On the centralizer of K in [formula omitted

Abstract

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 2 n . We also determine the variety, Nil K , of unstable points with respect to the action K on g and show that Nil K is already defined by A 2 n . 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 = A r where r = ( 2 n 2 ) dim p .