Non-stable \(K_1\)-functors of multiloop groups (Q2786457)

Let \(k\) be a field of characteristic 0. Let \(G\) be a reductive group over the ring of Laurent polynomials \(R=k[x_1^ {\pm 1} ,\ldots ,x_n^ {\pm 1} ]. \) Assume that \(G\) contains a maximal \(R\)-torus, and that every semisimple normal subgroup of \(G\) contains a two-dimensional split torus \textbf{G}\(_m ^2 .\) The author shows that the natural map of non-stable \(K_1\) -functors, also called Whitehead groups, \(K_1^G (R) \to K_1^G (k ((x _1 )) \cdots((x _n )))\) is injective, and an isomorphism if \(G\) is semisimple. As an application, she provides a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii-Neher) and the subgroup generated by exponential automorphisms.