An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras

Dmitri I. Panyushev

Published 2004-09-03Version 1

Let $\g$ be a simple Lie algebra of type A or C. We show that the coadjoint representation of any seaweed subalgebra of $\g$ has some properties similar to that of the adjoint representation of a reductive Lie algebra. Namely, a) the field of invariants is rational and b) there exists a generic stabiliser whose identity component is a torus. Our main tool for this is a result about coadjoint representations of some N-graded Lie algebras, which can be regarded as an extension of Rais' theorem for the index of semi-direct products. For all other simple types, we give a uniform description of a parabolic subalgebra such that its coadjoint representation has no generic stabiliser. The crucial property here is that if $\g$ is not of type A or C, then the highest root is fundamental. We also show that, for any parabolic subgroup, the ring of regular invariants of the coadjoint representation is trivial.