Periodic Automorphisms of Takiff Algebras, Contractions, and Θ-groups

Let G be a connected reductive algebraic group with Lie algebra g. The ground field k is algebraically closed and of characteristic zero. Fundamental results in invariant theory of the adjoint representation ofG are primarily associated with C. Chevalley and B. Kostant. Especially, one should distinguish the ”Chevalley restriction theorem” and seminal article of Kostant [5]. Later, Kostant and Rallis extended these results to the isotropy representation of a symmetric variety [6]. In 1975, E.B. Vinberg came upwith the theory of θ-groups. This theory generalises and presents in the most natural form invariant-theoretic results previously known for the adjoint representation and isotropy representations of the symmetric varieties. Let us remind the main construction and results of Vinberg’s article [15]. Let θ ∈ Aut(g) be a periodic (= finite order) automorphism of g. The order of θ is denoted by |θ|. Fix a primitive root of unity ζ = |θ| √ 1 and consider the periodic grading (or Z|θ|-grading)