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

Brüchert, Gerd

Canadian journal of mathematics, Tome 50 (1998) no. 5, pp. 972-1006

Résumé

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.

Détail

DOI : 10.4153/CJM-1998-049-3

Mots-clés : 22E65, 17B65

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