Semisimple elements and the little Weyl group of real semisimple $Z_m$-graded Lie algebras

TL;DR

This paper studies semisimple orbits in Vinberg $ heta$-representations over complex and real fields, establishing a technical equality of hyperplanes, classifying real Cartan subspaces via Galois cohomology, and characterizing orbits with real representatives.

Contribution

It provides a case-by-case analysis proving the equality of hyperplanes related to roots and reflections, and classifies real Cartan subspaces using Galois cohomology, extending previous results.

Findings

Equality of hyperplanes for roots and Weyl group reflections

Classification of real Cartan subspaces via Galois cohomology

Characterization of orbits with real representatives

Abstract

We consider the semisimple orbits of a Vinberg $\theta$-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}.