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Trace class elements and crosssections in KacMoody groups


Published:19981001
Printed: Oct 1998
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Abstract
Let $G$ be an affine KacMoody group, $\pi_0,\dots,\pi_r,\pi_{\delta}$
its fundamental irreducible representations and $\chi_0, \dots,
\chi_r, \chi_{\delta}$ their characters. We determine the set of all
group elements $x$ such that all $\pi_i(x)$ act as trace class
operators, \ie, such that $\chi_i(x)$ exists, then prove that the
$\chi_i$ are class functions. Thus, $\chi:=(\chi_0, \dots, \chi_r,
\chi_{\delta})$ factors to an adjoint quotient $\bar{\chi}$ for $G$.
In a second part, following Steinberg, we define a crosssection $C$
for the potential regular classes in $G$. We prove that the
restriction $\chi_C$ behaves well algebraically. Moreover, we obtain
an action of $\hbox{\Bbbvii C}^{\times}$ on $C$, which leads to a
functional identity for $\chi_C$ which shows that $\chi_C$ is
quasihomogeneous.