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 U (g) K 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 U (g) K. Generators of U (g) K may be given by generators of the symmetric algebra analogue S(g) K. Let S m (g) K , m ∈ Z + , be the subalgebra of S(g) K defined by K-invariant polynomials of degree at most m. For convenience write A = S(g) K and A m for the subalgebra of A generated by S m (g) K. Let Q and Q m be the respective quotient fields of A and A m. We prove that if n = dim g one has Q = Q 2n. We also determine the variety, N il K , of unstable points with respect to the action K on g and show that N il K is already defined by A 2n. As pointed out to us by Hanspeter Kraft this fact together with a result of Harm Derksen (See [D]) implies, indeed, that A = A r where r = 2n 2 dim p.