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Trace Class Elements and Cross-Sections in Kac-Moody Groups

Published online by Cambridge University Press:  20 November 2018

Gerd Brüchert
Gerd Brüchert*
Affiliation:
Mathematisches Seminar, Universität Hamburg, Bundesstrasse, 55, 20146 Hamburg, Germany email: bruechert@math.uni-hamburg.de
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Abstract

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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.

Keywords

22E6517B65
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 5 , 01 October 1998 , pp. 972 - 1006
Copyright
Copyright © Canadian Mathematical Society 1998

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