Complexity of actions over perfect fields

TL;DR

This paper investigates the complexity of actions of reductive groups over perfect fields on algebraic varieties, establishing finiteness results for certain orbits and introducing group actions that generalize known structures to broader field contexts.

Contribution

It generalizes Vinberg's equality on P-complexities to perfect fields and introduces Weyl group actions on P-invariant subvarieties and orbits in this setting.

Findings

Finiteness of P-orbits containing k-points for k-spherical varieties.

Equality of P-complexities for varieties and their P-invariant subvarieties.

Introduction of a Weyl group action on P-invariant subvarieties and orbits.

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.