Graham's Variety and Perverse Sheaves on the Nilpotent Cone

TL;DR

This paper explores Graham's variety and its associated perverse sheaf on the nilpotent cone, focusing on the structure of the sheaf's summands and their relation to affine pavings in type A.

Contribution

It introduces a new perspective on Graham's variety, analyzing the decomposition of the related perverse sheaf, especially in type A, using combinatorial affine pavings.

Findings

Graham's map is a universal covering over the principal orbit.

The perverse sheaf decomposes into summands related to affine pavings.

Results are specific to type A_n with combinatorial descriptions.

Abstract

In recent work, Graham has constructed a variety with a map to the nilpotent cone which is similar in some ways to the Springer resolution. One aspect in which Graham's map differs is that it is not in general an isomorphism over the principal orbit, but rather the universal covering map. This map gives rise to a certain semisimple perverse sheaf on the nilpotent cone, and we discuss here the problem of describing the summands of this perverse sheaf. For type $A_n$, a key tool is a known combinatorial description of an affine paving of Springer fibers.