Abstract view
On the Steinberg Map and Steinberg CrossSection for a Symmetrizable Indefinite KacMoody Group


Published:20010201
Printed: Feb 2001
Abstract
Let $G$ be a symmetrizable indefinite KacMoody group over $\C$. Let
$\Tr_{\La_1},\dots,\Tr_{\La_{2nl}}$ 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 $\chi := (\Tr_{\La_1},\dots,
\Tr_{\La_{2nl}})$ as well as the Steinberg cross section $C$,
together with a natural parametrisation $\omega \colon \C^{n} \times
(\C^\times)^{\,nl} \to C$. 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~$C$.