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# Rationality and Orbit Closures

Published:2003-06-01
Printed: Jun 2003
• Jason Levy
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## Abstract

Suppose we are given a finite-dimensional vector space $V$ equipped with an $F$-rational action of a linearly algebraic group $G$, with $F$ a characteristic zero field. We conjecture the following: to each vector $v\in V(F)$ there corresponds a canonical $G(F)$-orbit of semisimple vectors of $V$. In the case of the adjoint action, this orbit is the $G(F)$-orbit of the semisimple part of $v$, so this conjecture can be considered a generalization of the Jordan decomposition. We prove some cases of the conjecture.
 MSC Classifications: 14L24 - Geometric invariant theory [See also 13A50] 20G15 - Linear algebraic groups over arbitrary fields