Koszul duality in representation theory

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The Koszul duality conjecture of the applicant, hopefully soon a theorem by Nadler and Ben-Zvi, conjectures a way to lift the local Langlands correspondence to the level of an equivalence of triangulated categories between roughly speaking the derived category of a category of representations and the equivariant derived category of the Langlands parameter space, more precisely it's version by Adams, Barbasch and Vogan. Our goal is to understand what the operations of tensoring with finite dimensional representation and induction correspond to on the the other side of this equivalence, i.e. in the equivariant derived category of the Langlands parameter space.