Nilpotent orbits for Borel subgroups of SO5(k)

Burkhart, Madeleine; Vella, David

doi:10.2140/involve.2019.12.451

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Nilpotent orbits for Borel subgroups of $\mathrm{SO}_{5}(k)$

Madeleine Burkhart and David Vella

Vol. 12 (2019), No. 3, 451–462

DOI: 10.2140/involve.2019.12.451

Abstract

Let $G$ be a quasisimple algebraic group defined over an algebraically closed field $k$ and $B$ a Borel subgroup of $G$ acting on the nilradical $n$ of its Lie algebra $b$ via the adjoint representation. It is known that $B$ has only finitely many orbits in only five cases: when $G$ is type $A_{n}$ for $n \leq 4$ , and when $G$ is type $B_{2}$ . We elaborate on this work in the case when $G = {SO}_{5} (k)$ (type $B_{2}$ ) by finding the defining equations of each orbit. We use these equations to determine the dimension of the orbits and the closure ordering on the set of orbits. The other four cases, when $G$ is type $A_{n}$ , can be approached the same way and are treated in a separate paper.

Keywords

nilpotent orbits, Borel subgroups, modality

Mathematical Subject Classification 2010

Primary: 17B08, 20G05

Milestones

Received: 16 August 2017

Revised: 8 February 2018

Accepted: 10 July 2018

Published: 14 December 2018

Communicated by Kenneth S. Berenhaut

Authors

Madeleine Burkhart
Mathematics Department University of Washington Seattle, WA United States

David Vella
Mathematics and Statistics Department Skidmore College Saratoga Springs, NY United States

Vol. 12, No. 3, 2019