Non-stable K1-functors of Multiloop Groups

Abstract

AbstractLet k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials . Assume that G contains amaximal R-torus, and that every semisimple normal subgroup of G contains a two-dimensional split torus G2m. We show that the natural map of non-stable K1-functors, also called Whitehead groups, KG1(R) → KG1 ( k((x1)) … ((xn))) is injective, and an isomorphism if G is semisimple. As an application, we provide 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.