Semisimple elements and the little Weyl group of real semisimple Z_m-graded Lie algebras
arxiv(2022)
摘要
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 .
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要