The module structure of the equivariantK-theory of the based loop group ofSU(2)
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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.
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