For geometric representation theorists down here.


Consider the Beilinson-Bernstein theorem: 



> 
> Functor of global sections establishes
>  the correspondence between twisted
>  D-modules with fixed twist θ on the
>  flag variety and g-representations
>  with fixed central character. These
>  are modules over the same algebra D[θ]
>  = U /(Z −χ). This correspondence respects the structure of abelian
>  category. It takes K-equivariant
>  D-modules to (g, K)-admissible
>  modules.
> 
> 
> 


Why do people refer to its derived version as the Koszul duality?


How is this related to Soergel's conjecture?



Beilinson-Bernstein and Koszul duality
Well, *I* don't call its derived version "Koszul duality." Koszul duality is a relationship between pairs of graded algebras (you could upgrade this to graded categories, but why bother?). Given a positively graded algebra A, consider the zero-degree part A0 as a module. Then B=Ext\*A(A0,A0) is a new bigraded algebra: it has a homological grade, and one induced by taking a graded free resolution of A0.


If these coincide, we say that A is Koszul and B is its Koszul dual. The functor of Ext\*A(A0,-) induces an equivalence between the derived categories of graded modules (with strange behavior on the grading!) over A and B (there's a version for ungraded modules of A, but those are sent to dg-modules of B and vice versa).


One of the strange facts of geometric representation theory is that categories which seem to a priori have no good reason to be Koszul dual actually are. For example, the categories of (g,N)-admissible modules with fixed generic central character for one Lie algebra is Koszul dual to the corresponding category for its Langlands dual. Soergel's conjecture is an version of this statement to (g,K) admissible modules for various symmetric subgroups K (which is much harder than the version for N).


What this has to do with Beilinson-Bernstein is that all these theorems are most naturally studied from a geometric perspective. For example, Koszulity tends to be related to purity of intersection cohomology.


