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Trace Class Elements and Cross-Sections in Kac-Moody Groups
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $G$ be an affine Kac-Moody group,
${{\pi }_{0}},...,{{\pi }_{r}},{{\pi }_{\delta }}$ its fundamental irreducible representations and
${{\chi }_{0}},...,{{\chi }_{r}},{{\chi }_{\delta }}$ their characters. We determine the set of all group elements
$x$ such that all
${{\pi }_{i}}(x)$ act as trace class operators, i.e., such that
${{\chi }_{i}}(x)$ exists, then prove that the
${{\chi }_{i}}$ are class functions. Thus, (
$\chi \,:=\,({{\chi }_{0}},...,{{\chi }_{r}},\,{{\chi }_{\delta }})$)factors to an adjoint quotient
$\bar{\chi }$ for
$G$. In a second part, following Steinberg, we define a cross-section
$C$ for the potential regular classes in
$G$. We prove that the restriction
$\chi \text{ }\!\!|\!\!\text{ }c$ behaves well algebraically. Moreover, we obtain an action of
${{\mathbb{C}}^{\times }}$ on
$C$, which leads to a functional identity for
$\text{ }\chi |\text{ c}$ which shows that
$\text{ }\chi |\text{ c}$ is quasi-homogeneous.
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- Copyright © Canadian Mathematical Society 1998
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