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EQUIVARIANT K-THEORY OF COMPACT LIE GROUP ACTIONS WITH MAXIMAL RANK ISOTROPY
 

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. 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 fixed-point 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 n-tuples in compact matrix groups such as SU(r), U(q) and Sp(k) with
the conjugation action.

  

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

 

Collections: Mathematics