Geometric and combinatorial properties of extended Springer fibers

TL;DR

This paper extends the Springer resolution to a generalized form, revealing that its fibers' geometry is governed by tableaux combinatorics, with implications for Lusztig's generalized Springer correspondence.

Contribution

It proves that fibers of the extended Springer resolution are paved by affines and provides combinatorial formulas for their Betti numbers, connecting geometry with tableaux combinatorics.

Findings

Fibers are paved by affines up to a finite group action.

Betti numbers of fibers are given by combinatorial formulas.

Dimensions of stalks of Lusztig sheaves are explicitly computed.

Abstract

We consider a generalization of the Springer resolution studied in earlier work of the authors, called the extended Springer resolution. In type $A$, this map plays a role in Lusztig's generalized Springer correspondence comparable to that of the Springer resolution in the Springer correspondence. The fibers of the Springer resolution play a key part in the latter story, and connect the combinatorics of tableaux to geometry. Our main results prove the same is true for fibers of the extended Springer resolution -- their geometry is governed by the combinatorics of tableaux. In particular, we prove that these fibers are paved by affines, up to the action of a finite group, and give combinatorial formulas for their Betti numbers. This yields, among other things, a simple formula for dimensions of stalks of the Lusztig sheaves arising in the study of the generalized Springer correspondence, and shows that there is a close resemblance between each Lusztig sheaf and the Springer sheaf for a smaller group.