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

Question

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)

Answer 1

I'm far from being an expert on BBD and related algebraic geometry, but the question about "construction" of irreducible representations in the infinite dimensional context has to be approached with great care. Although direct constructions are possible in a few special cases, the main problems for Lie groups or their Lie algebras usually require an indirect approach.

For example, in the BGG category (say with integral weights) there is an easy construction of Verma modules using induction methods; the formal character is also easy to exhibit. But the unique simple quotient cannot in general be constructed even by sophisticated methods. Instead you try to imitate the BGG approach to the finite dimensional character formulas: in effect, express the unknown formal character as a $\mathbb{Z}$-linear combination of the known Verma characters. This can provide an effective algorithm based on the partial ordering of weights. The integral coefficients here still need to be found. Kazhdan-Lusztig predicted these in terms of recursively computable polynomial values at 1, but so far only the geometric methods of Beilinson-Bernstein or Brylinski-Kashiwara have been able to prove this.

Similar predictions are made by Lusztig in prime characteristic for representations of semisimple algebraic groups, but only proved for large primes (and with no definite prediction for primes less than the Coxeter number). Analogues for quantum enveloping algebras at a root of unity have by now been attacked successfully using the characteristic 0 geometric methods. In the Lie group direction, Vogan and others have pressed further with partial success in the spirit of the KL Conjecture. But all of these problems are extremely difficult. At the end of the day, very few concrete constructions of irreducible representations are found. Usually one settles for some kind of "character" information. While the classical Borel-Weil theorem inspires some of the later moves, the story gets much more complicated.

Concerning the added "further question", my remarks apply equally well to affine Lie algebras in most cases. But the situation there for the critical level or for the somewhat parallel finite dimensional modular theory mentioned above is less settled. There has been a lot of recent progress in both cases, for example in the modular theory by work of Bezrukavnikov, Mirkovic, Rumynin. In all cases, the results obtained by localization or other geometric methods lead mainly to multiplicity formulas and recursive computation of characters rather than explicit construction of simple modules. But in finite dimensional cases, even their dimensions have been elusive.

ADDED: Going back to the questions asked, the work of Beilinson-Bernstein and others does not directly construct new representations. But it's an essential part of the working out of character formulas for various classes of irreducible representations, translated into composition factor multiplicity problems for induced modules such as Verma modules. This is where the reformulation of problems in the language of algebraic geometry and sheaf theory has been important for representation theory. No algebraic method is known (or expected) for giving an explicit construction of the infinite dimensional irreducibles. Characterizing them abstractly as quotients of Verma modules or the like gives very litle information about the characters and can't be viewed as a construction.

Answer 2

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."

Answer 3

There could be different ways to give meaning to the phrase "explicit construction".

In an algebro-geometric sense, an expicit construction comes from more classical Borel-Weil-Bott theorem of which BDD is an abstract generalization. There's a number of proof in the literature, e.g the one by Jacob Lurie (on his home page).

According to the BWB, you can get the (finite-dimensional) representation by taking the global sections of one of the equivariant bundles $ \mathcal O(\lambda)$.

Another way to construct the representation would be to start with some simple $\mathfrak g$-modules and combine them to get your represenation. In this way, BDD helps by establishing a correspondence between simple equivariant D-modules and Verma modules. Therefore, the resolution for a bundle $\mathcal O(\lambda)$ corresponds to a construction in the category of $\mathfrak g$-modules, the one called Bernstein-Gelfand-Gelfand resolution, giving rise to Weyl character formula

As an example, the $\mathfrak{sl}_2$ modules correspond to equivariant D-modules on a $\mathbb P^1$, which has two cells. Therefore, a BGG resolution for an $\mathfrak{sl}_2$-module has two terms. Since a Verma module for $\mathfrak{sl}_2$ with an integer weight $\lambda$ has (I think) exactly one vector of each weight $\lambda' \le \lambda$, you can picture it as a ray on a weight lattice; the picture then becomes [segment] = [ray] - [ray].