Koszul duality for generalized steinberg representations of p-adic groups

In this paper we prove a novel result on two categories that appear in the local Langlands correspondence, for generalized Steinberg representations. Let G be a semisimple reductive group split over a p-adic field F. The main result of this paper shows that category of modules over the extension algebra of generalized Steinberg representations of $G(F)$ appears as a full subcategory of equivariant perverse sheaves on the variety of Langlands parameters for these representations.