正特征约化李代数的对合
Involutions of reductive Lie algebras in positive characteristic
摘要：
设$G$是特征为$\neq 2$的域$k$上的约化群，${\mathfrak g}=\Lie(G)$，$θ$是$G$的对合自同构，${\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}$是相应的对称空间分解.对于特征为零的基场的情况，各向同性群$G^θ$对${\mathfrak p}$的作用是众所周知的，因为Kostant和Rallis的著名论文。这样一种具有积极特征的理论被证明更难发展。在这里，我们使用的方法的基础上的一些工具，从几何不变的理论建立相应的结果（好）的正特征。证明了${\mathfrak p}$的幂零元簇${\cal N}$有稠密开轨道，并且对商映射${\mathfrak p}\to{\mathfrak p}/G^θ$的每一个纤维也是如此.然而，我们表明，相应的声明$G$，由Richardson，是不正确的。我们提供了一个新的，（大部分）计算免费证明的不可约组件的${\cal N}$的数量，延长了$k={\mathbb C}$的Sekiguchi的结果。最后，我们应用Skryabin定理来描述无穷小不变量$k[{\mathfrak p}]^{\mathfrak k}$。
英文摘要：
Let $G$ be a reductive group over a field $k$ of characteristic $\neq 2$, let ${\mathfrak g}=\Lie(G)$, let $θ$ be an involutive automorphism of $G$ and let ${\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}$ be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group $G^θ$ on ${\mathfrak p}$ is well-understood, since the well-known paper of Kostant and Rallis. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic. Among other results, we prove that the variety ${\cal N}$ of nilpotent elements of ${\mathfrak p}$ has a dense open orbit, and that the same is true for every fibre of the quotient map ${\mathfrak p}\to{\mathfrak p}/G^θ$. However, we show that the corresponding statement for $G$, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of ${\cal N}$, extending a result of Sekiguchi for $k={\mathbb C}$. Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants $k[{\mathfrak p}]^{\mathfrak k}$.
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