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

http://dx.doi.org/10.4153/CJM-2006-026-7
Canad. J. Math. 58(2006), 625-642

  Published:2006-06-01
 Printed: Jun 2006

In this paper we generalise the concept of a Steinberg cross section to non-connected affine Kac--Moody groups. This Steinberg cross section is a section to the restriction of the adjoint quotient map to a given exterior connected component of the affine Kac--Moody group. (The adjoint quotient is only defined on a certain submonoid of the entire group, however, the intersection of this submonoid with each connected component is non-void.) The image of the Steinberg cross section consists of a ``twisted Coxeter cell'', a transversal slice to a twisted Coxeter element. A crucial point in the proof of the main result is that the image of the cross section can be endowed with a $\Cst$-action.