## D-modules supported on the nilpotent cone

I am reading some papers which involve D-modules on a Lie algebra g, which are supported on the nilpotent cone n. They are equivariant for the action of G. (In particular, I consider g=sl_n).

It was explained to me (the statement, not the proof) that the category of such D-modules is semisimple, and that the simple objects are given by (the intermediate extension) of the constant sheaf sheaf of functions on each g-orbit on n, so they are in bijection with Jordan decompositions with all zeroes, so just partitions of n.

I'm not strong with D-modules (I'm learning!). My question is this: Since g is affine, D(g) is an associative algebra, namely the Weyl algebra on the vector space g. How can I describe the D(g)-module M corresponding to partition \lambda explicitly as a module over D(g)?

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Watch out that there are more simple objects than it looks like at first glance, even for sl_n. Although the orbits are parameterized by partitions, they can carry nontrivial local systems whose intermediate extensions to the nilpotent cone will be new simple D-modules.

I have heard that the general problem of writing down intermediate extensions explicitly, say by generators and relations, is weirdly difficult. Kari Vilonen did this for isolated singularities in his thesis. For the nilpotent cone I wonder how well Ben's suggestion works: is it a simple matter to pick out the isotypic components of this pushforward D-module, using the bare fact (geometric magic) that it's an intermediate extension of a D-module you know how to decompose?

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 Are there really non-trivial local systems in the sl_n case (I knot they happen in other types)? I was under the impression that in that case the orbits were all simply connected (since the centralizer of a nilpotent is connected). – Ben Webster♦ Oct 28 2009 at 14:28 Are lower-case-g equivariant D-modules more like imposing equivariance with respect to a simply connected group or an adjoint group? SL_n (even SL_2) has a center, so there is always some equivariant local system on the regular orbit. In SL_2, this regular orbit is topologically C x C^*, and I think that SL_2 equivariance means that the monodromy around that C^* squares to 1. I don't know what happens for GL_n or PGL_n. – David Treumann Oct 28 2009 at 16:36 Sorry, I think I introduced the confusion here. It should be G equivariant not g equivariant. A lot of difference a shift makes. As it was explained to me, Ben's point about the orbits being simply connected for sl_n is important here. – David Jordan Oct 31 2009 at 15:11 They aren't simply connected! Even for SL_2. The centralizer of [[0,0],[1,0]] has one component of the form [[1,0][*,1]] and one of the form [[-1,0][*,-1]]. – David Treumann Oct 31 2009 at 18:47 yes, sorry i understand now what you said: the centralizer any non-zero nilpotent element has two components each equal to \CC. – David Jordan Oct 31 2009 at 21:03
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I think saying "constant sheaf" isn't quite right. You want to take the functions on each orbit, and do the intermediate extension of that D-module.

Now, that's a bit unexplicit, so let me try a different description. Recall that there is a Grothendieck simultaneous resolution, a map from G x_B b (the adjoint bundle for b on G/B) to g extending the inclusion of b in the obvious way. If one takes functions on G x_B b, and pushes them forward by this map, then one gets a D-module on g. The isotypic summands of this D-module are in bijection with the representations of S_n. Over regular semi-simple elements, the map G x_B b -> g is a Galois S_n-cover, and so we're just decomposing the local system. By some geometric magic, this decomposition extends over the rest of the Lie algebra.

Now, we have to do Fourier transform, which for D-modules, just means change your mind about which variables in the Weyl algebra are coordinate and which ones are differentiations. Then the summand corresponding to the S_n representation for a Young diagram becomes exactly the intermediate extension for the orbit with corresponding Jordan type (or is it the transpose? I always forget).

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 Thanks, Ben! What I'd really like to do now is to decode what you wrote into what the space M of global sections of the D-module is, and how the vector fields act (I guess the action of functions on g will be obvious once I know the vector space?) If you have any insight here, do share; otherwise I'll work through it. – David Jordan Oct 28 2009 at 2:43

There is a paper of Hotta and Kashiwara which identifies the "geometric" constructions people have mentioned with an "algebraic" one which came from Harish-Chandra's study of characters (invariant eigendistributions is the term I think). This gives you an explicit description of the D-module Rf_*(O) associated to the Springer resolution, one for the D-modules on the whole Lie algebra associated to the Grothendieck resolution on the whole Lie algebra, and proves how the two are related via the Fourier transform.

For sl_n this sheaf decomposes into simple summands corresponding to the middle extension of the structure sheaf of each nilpotent orbit. At least that these modules are the only possibilities follows from the GL_n equivariance of the Springer resolution: the constituents therefore have to be GL_n equivariant, and the equivariant fundamental groups are trivial in this case. If you work with SL_n, then on the regular orbit the equivariant fundamental group is isomorphic to the centre of SL_n, and in some sense this case is responsible for all the nontrivial equivariant local systems (Lusztig's generalized Springer correspondence arose from trying to understand related things).

The upshot though is that the simple GL_n-equivariant D-modules on the nilcone are the simple summands of the Springer sheaf. Now one way of saying Springer theory is that the Springer sheaf's decomposition into irreducibles is governed by the action of W the Weyl group on it (i.e. by the isotypic components of this action). Thus understanding the W-action on this sheaf gives a way of trying to understand these simple D-modules which is different from simply trying to calculate middle extensions.

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 Thanks, I'll look into this paper. For what I am studying it's very likely that a more "algebraic" approach will be useful (because one wants to study a certain quantization using "quantum D-modules", and the geometry mostly disappears (so far) in the understanding of constructions there. So it is helpful to have algebraic descriptions of everything. In theory, this should always be possible, but sometimes it is harder than others to do the translation. – David Jordan Nov 21 2009 at 15:44