Let G=SU(2) and let Omega G denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H^*(Omega G) is a divided polynomial algebra Gamma[x]. T
he algebra Gamma[x] can be described as an inverse limit as k goes to infinity of the symmetric subalgebra in the exterior algebra Lambda(x_1, ...,x_k) in the variables x_1, ..., x_k. We compute the R(G)-algebra structure of the G-equivariant K-theory of Omega G in a way which naturally generalizes the classical computation of the ordinary cohomology ring of Omega G as a divided polynomial algebra Gamma[x]. Specifically, we prove that K^*_G(Omega G) is an inverse limit of the symmetric (S_{2r}-invariant) subalgebra of K^*_G((P^1)^{2r}), where the symmetric group S_{2r} acts in the natural way on the factors of the 2r-fold product (P^1)^{2r} and G acts diagonally via the standard action on each complex projective line P^1.
Let $G=SU(2)$ and let $Omega G$ denote the space of based loops in SU(2). We explicitly compute the $R(G)$-module structure of the topological equivariant $K$-theory $K_G^*(Omega G)$ and in particular show that it is a direct product of 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 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.
