 
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. If the fixedpoint set XT
has the homotopy type of a finite WCW complex, we prove that the rationalized complex
equivariant Ktheory of X is a free module over the representation ring of G. Given additional
conditions on the Waction on the fixedpoint set XT
we show that the equivariant Ktheory of
X is free over R(G). We use this to provide computations for a number of examples, including
the ordered ntuples of commuting elements in G with the conjugation action.
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. In this article we study the problem of computing K
G(X), the complex
Gequivariant Ktheory of X. Our work is motivated by the examples given by spaces of
ordered commuting ntuples in compact matrix groups such as SU(r), U(q) and Sp(k) with
the conjugation action.
