A new approach to the generalized Springer correspondence

TL;DR

This paper generalizes the Springer resolution to produce a broader correspondence between Weyl group representations and perverse sheaves, confirming predictions in type A cases.

Contribution

It introduces a new generalized Springer resolution and demonstrates its effectiveness in recovering all simple perverse sheaves in type A.

Findings

All simple perverse sheaves on the nilpotent cone are obtained in type A.

The generalized approach aligns with Lusztig's predictions.

The method extends Springer correspondence to a broader setting.

Abstract

The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the Springer resolution, establishing an injective map from the set of irreducible Weyl group representations to simple equivariant perverse sheaves on the nilpotent cone. In this manuscript, we consider a generalization of the Springer resolution using a variety defined by the first author. Our main result shows that in the type A case, applying the decomposition theorem to this map yields all simple perverse sheaves on the nilpotent cone with multiplicity as predicted by Lusztig's generalized Springer correspondence.