TITLE: Beilinson-Bernstein and Koszul duality
QUESTION [7 upvotes]: 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?

REPLY [5 votes]: Some general discussions related to Soergel's conjecture and Beilinson-Bernstein can also be found in the introductions to my two papers with David Nadler on the arXiv,  0706.0322 (which is about to be split into two much improved papers) and 0904.1247.