The Real K-Theory Of Compact Lie Groups

Abstract

Let G be a compact, connected, and simply-connected Lie group, equipped with a Lie group involution [sigma]G and viewed as a G-space via the conjugation action. In this thesis we compute Atiyah's Real K-theory of G in several contexts. We first obtain a complete description of the algebra structure of the equivariant KR-theory of both (G, [sigma]G ) and (G, [sigma]G ? inv), where inv means group inversion, by utilizing the notion of Real equivariant formality and drawing on previous results on the module structure of the KR-theory and the ring structure of the equivariant K-theory. The Freed-Hopkins-Teleman Theorem (FHT) asserts a canonical link between the equivariant twisted K-homology of G and its Verlinde algebra. In the latter part of the thesis we give a partial generalization of FHT in the presence of a Real structure of G. Along the way we develop preliminary materials necessary for this generalization, which are of independent interest in their own right. These include the definitions of Real Dixmier-Douady bundles, the Real their cohomology group which is shown to classify the former, and Real Spinc structures.