Parabolic Actions in type A and their Eigenslices

Abstract

The main theme of this paper is that many of the remarkable properties of invariant theory pertaining to semisimple Lie algebras carry over to parabolic subalgebras even though the latter have less structure. This includes the polynomiality of the invariant subalgebra of the symmetric algebra of a (truncated) parabolic subalgebra, the existence of a slice to the regular coadjoint orbits and the construction of maximal Poisson commutative polynomial subalgebras by "shift of argument". The first of these properties was established for most parabolics in [FJ1]. Here the existence of a slice to (most) regular coadjoint orbits is established for parabolics in type A which are invariant under the Dynkin diagram involution. In a subsequent paper [JL] maximal Poisson commutative polynomial subalgebras are described for those (truncated) parabolics in sl(n) having index n - 1.