Rationality and Orbit Closures

http://dx.doi.org/10.4153/CMB-2003-021-6
Canad. Math. Bull. 46(2003), 204-215

  Published:2003-06-01
 Printed: Jun 2003

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.