The product structure of the equivariant \(K\)-theory of the based loop group of \(SU(2)\) (Q2922436)

Let \(G\) be a compact connected Lie group with the conjugation action of \(G\) on itself, \(\Omega G\) the space of continuous based loops in \(G\), equipped with the pointwise conjugation action of \(G\). The main result in this paper is a concrete computation of the equivariant \(K_G^{*}\) -algebra structure of \(G\)-equivariant \(K_G^{*}(G)\) for the specific case of the group of special unitary matrices of order 2, \(G = SU(2)\). The authors prove that \(K_G^{*}(\Omega G)\) is the inverse limit of the system of symmetric \(S_{2r}\)-invariant subalgebras \(K_G((\mathbb P^1)^{2r})^{S_{2r}})\) of \(K_G((\mathbb P^1)^{2r})\) with the natural action of the symmetric group \(S_{2r}\) on factors of the product of projective lines \((\mathbb P^1)^{2r}\) (Theorems 1.1, 6.1).