THE PRODUCT STRUCTURE OF THE EQUIVARIANT K-THEORY OF THE BASED LOOP GROUP OF SU(2)

Let G=SU(2) and let Ω 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*(Ω G) is a divided polynomial algebra Γ[x]. The algebra Γ[x] can be described as an inverse limit as k→∞ of the symmetric subalgebra in Λ(x1, …, xk), where Λ(x1, …, xk) is the usual exterior algebra in the variables x1, …, xk. We compute the R(G)-algebra structure of the G-equivariant K-theory K*G(Ω G) which naturally generalizes the classical computation of H*(Ω G) as Γ[x]. Specifically, we prove that K*G(Ω G) is an inverse limit of the symmetric (S2r-invariant) subalgebra (K*G((ℙ1)2r))S2r of K*G((ℙ1)2r), where the symmetric group S2r acts in the natural way on the factors of the product (ℙ1)2r and G acts diagonally via the standard action on each factor.