Point stabilisers for the enhanced and exotic nilpotent cones

TL;DR

This paper provides a detailed decomposition of point stabilisers in enhanced and exotic nilpotent cones, deriving formulas for orbit point counts over finite fields, confirming a recent conjecture.

Contribution

It introduces a semi-direct product decomposition of point stabilisers and derives explicit formulas for orbit sizes, advancing understanding of nilpotent cone structures.

Findings

Semi-direct product decomposition of point stabilisers

Formulas for orbit point counts over finite fields

Confirmation of Achar and Henderson's conjecture

Abstract

We give a semi-direct product decomposition of the point stabilisers for the enhanced and exotic nilpotent cones. In particular, we arrive at formulas for the number of points in each orbit over a finite field. This is in accordance with a recent conjecture of Achar and Henderson.