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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On bounded elements of linear algebraic groups

Author: Kwan-Yuk Claire Sit
Journal: Trans. Amer. Math. Soc. 209 (1975), 185-198
MSC: Primary 22E20; Secondary 20G25
MathSciNet review: 0379750
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Abstract: Let $ F$ be a local field of characteristic zero and $ {\text{G }}$ a connected algebraic group defined over $ F$. Let $ G$ be the locally compact group of $ F$-rational points. One characterizes the group $ B(G)$ of $ g \in G$ whose conjugacy class is relatively compact. For instance, if $ {\text{G}}$ is $ F$-split or reductive without anisotropic factors then $ B(G)$ is the center of $ G$. If $ H$ is a closed subgroup of $ G$ such that $ G/H$ has finite volume, then the centralizer of $ H$ in $ G$ is contained in $ B(G)$. If, moreover, $ H$ is the centralizer of some $ x \in G$ then $ G/H$ is compact.

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Keywords: Local fields, automorphisms of bounded displacement, bounded elements, homogeneous spaces of finite volume, density theorem
Article copyright: © Copyright 1975 American Mathematical Society