# Nilpotent orbits of height 2 and involutions in the affine Weyl group

## TL;DR

This paper classifies and describes the structure of certain nilpotent orbits of height 2 in Lie algebras of algebraic groups, linking their orbit closures to involutions in the affine Weyl group.

## Contribution

It provides a parametrization of B-orbits in nilpotent elements of height 2 and describes their closure relations via affine Weyl group involutions.

## Key findings

- Parametrization of B-orbits using orthogonal root subsets
- Complete description of orbit closure order via affine Weyl group involutions
- Finiteness of B-orbits on nilpotent elements of height ≤ 2

## Abstract

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in g whose height is at most 2. We provide a parametrization of the B-orbits in N_2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g.