A Steinberg Cross-Section for Non-Connected Affine Kac-Moody Groups

Abstract

We generalise the concept of a Steinberg cross-section to non-connected Kac-Moody group. As in the connected case, which was treated by G. Br\"uchert, a quotient map w.r.t the conjugacy action exists only on a certain submonoid of the Kac-Moody group. Non-connected Kac-Moody groups appear naturally as semidirect product of \C^* with a central extension of loop groups LG, where the underlying simple group G is no longer simply connected and might even be non-connected. In contrast to the connected case, the understanding of central extensions of non-connected loop groups is a rather complicated issue. Following the approach of V. Toledano Laredo, who dealt with the case of automorphisms coming from the fundamental group pi_1(G), we classify all of these central extensions for cyclic component group of LG. Then, we define the quotient map w.r.t conjugacy action. Furthermore, we construct the cross-section in every connected component of LG and show that, due the one-dimensional centre, it carries a natural \C^*-action which does not exist in the finite dimensional case.