D-modules supported on the nilpotent cone

Question

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)?

Answer 1

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?

Answer 2

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).

Answer 3

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.

Answer 4

This is a very old question, but for future visitors, I would like to point out that the answer is described in section 3 of:

T. Levasseur, "Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra", 1998.

It isn't clear to me that all of the simple objects are covered by the constructions in this paper, but at least it seems that the type you're considering are - i.e. those arising from minimal extension of functions on a nilpotent orbit.