Equivariant twisted Real K-theory of compact Lie groups

Let G be a compact, connected, and simply-connected Lie group viewed as a G-space via the conjugation action. The Freed-Hopkins-Teleman Theorem (FHT) asserts a canonical link between the equivariant twisted K-homology of G and its Verlinde algebra. In this paper, we give a 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 third cohomology group which is shown to classify the former, and Real Spinc structures.