 
Summary: EQUIVARIANT KTHEORY OF COMPACT LIE GROUP ACTIONS WITH
MAXIMAL RANK ISOTROPY
ALEJANDRO ADEM
AND JOS´E MANUEL G´OMEZ
Abstract. Let G denote a compact connected Lie group with torsionfree fundamental group
acting on a compact space X such that all the isotropy subgroups are connected subgroups of
maximal rank. Let T G be a maximal torus with Weyl group W. In this article we derive
conditions on the induced action of W on the fixedpoint set XT
which imply that the complex
equivariant Ktheory of X is a free module over the representation ring of G. We use this to
provide computations of equivariant Ktheory for key examples, including the ordered ntuples
of commuting elements in G with the conjugation action as well as certain linear Gspheres.
1. Introduction
Let G denote a compact connected Lie group with torsionfree fundamental group. Suppose
that G acts on a compact space X so that each isotropy subgroup is a connected subgroup
of maximal rank; that is, Gx contains a maximal torus T G for every x X. In this
article we study the problem of computing K
G(X), the complex Gequivariant Ktheory of
X. Our work is primarily motivated by the examples given by spaces of ordered commuting
ntuples in compact matrix groups such as U(m), SU(m) and Sp(m) with the conjugation
