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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)

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# How to compute irreducible representation of Lie algebra in the framework of BBD

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?