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

Published:2006-06-01
Printed: Jun 2006
• Stephan Mohrdieck
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## Abstract

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.
 MSC Classifications: 22E67 - Loop groups and related constructions, group-theoretic treatment [See also 58D05]

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