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

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

2004-09-03

arxiv.org/abs/math/0409055v1

Mathematics

Algebraic Geometry

17 pages

Scientific paper

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.

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