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.