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

Willem de Graaf, Hông Vân Lê

Published 2022-12-07Version 1

In this paper we introduce the notion of the algebraic $\theta$-group of a semisimple $Z_m$-graded Lie algebra over a field $k$ of characteristic $0$ whose identity component is the $\theta$-group. Using this, we extend a number of results concerning structure of semisimple elements in complex semisimple $Z_m$-graded Lie algebras and some results concerning real (and complex) semisimple Lie algebras to the case of real (and complex) semisimple $Z_m$-graded Lie algebras, using Galois cohomology theory and ad hoc techniques. In particular, we parameterize the conjugacy classes of Cartan subspaces in a real semisimple $Z_m$-graded Lie algebra $g$ in terms of Galois cohomology. We prove that if $m$ is a prime number or $g^c$ is a complex semisimple $Z_m$-graded Lie algebra of maximal rank then the centralizer $Z_{g^c}(p)$ of a homogeneous semisimple element $p$ and the stabilizer $W_p$ of $p$ under the Weyl group action define each other. As a result, we obtain a number of consequences on the conjugacy classes of homogeneous semisimple elements in $g^c$ and in its real forms.