A transitivity result for ad-nilpotent ideals in type A

TL;DR

This paper establishes a transitivity property for ad-nilpotent ideals in type A, linking their equivalence classes to their largest nilpotent orbits, with applications in cohomology and Kazhdan-Lusztig theory.

Contribution

It proves that under a specific equivalence relation, ad-nilpotent ideals with the same largest nilpotent orbit form classes, advancing understanding of their structure and applications.

Findings

Equivalence classes correspond to ideals with the same largest nilpotent orbit

Applications to vanishing cohomology of vector bundles on flag varieties

Connections to Kazhdan-Lusztig cells in affine Weyl groups

Abstract

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal meets a unique largest nilpotent orbit in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals with the same largest nilpotent orbit. We include two applications of the result, one to the higher vanishing of cohomology groups of vector bundles on the flag variety and another to the Kazhdan-Lusztig cells in the affine Weyl group of the symmetric group. Finally, some combinatorial results are discussed.