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

Abstract

Abstract. 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 ⊂ 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, 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 XT 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.

On Spaces of Commuting Elements in Lie Groups

“…Consequently the claim (5) follows from (6) and (8), since graded traces are multiplicative on graded tensor products. This completes the proof.…”

“…Their work was followed by work of T. Baird [9], who studied ordinary and equivariant cohomology, Baird-JeffreySelick [10],Ádem-Gómez [6,8,7],Ádem-Cohen-Gómez [3,4], Sjerve-Torres-Giese [27],Ádem-Cohen-Torres-Giese [5], Pettet-Suoto [26], Gómez-Pettet-Suoto [19], Okay [25]. Most of this work has been focused on the study of invariants such as cohomology, K-theory, connected components, homotopy type and stable decompositions.…”

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

“…In this case the homomorphism φ : π 1 (X T ) → Hom(T, S 1 ) corresponding to p is also trivial. Therefore we have a natural isomorphism H p π 1 (X T ) ( X T ; R Q ) W ∼ = H p (X T ; R(T ) Q ) W and the latter is isomorphic to H * (X T ; Q) ⊗ R(G) as a module over R(G) Q by [1,Theorem 4.3]. In this case the spectral sequence collapses at the E 2 -term without extension problems by [1,Theorem 5.4].…”

On the space of commuting orthogonal matrices

“…More recently, in [4], Adem and Gómez have described all compact simply connected Lie groups that have connected C k .G/ for all k as the finite products of the groups Sp.n/ and SU.m/.…”

“…Then C k .G/ D S T DˆT .G T k / and so it is connected since G T k is connected. 4 The connected components of C k .O.n// In this section we will apply the strategy described in the previous section to prove Theorem 1.1. It follows directly from the next theorem.…”