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)
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