Langlands' philosophy and Koszul duality

In this article I will formulate conjectures relating representation theory to geometry and prove them in some cases. I am trying to extend work of Adams-Barbasch-Vogan and some joint work of myself with Beilinson and Ginzburg. All this should be regarded as a contribution to (the local case of) Langlands' philosophy. The localization of Beilinson-Bernstein also establishes a relation between representation theory and geometry. This is however very different from the relation to be investigated in this article: Whereas localization leads to geometry on the group itself (or its flag manifold), the results of Adams-Barbasch-Vogan to be extended in this article lead to geometry on the Langlands dual group.