On divisible weighted Dynkin diagrams and reachable elements

Abstract

Let e be a nilpotent element of a complex simple Lie algebra $\mathfrak{g}$. The weighted Dynkin diagram of e, $\mathcal{D}(e)$, is said to be divisible if ${{{\mathcal{D}(e)}} \left/ {2} \right.}$ is again a weighted Dynkin diagram. The corresponding pair of nilpotent orbits is said to be friendly. In this paper we classify the friendly pairs and describe some of their properties. Any subalgebra $\mathfrak{s}{\mathfrak{l}_3}$ in $\mathfrak{g}$ gives rise to a friendly pair; such pairs are called A₂-pairs. If Gx is the lower orbit in an A₂-pair, then $x \in \left[ {{\mathfrak{g}^x},{\mathfrak{g}^x}} \right]$, i.e., x is reachable. We also show that ${\mathfrak{g}^x}$ has other interesting properties. Let ${\mathfrak{g}^x} = { \oplus_{i \geqslant 0}}{\mathfrak{g}^x}(i)$ be the $\mathbb{Z} - {\text{grading}}$ determined by a characteristic of x. We prove that ${\mathfrak{g}^x}$ is generated by the Levi subalgebra ${\mathfrak{g}^x}(0)$ and two elements of ${\mathfrak{g}^x}(1)$. In particular, the nilpotent radical of ${\mathfrak{g}^x}$ is generated by the subspace ${\mathfrak{g}^x}(1)$.