Rationality and Orbit Closures

[摘要] Suppose we are given a finite-dimensional vector space $V$ equippedwith an $F$-rational action of a linearly algebraic group $G$, with$F$ a characteristic zero field. We conjecture the following: to eachvector $vin V(F)$ there corresponds a canonical $G(F)$-orbit ofsemisimple vectors of $V$. In the case of the adjoint action, thisorbit is the $G(F)$-orbit of the semisimple part of $v$, so thisconjecture can be considered a generalization of the Jordandecomposition. We prove some cases of the conjecture.

[学科分类] 数学（综合）

[关键词] Frankel's Theorem