Nilpotent Orbits for Borel Subgroups of $SO_{5}(k)$

TL;DR

This paper investigates the nilpotent orbits of Borel subgroups acting on the Lie algebra of SO(5), providing explicit equations, orbit dimensions, and closure relations, extending known classifications for type B2.

Contribution

It explicitly determines polynomial equations, orbit dimensions, and closure orderings for nilpotent orbits in SO(5), advancing the classification of Borel subgroup actions in this case.

Findings

Derived polynomial equations for each orbit.

Calculated dimensions of the orbits.

Established closure ordering among orbits.

Abstract

Let $G$ be a quasi-simple algebraic group defined over an algebraically closed field $k$ and $B$ a Borel subgroup of $G$ acting on the nilradical $\mathfrak{n}$ of its Lie algebra $\mathfrak{b}$ via the Adjoint representation. It is known that $B$ has only finitely many orbits in only five cases: when $G$ is of type $A_{n}$ for $n \leq 4$, and when $G$ is type $B_{2}$. In this paper, we elaborate on this work in the case when $G =SO_{5}(k)$ (type $B_{2})$ by finding the polynomial 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.