Coadjoint orbits of reductive type of seaweed Lie algebras

TL;DR

This paper studies the structure of coadjoint orbits of reductive type in certain algebraic groups, focusing on quasi-reductive parabolic and seaweed subalgebras, and describes their maximal reductive stabilisers.

Contribution

It provides a classification of maximal reductive stabilisers for coadjoint orbits of quasi-reductive parabolic and seaweed subalgebras in complex Lie algebras.

Findings

Classification of maximal reductive stabilisers for parabolic subalgebras.

Description of maximal reductive stabilisers for seaweed subalgebras of gl(n).

Extension of Duflo's results to new classes of subalgebras.

Abstract

A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of its dual of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup of GL(q), where q=Lie(Q). Due to results of M. Duflo, coadjoint representation of a quasi-reductive Q possesses a so called maximal reductive stabiliser and knowing this subgroup, defined up to a conjugation in Q, one can describe all coadjoint orbits of reductive type. In this paper, we consider quasi-reductive parabolic subalgebras of simple complex Lie algebras as well as all seaweed subalgebras of gl(n) and describe the classes of their maximal reductive stabilisers.