论文标题
对完美领域的动作的复杂性
Complexity of actions over perfect fields
论文作者
论文摘要
让$ g $成为一个连接的还原组，比在代数品种$ x $上作用的完美字段$ k $，让$ p $是$ g $的最小抛物线子亚组。对于$ k $ -spherical $ g $ -Varieties，我们证明$ p $ -Orbits包含$ k $ - 点的有限结果。这是$ p $ complexities $ x $的平等，以及$ x $中的任何$ p $ invariant $ k $ nendens subvariety的任何相应的结果，在代数封闭的field $ k $的情况下，$ x $中的E.B.Vinberg的相应结果。另外，我们还介绍了受限的Weyl Group $ w $的动作，$ k $ - 浓度$ p $ p $ - invariant封闭的封闭subvarieties $ x $的最大$ p $ - complexity和$ k $ rank在$ {\ rm char} \ k = 0 $上的$ k $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - F.Knop在代数封闭的现场案例中引入的$ B $ -Orbits。我们还介绍了与此动作相关的一个小组，并根据$ w $的发电机描述其发电机，该发电机概括了代数封闭的现场案例中对M.Brion的描述。
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.
