Summary: EQUIVARIANT K-THEORY 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
n-tuples in compact matrix groups such as U(m), SU(m) and Sp(m) with the conjugation

Source: Adem, Alejandro - Department of Mathematics, University of British Columbia

Collections: Mathematics