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

Abstract

Let G=SU(2) and let ΩG 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^*(ΩG) is a divided polynomial algebra Γ[x]. The algebra Γ[x] can be described as an inverse limit as k goes to infinity of the symmetric subalgebra in the exterior algebra Λ(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 ΩG in a way which naturally generalizes the classical computation of the ordinary cohomology ring of ΩG as a divided polynomial algebra Γ[x]. Specifically, we prove that K^*_G(ΩG) 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.