论文标题
半元素元素和真实半imple $ z_m $ $ $ $ lie代数的小元素
Semisimple elements and the little Weyl group of real semisimple $Z_m$-graded Lie algebras
论文作者
论文摘要
我们考虑Vinberg $θ$代表的半圆柱。首先，我们将复数作为基础场。通过案例分析，我们显示了一个技术结果，表明两组超平面的平等，一个是对应于cartan子空间的限制根，另一个对应于（小）Weyl组中的复数反射。半神经轨道的代表中有有限数量的集合，这些集合对应于（小）Weyl组的反射亚组。我们技术结果的后果之一是，固定集合中的要素在表演组中都具有相同的稳定器。其次，我们研究基本场是实际数字时会发生什么。我们查看Cartan子空间，并表明可以通过固定真实Cartan子空间的标准器的第一个Galois共同体组来对真正的Cartan子空间进行分类。在实际情况下，轨道可以使用Galois共同体进行分类。但是，为了这样做，我们需要知道哪些轨道具有真正的代表。我们展示的定理表征了确实具有如此真实代表的均匀半神经元素的轨道。这紧随其后，并从\ cite {bgl}概括了一个定理。
We consider the semisimple orbits of a Vinberg $θ$-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one corresponding to the restricted roots of a Cartan subspace, the other corresponding to the complex reflections in the (little) Weyl group. The semisimple orbits have representatives in a finite number of sets that correspond to reflection subgroups of the (little) Weyl group. One of the consequences of our technical result is that the elements in a fixed such set all have the same stabilizer in the acting group. Secondly we study what happens when the base field is the real numbers. We look at Cartan subspaces and show that the real Cartan subspaces can be classified by the first Galois cohomology set of the normalizer of a fixed real Cartan subspace. In the real case the orbits can be classified using Galois cohomology. However, in order for that to work we need to know which orbits have a real representative. We show a theorem that characterizes the orbits of homogeneous semisimple elements that do have such a real representative. This closely follows and generalizes a theorem from \cite{bgl}.
