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# On the Steinberg Map and Steinberg Cross-Section for a Symmetrizable Indefinite Kac-Moody Group

Published:2001-02-01
Printed: Feb 2001
• Claus Mokler
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## Abstract

Let $G$ be a symmetrizable indefinite Kac-Moody group over $\C$. Let $\Tr_{\La_1},\dots,\Tr_{\La_{2n-l}}$ be the characters of the fundamental irreducible representations of $G$, defined as convergent series on a certain part $G^{\tralg} \subseteq G$. Following Steinberg in the classical case and Br\"uchert in the affine case, we define the Steinberg map $\chi := (\Tr_{\La_1},\dots, \Tr_{\La_{2n-l}})$ as well as the Steinberg cross section $C$, together with a natural parametrisation $\omega \colon \C^{n} \times (\C^\times)^{\,n-l} \to C$. We investigate the local behaviour of $\chi$ on $C$ near $\omega \bigl( (0,\dots,0) \times (1,\dots,1) \bigr)$, and we show that there exists a neighborhood of $(0,\dots,0) \times (1,\dots,1)$, on which $\chi \circ \omega$ is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of $G$ on~$C$.
 MSC Classifications: 22E65 - Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] 17B65 - Infinite-dimensional Lie (super)algebras [See also 22E65]

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