On Invariant Theory of Θ-groups

The ground field k is algebraically closed and of characteristic zero. Throughout, G is a connected and simply connected semisimple algebraic group, g is its Lie algebra, and Φ is the Cartan–Killing form on g; l = rk g. Int g (resp. Aut g) is the group of inner (resp. all) automorphisms of g; N is the nilpotent cone in g. For x ∈ g, z(x) is the centraliser of x in g. Let g = ⊕i∈Zmgi be a periodic grading of g and θ the corresponding m th order automorphism of g. Let G0 denote the connected subgroup of G with Lie algebra g0. Invariant Theory of θ-groups deals with orbits and invariants of G0 acting on g1. Its main result is that there is a subspace c ⊂ g1 and a finite reflection group W (c, θ) in c (the little Weyl group) such that k[g1] G0 ≃ k[c] . We say that the grading is N-regular (resp. S-regular) if g1 contains a regular nilpotent (resp. semisimple) element of g. The grading is locally free if there is x ∈ g1 such that z(x) ∩ g0 = {0}. The same terminology also applies to θ. In this paper, we obtain some structural results for gradings with these properties and study interrelations of these properties. Section 1 contains some preliminary material on θ-groups and regular elements. In Section 2, we begin with a dimension formula for semisimple G0-orbits in g1. We also prove two “uniqueness” theorems. Recall that Int g is the identity component of Aut g, and it operates on Aut g via conjugations. Given m ∈ N, we prove that each connected component of Aut g contains at most one Int g-orbit consisting of automorphisms of orderm that are either N-regular or S-regular and locally free. In Section 3, we show that θ-groups corresponding to N-regular gradings enjoy a