The real \(K\)-theory of compact Lie groups (Q2447880)

The two main results of the paper formulate and prove analogues of known computations of the ring structure of the equivariant \(K\)-theory due to \textit{J.-L. Brylinski} and \textit{B. Zhang} [\(K\)-Theory 20, No. 1, 23--26 (2000; Zbl 0971.19003)]. The theory addressed in this paper is the real \(K\)-theory or \(KR\)-theory defined by Atiyah, which is a version of topological \(K\)-theory for spaces equipped with an involution. One of the necessary contributions is a careful definition of the equivariant theory in this setting based on what the author calls ``real equivariant formality''. The author then describes the ring structure of the equivariant \(KR\)-theory of any compact, connected and simply-connected Lie group in terms of relations of generators associated to real representations of \(G\). To contrast, the previous work of Brylinski/Zhang showed that for that the equivariant \(K\)-theory of a compact connected Lie group \(G\), which acts on itself by conjugation, is isomorphic to the ring of Grothendieck differentials of the complex representation ring \(R(G)\). The author points out that in general the equivariant \(KR\)-theory is not a ring of Grothendieck differentials. However, in certain cases, the result is an exterior algebra over the localized coefficient ring of equivariant \(KR\)-theory after inverting 2 in the equivariant \(KR\)-theory ring.