IJPAM: Volume 47, No. 2 (2008)

A COMPACTNESS PROPERTY ASSOCIATED WITH
COVERINGS OF REAL ALGEBRAIC GROUPS

Martin Moskowitz
Ph.D. Program in Mathematics
Graduate Center
City University of New York
365, Fifth Avenue, New York, NY 10016-4309, USA
e-mail: martin.moskowitz@gmail.com


Abstract.In this note we show that for any group covering $\pi:G\rightarrow G^*$, where $G$ is a connected Lie group and $G^*$ is a real algebraic group, if the $G^*$-conjugacy class of an $x^*\in G^*$ is compact and $x\in G$ maps onto $x^*$, then so is the $G$-conjugacy class of $x$. It follows that if the $G^*$-conjugacy class of $x^*$ has a finite $G^*$-invariant measure, then the $G$-conjugacy class of $x$ also has a finite $G$-invariant measure. Account has to be taken of both the Zariski and Euclidean topologies.

Received: June 19, 2008

AMS Subject Classification: 22E15, 11F23

Key Words and Phrases: covering group, real algebraic group, conjugacy class, and centralizer of an element, compact orbit, finite invariant measure, adjoint representation, finite sheeted covering, equivariance, the Zariski and Euclidean topologies

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 47
Issue: 2