Stability for biparabolic subalgebras of simple Lie algebras

TL;DR

This paper proves that for biparabolic subalgebras of simple Lie algebras, stability and quasi-reductivity are equivalent, confirming a conjecture by Panyushev.

Contribution

It establishes the equivalence between stability and quasi-reductivity specifically for biparabolic subalgebras of simple Lie algebras, resolving a conjecture.

Findings

Confirmed Panyushev's conjecture for biparabolic subalgebras.

Showed stability implies quasi-reductivity in this class.

Provided a characterization of stability in the context of biparabolic subalgebras.

Abstract

Let K be an algebraically closed field of characteristic 0. It is well known that any quasi-reductive Lie algebra is stable. However, there are stable Lie algebras which are not quasi-reductive. This raises the question, if for some particular class of non-reductive Lie algebras, there is equivalence between stability and quasi-reductivity. In particular, it was conjectured by Panyushev that these two notions are equivalent for biparabolic subalgebras of a reductive Lie algebra. In this paper, we give a positive answer to this conjecture.