  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.
