Existence of Richardson elements for seaweed Lie algebras of finite type

Abstract

Seaweed Lie algebras are a natural generalisation of parabolic subalgebras of reductive Lie algebras. The well-known Richardson Theorem says that the adjoint action of a parabolic group has a dense open orbit in the nilpotent radical of its Lie algebra \cite{richardson}. We call elements in the open orbit Richardson elements. In \cite{JSY} together with Yu, we generalized Richardson's Theorem and showed that Richardson elements exist for seaweed Lie algebras of type $\mathbb{A}$. Using GAP, we checked that Richardson elements exist for all exceptional simple Lie algebras except $\mathbb{E}_8$, where we found a counterexample. In this paper, we complete the story on Richardson elements for seaweeds of finite type, by showing that they exist for any seaweed Lie algebra of type $\mathbb{B}$, $\mathbb{C}$ and $\mathbb{D}$. By decomposing a seaweed into a sum of subalgebras and analysing their stabilisers, we obtain a sufficient condition for the existence of Richarson elements. The sufficient condition is then verified using quiver representation theory. More precisely, using the categorical construction of Richardson elements in type $\mathbb{A}$, we prove that the sufficient condition is satisfied for all seaweeds of type $\mathbb{B}$, $\mathbb{C}$ and $\mathbb{D}$, except in two special cases, where we give a directproof.