On the centralizer of K in

Let be a complexified Cartan decomposition of a complex semisimple Lie algebra and let K be the subgroup of the adjoint group of corresponding to . If H is an irreducible Harish-Chandra module of , then H is completely determined by the finite-dimensional action of the centralizer on any one fixed primary 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 . Generators of may be given by generators of the symmetric algebra analogue . Let , , be the subalgebra of defined by K-invariant polynomials of degree at most m. For convenience write and Am for the subalgebra of A generated by . Let Q and Qm be the respective quotient fields of A and Am. We prove that if one has Q=Q2n. We also determine the variety, NilK, of unstable points with respect to the action K on 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 .