# Beilinson-Bernstein and Koszul duality

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?

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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.

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This is quite cool. I'm trying to understand everything in the answer, could you help especially with the last paragraph? –  Ilya Nikokoshev Oct 12 '09 at 18:36
And the way I understand it, knowing B-B you should be somehow proving/trying to prove Soergel's conjecture via geometry of something about G and something about {}^L G, right? –  Ilya Nikokoshev Oct 12 '09 at 18:37
For the last paragraph, I would read "Koszul duality patterns in representation theory" by Beilinson, Ginzburg and Soergel. They don't explicitly discuss Beilinson-Bernstein; instead they wrap it up in the equivalence between category O and certain perverse sheaves on the flag variety, but it's really the same theorem. –  Ben Webster Oct 12 '09 at 19:30

I was first introduced to Soergel's conjecture as a "categorification" of Vogan duality which identifies blocks of representations for a real reductive Lie group (a few more adjectives should be included here) with with the representations of a real group (determined by the block) in the dual inner class. Dual here is just root system duality for the associated complex groups, although the inner class does also determine the Langlands dual. Jeff Adams gave a nice series of lectures on inner classes/real forms for real groups and their duals this summer at the University of Utah. Lecture notes can be found here: http://www.math.utah.edu/realgroups/schedule.html. Adams' and Vogan's lectures on this site should be thought of as an introduction to their book with Dan Barbasch The Langlands Classification and Irreducible Characters for Real Reductive Lie Groups". This book already (in the introduction, even) identifies the K-groups of categories of representations of (strong real forms of) a real group with dual-equivariant perverse sheaves on a variety associated to that form (again, many more adjectives are needed). Soergel's conjecture gives an analogous categorical equivalence, and in its derived/dg version looks at the derived category of representations in this picture, and on the dual side a dg-category associated to the dual-equivariant sheaves. This is the setting in which he makes a claim about the equivalence coming from a derived version of Koszul duality.

I second Ben's statement about the geometric perspective. The known proofs of the Kazhdan-Lusztig conjectures are geometric, vanishing results for cohomological induction are trivial geometrically, and the parameterization of irreducible objects in the category you describe is also quite easy.

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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.

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Thanks! You might notice I quote both of these papers in other questions and it would be great to be able to read 0706.0322 –  Ilya Nikokoshev Oct 22 '09 at 7:42