Chikwong Fok

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 $(G, \sigma_G\circ\text{inv})$, where $\text{inv}$ 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 third cohomology group which is shown to classify the former, and Real $\text{Spin}^c$ structures.