  For a complex reductive Lie group G with Lie algebra g, Cartan subalgebra h
and Weyl group W, we describe the category of perverse sheaves on h/W smooth
w.r.t the natural stratification. The answer is given in terms of mixed Bruhat
sheaves, which are certain mixed sheaf-cosheaf data on cells of a natural cell
decomposition of h/W. Using the parabolic Bruhat decomposition, we relate mixed
Bruhat sheaves with the properties of various procedures of parabolic induction
and restriction that connect different Levi subgroups in G.
