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Instabile rationale Konjugationsklassen in reduktiven Gruppen

Horn, Tobias

English Title: Instable rational Conjugacy Classes in reductive Groups

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Abstract

Sei $G$ eine zusammenhängende quasizerfallende reduktive Gruppe über einem perfekten Körper $K$ und sei die Charakteristik von $K$ gut. Steinberg und Kottwitz untersuchen die Frage, ob eine rationale Konjugationsklasse $x$ ein Element aus $G(K)$ enthält. Hierbei studieren sie die Konjugationsklasse in der einfach zusammenhängenden Überlagerung $G_{\operatorname{sc}}$ von $G$. Das Ziel dieser Arbeit ist es, einen weiteren Zugang für quasizerfallende halbeinfache Gruppen anzugeben. Zu allererst beweisen wir, daß eine rationale Konjugationsklasse in einer zusammenhängenden zerfallenden Gruppe $G$ ein Element aus $G(K)$ besitzt. Dieses Resultat führt zu einer Klassifikation aller rationaler Konjugationsklassen in zusammenhängenden, quasizerfallenden, einfachen Gruppen, die kein rationales Element enthalten. Das Cupprodukt beider Zugänge beschreibt dies.

Translation of abstract (English)

Let $G$ be a connected, quasi-split, reductive group over a perfect field $K$ of good characteristic. Steinberg and Kottwitz study whether a rational conjugacy class $x$ in $G$ contains an element of $G(K)$. Their approach investigates the conjugacy class $x$ in the simply connected cover $G_{\operatorname{sc}}$ of $G$. The main purpose of this paper is to give a different second approach for quasi-split, semisimple groups. First of all we show that each rational conjugacy class of a connected, {\emph split}, semisimple group contains an element of $G(K)$. This result leads to a classification of all rational conjugacy classes in connected, quasi-split, simple groups which have no element of $G(K)$. They are classified by the cup product of both approaches.

Item Type: Dissertation
Supervisor: Weissauer, Prof. Dr. Rainer
Date of thesis defense: 4 February 2005
Date Deposited: 10 Feb 2005 10:45
Date: 2004
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Mathematics
Subjects: 510 Mathematics
Controlled Keywords: Reduktive Gruppe, Konjugationsklasse, Steinberg, Robert
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