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

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