The module structure of the equivariant K-theory of the based loop group of SU(2)

Let $G=SU(2)$ and let $\mathrm{\Omega }G$ denote the space of based loops in SU(2). We explicitly compute the $R(G)$-module structure of the topological equivariant $K$-theory $K_G^{\ast }(\mathrm{\Omega }G)$ and in particular show that it is a direct product of copies of $K_G^{\ast }(\text{pt})\cong R(G)$. (We intend to describe in detail the $R(G)$-algebra (i.e. product) structure of $K_G^{\ast }(\mathrm{\Omega }G)$ in a forthcoming companion paper.) Our proof uses the geometric methods for analyzing loop spaces introduced by Pressley and Segal (and further developed by Mitchell). However, Pressley and Segal do not explicitly compute equivariant $K$-theory and we also need further analysis of the spaces involved since we work in the equivariant setting. With this in mind, we have taken this opportunity to expand on the original exposition of Pressley-Segal in the hope that in doing so, both our results and theirs would be made accessible to a wider audience.