TITLE: How to compute irreducible representation of Lie algebra in the framework of BBD
QUESTION [8 upvotes]: We know Beilinson-Bernstein established the following famous equivalence:
$D-mod_{G/B}\rightarrow U(g)-mod_{\lambda}$,where $G$ is algebraic group and $B$ is Borel subgroup, $G/B$ is flag variety of finite dimensional Lie algebra $g$, $\lambda$ is central character.
This equivalence means that one can study representations of Lie algebra $g$ via D-modules. But How? 
My question:
Is there machinery in the framework of BBD to construct irreducible representations of $g$  explicitly? 
I am aware that there is Riemann-Hilbert correspondence to describe the correspondence between Perverse sheaves and holonomic D-modules. It seems that it is possible to know the irreducible objects in category of Perverse sheaves.(I guess in this case,we will know the irreducible representations corresponding to holonomic modules) But even in this case, I did not find appropriate reference. I wonder whether somebody compute some concrete examples such as flag variety of $sl_2$($P^1$). 
Further question: I also want to know the answers for affine Lie algebra case(Frenkel-Gaitsgory established analogue of BB-equivalence for critical level affine Lie algebra). Does this work give new class of irreducible representations of affine Lie algebra?
REMARK: What I want to know is the advantage to use D-module theory to construct representations.(if we can)For example, for some general Lie algebra $g$, we consider flag variety $X$. Then we consider $D-mod_{X}$, how to use algebraic geometric machinery on this category to construct irreducible representations of $g$ explicitly?(I would like to know is there any construction(in BBD)which can describe representations)

REPLY [6 votes]: Hello, I want to try to answer the first question you pose (at least I can answer the question I believe you are asking) -- sorry it's so far past when you asked it, but perhaps (if you know the answer yourself now) it will be useful to someone else to just have this information up anyway?  It sounds like you are asking "what is the D-module version of IC-sheaves?" In the BGG setting, Beilinson-Bernstein localization gives an equivalence between $B$-equivariant $D_{G/B}$-modules and highest weight $\mathcal{U}(\mathfrak{g})$-modules with trivial infinitesimal character. We can twist the D-module picture I am about to describe to obtain modules with other infinitesimal characters.  Irreducible $B$-equivariant $D_{G/B}$-modules are constructed as follows:  Let $Q$ be a $B$-orbit on $G/B$ and $i: Q\to G/B$ its inclusion.  Let $\tau$ be the structure sheaf of $Q$ (we can replace $B$ by more general groups in which case $\tau$ must range over all irreducible connections on $Q$, by which I mean specifically "line bundle with connection" so that we are working with $D_Q$-modules).  The D-module direct image $i_+ \tau$ has a unique irreducible submodule $L(\tau)$, and this construction in fact establishes a bijection between orbits $Q$ (more generally, pairs $(Q,\tau)$) and irreducible objects $L(\tau)$. Under Riemann-Hilbert, the $L(\tau)$ are precisely the IC sheaves, and upon taking global sections we recover irreducible representations.  The sheaves $i_+\tau$ have the co-Verma modules as their global sections and you can even recover the (co-)BGG resolution using D-module constructions alone (it's the Cousin resolution -- see Hartshorne's Residues and Duality). 
A good reference which has this all written down in the context of Harish-Chandra modules (and for arbitrary twists) is the paper "Localization and standard modules for real semi-simple Lie groups I: The duality theorem" -- the BGG version of the entire paper should work exactly the same but with $B$ instead of $K$ and "co-Verma" instead of "co-standard."