Equivariant K-theory of compact Lie group actions with maximal rank isotropy

Let G denote a compact connected Lie group with torsion-free fundamental group acting on a compact space X such that all the isotropy subgroups are connected subgroups of maximal rank. Let T subset of G be a maximal torus with Weyl group W. If the fixed-point set X-T has the homotopy type of a finite W-CW complex, then we prove that the rationalized complex equivariant K-theory of X is a free module over the representation ring of G. Given additional conditions on the W-action on the fixed-point set X-T, we show that the equivariant K-theory of X is free over R(G). We use this to provide computations for a number of examples, including the ordered n-tuples of commuting elements in G with the conjugation action.