On Hecke algebras and Z-graded twisting, shuffling and Zuckerman functors

Let g be a complex semisimple Lie algebra with Weyl group W. Let H(W) be the Iwahori-Hecke algebra associated to W. For each w ∈ W, let T_w and C_w be the corresponding Z-graded twisting functor and Z-graded shuffling functor, respectively. In this paper, we present a categorical action of H(W) on the derived category D^b( 0^Z₀) of the Z-graded BGG category 0^Z₀ via derived twisting functors as well as a categorical action of H(W) on D^b(O^Z₀) via derived shuffling functors. As applications, we get graded character formulae for T_sL(x) and C_sL(x) for each simple reflection s.