On the centralizer of K in U(g)

Let g=k+p be a complexified Cartan decomposition of a complex semisimple Lie algebra g and let K be the subgroup of the adjoint group of g corresponding to k. If H is an irreducible Harish-Chandra module of U(g), then H is completely determined by the finite-dimensional action of the centralizer UK(g) on any one fixed primary k component in H. This original approach of Harish-Chandra to a determination of all H has largely been abandoned because one knows very little about generators of UK(g). Generators of UK(g) may be given by generators of the symmetric algebra analogue SK(g). Let SmK(g), m∈Z+, be the subalgebra of SK(g) defined by K-invariant polynomials of degree at most m. For convenience write A=SK(g) and Am for the subalgebra of A generated by SmK(g). Let Q and Qm be the respective quotient fields of A and Am. We prove that if n=dimg one has Q=Q2n.We also determine the variety, NilK, of unstable points with respect to the action K on g and show that NilK is already defined by A2n. As pointed out to us by Hanspeter Kraft, this fact together with a result of Harm Derksen (see [H. Derksen, Polynomial bounds for rings of invariants, Proc. Amer. Math. Soc. 129 (4) (2001) 955–963]) implies, indeed, that A=Ar where .