  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.
