Smoothness properties of the Springer fiber

Question

The Springer fiber, recall, is defined (briefly) with reference to a chosen unipotent matrix $U \in \mathrm{GL}_n$, and consists of all flags $0 = F_0 \subset F_1 \subset \dots \subset F_n = \mathbb{C}^n$ that are $U$-invariant term-by-term. It is of course the subject of a famous construction (by Springer) of representations of $S_n$, or more generally of any Weyl group with a suitably modified definition, and this representation can be made more or less explicit in the action of the simple transpositions (simple reflections). Namely, it acts on the vector space spanned by the irreducible components of the Springer fiber and a formula was given in Hotta's paper "On Joseph's construction of Weyl group representations" that expresses the matrix coefficients of the simple transpositions in terms of some geometric quantities connected with these components. In particular, it relies on knowledge of their normalizations.

It is known (see the paper of Fresse and Melnikov "On the singularity of the irreducible components of a Springer fiber in $\mathfrak{sl}_n$") that these components may or may not be nonsingular, and it has been computed in some special situations when they are. It has also been shown by Perrin and Smirnov ("Springer fibers in the two columns case for types A and D are normal") that for some very particular situations where explicitly non-smooth components have been identified, they are nonetheless normal. However, no general result seems to have been published (the Perrin-Smirnov paper is from 2010, for example).

My question: is anything more known about the normality or the normalization or even the desingularization of the Springer fiber?

I suppose a secondary (or more primitive) question would be if anything more than Hotta's formula is known in the direction of an explicit and practical expression for the matrix coefficients of the Springer representation.

Answer 1

Some added observations on what is likely to be a very complicated problem:

1. The 2010 Selecta Math. paper by Fresse and Melnikov which you mention was posted here, but the published version is somewhat longer. I haven't checked the precise differences, but this is always a hazard when consulting arXiv preprints. The joint work of Melnikov with various people has uncovered many of the known results about singularities of irreducible components of Springer fibers. This builds on work of Francis Fung in type $A$, for example, and mostly deals with aspects that can be formulated combinatorially.

2. As far as I can see, the geometry of Springer fibers is little understood, beyond the older work of Spaltenstein which determined in general the dimensions of the fibers and proved the equidimensionality of their components. So the refined study of singularities (or normality) is likely to require some new ideas.

3. While I agree with what Alexander Woo says in his first paragraph, I don't agree with his reading of the 1979 Kazhdan-Lusztig paper. As their introduction indicates, they were motivated by several different-looking problems including Springer's representations of Weyl groups (on cohomology of fibers in the Springer desingularization of the unipotent variety), the indirectly related problem of working out singularities of Schubert varieties, and the even more indirectly related problem of understanding algebraic work on Verma modules and primitive ideals in enveloping algebras. While those subjects do have connections, they are rather subtle, as Geordie's comment indicates.

4. One aspect which remains mysterious to me is the rough analogy between two pictures: Schubert varieties in the flag variety and their partial ordering (corresponding to the Chevalley-Bruhat ordering in the Weyl group); unipotent classes or nilpotent orbits and their closure ordering. In each case you have varieties which are sometimes not smooth, and might or might not even be normal.

Moreover, the two pictures are connected indirectly by the fact that the Springer fibers live in the cotangent bundle of the flag variety. All of this requires unpacking, even in type $A$ where the combinatorics is better-behaved.

5. Concerning Springer fibers and their irreducible components, the cohomology of a fiber turns out to be important in representation theory of different flavors. On the other hand, Slodowy's work on slices to nilpotent orbits leads to smooth varieties in the cotangent bundle which have twice the dimension of the corresponding Springer fiber but lead to the same equivariant K-theory in Lusztig's further work. He has studied these equivariant cohomology groups and their canonical bases extensively. In some ways this may be a more fruitful aspect of Springer fibers than the nonsingularity of their components, but there are many unknowns.

Answer 2

I am pretty sure what you are aware of is all that is known (publicly at least) about Springer components. You may want to try asking Fresse or Melnikov or Perrin directly. You should not expect a general result, but they may be making further progress on special cases.

As for your secondary question, my understanding (keeping in mind that I only understand this vaguely) is that Kazhdan--Lusztig polynomials were originally defined to answer precisely this question. In particular, the coefficients of $q^{(\ell(w)-\ell(v)-1)/2}$ (the largest possible by definition) in $P_{v,w}(q)$ give these matrix coefficients. (In particular, the Kazhdan--Lusztig cell representations of the Hecke algebra are the Springer representations.) Either you consider this an explicit and practical expression, or, given the state of explicit combinatorial knowledge about Kazhdan--Lusztig polynomials, you consider this as evidence the problem is impossible.

Answer 3

The paper by Kazhdan and Lusztig "A topological approach to Springer representation" may provide an answer to the question at the end of the post on an explicit form of Springer representation. Also, I think this question is related to Lusztig's homomorphism from the Hecke algebra $H$ to the asymptotic Hecke algebra $J$. To simplify the statement, assume $G=SL(n)$ (as the author of the question does). Then the matrix of the Springer representation in the canonical basis in the top homology of Springer fiber can be read off that homomorphism (specialized at $q=1$): the algebra $J$ in this case is a sum of matrix algebras, taking the summand corresponding to the given nilpotent we get the answer. The homomorphism is rather explicit, so this answer is as explicit as the canonical basis is.