Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

TL;DR

This paper explores generalized Springer resolutions involving nilpotent Hessenberg varieties, demonstrating their geometric properties, cohomological behaviors, and a new modular law relation, with implications for representation theory and symmetric functions.

Contribution

It introduces a new class of resolutions with fibers as Hessenberg varieties, proves their cohomological properties, and establishes a geometric modular law linking geometry and combinatorics.

Findings

Fibers have vanishing odd-degree cohomology.

The Springer sheaf decomposes into intersection cohomology sheaves from the Springer correspondence.

The geometric modular law relates cohomology of Hessenberg varieties to combinatorial symmetric functions.

Abstract

We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This leads to several implications for the dual picture, where we consider maps that generalize the Grothendieck-Springer resolution of the whole Lie algebra. In particular we are able to prove a conjecture of Brosnan. As we vary the maps, the cohomology of the corresponding nilpotent Hessenberg varieties often satisfy a relation we call the geometric modular law, which also has origins in the work on De Concini, Lusztig, and Procesi. We connect this relation in type $A$ with a combinatorial modular law defined by Guay-Paquet that is satisfied by certain symmetric functions and deduce some consequences of that connection.