Complexity of actions over perfect fields

Abstract: Let $G$ be a connected reductive group over a perfect field $k$ acting on an algebraic variety $X$ and let $P$ be a minimal parabolic subgroup of $G$. For $k$-spherical $G$-varieties we prove finiteness result for $P$-orbits that contain $k$-points. This is a consequence of an equality on $P$-complexities of $X$ and of any $P$-invariant $k$-dense subvariety in $X$, which generalizes a corresponding result of E.B.Vinberg in the case of algebraically closed field $k$. Also we introduce an action of the restricted Weyl group $W$ on the set of $k$-dense $P$-invariant closed subvarieties of $X$ of maximal $P$-complexity and $k$-rank in the case of ${\rm char}\ k =0$ and on the set of all $k$-dense $P$-orbits in the case of real spherical variety which generalizes the action on $B$-orbits introduced by F.Knop in the algebraically closed field case. We also introduce a little Weyl group related with this action and describe its generators in terms of the generators of $W$ which generalize the description of M.Brion in algebraically closed field case.