible sheaves live on certain varieties of
Langlands parameters. The involved conditions are checked for substantial
classes of groups and representations.
