Proofs of Beilinson-Bernstein

Question

The Beilinson-Bernstein localization theorem states roughly that the category of $D$-modules on the flag variety $G/B$ is equivalent to the category of modules over the universal enveloping algebra $U\mathfrak{g}$ with zero Harish-Chandra character. Here $G$ is any semisimple algebraic group over $\mathbb{C}$, $\mathfrak{g}$ its Lie algebra, and $B$ a Borel in $G$. (Really I should probably generalize to modules over a twist $D_\lambda$ of the ring of differential operators and other Harish-Chandra characters here.)

I think it's fair to say that this is one of the most important theorems in geometric representation theory. One sign of this is the huge number of generalizations to different settings (arbitrary Kac-Moody, affine at negative and critical levels, characteristic p, quantum, other symplectic resolutions...), each of which plays an key role in the corresponding representation theory.

A lot of the papers in this area seem to contain phrases like "Here we are generalizing a proof of the original BB theorem", but they seem to be all referring to different proofs. Unfortunately, I only know one proof, more or less the one that appears in the original paper of Beilinson and Bernstein, "Localisation de $\mathfrak{g}$-modules." So my question is

What other approaches are there to proving the localization theorem?

To be a little more precise, let's separate BB localization into three parts:

1. The computation of $R\Gamma(G/B,\mathcal{D}_\lambda)$ (for $\lambda$ with $\lambda+\rho$ dominant)
2. The exactness of $R\Gamma$ (true for all weights $\lambda$ with $\lambda+\rho$ dominant)
3. The conservativity of $R\Gamma$ (true for $\lambda$ with $\lambda+\rho$ dominant regular).

I'm mainly interested in proofs of parts $2$ and/or $3$ assuming part $1$ (which I know many proofs of.)

Standard disclaimer: I'm not an expert and all of this could be wrong.

Answer 1

In the introduction (section 1.3) of the paper

• Ed Frenkel and Dennis Gaitsgory, $D$-modules on the affine Grassmannian and representations of affine Kac-Moody algebras, Duke Math. J. 125(2) (2004) pp 279–327, doi:10.1215/S0012-7094-04-12524-2, arXiv:math/0303173,

they give a new proof of part (2) above. Their idea is to factor $ \Gamma : D^\lambda_{G/B}\text{-}mod \rightarrow U\mathfrak g\text{-}mod $ in two steps: $$ D^\lambda_{G/B}\text{-}mod \rightarrow U\mathfrak g\text{-}bimod \rightarrow U\mathfrak g\text{-}mod $$ where the first functor is given by $ \Gamma(G, \pi^* \mathcal F) $ (using the projection $ G \rightarrow G/B $) and the second functor is given by $ Hom(M(\lambda), M )$ where $M(\lambda) $ is the Verma module.

In this way, $\Gamma $ is the composition of two exact functors and thus is exact.

I don't know how this compares to the original proof of exactness.

Answer 2

I have sketched several possible proofs in this document: http://math.mit.edu/~bezrukav/localization_notes.pdf

Please let me know if there are comments or questions.