The Weyl group of a graded Lie algebra

The action of the group $G_0$ of fixed points of a semisimple automorphism $\theta$ of a reductive algebraic group $G$ on an eigenspace $V$ of this automorphism in the Lie algebra $\mathfrak g$ of the group $G$ is considered. The linear groups which are obtained in this manner are called $\theta$-groups in this paper; they have certain properties which are analogous to properties of the adjoint group. In particular, the notions of Cartan subgroup and Weyl group can be introduced for $\theta$-groups. It is shown that the Weyl group is generated by complex reflections; from this it follows that the algebra of invariants of any $\theta$-group is free.
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