Torsion automorphisms of simple Lie algebras

An automorphism σ of a simple finite dimensional complex Lie algebra g is called torsion, if σ has finite order in the group Aut(g) of all automorphisms of g. The torsion automorphisms of g were classified by Victor Kac in [12], as an application of his results on infinite dimensional Lie algebras. Those torsion automorphisms contained in the identity component G = Aut(g)◦ are called inner; they were classified in 1927 by Élie Cartan [6] who used (and perhaps introduced) the affine Weyl group and the geometry of alcoves for this purpose. This paper extends Cartan’s method to cover all torsion automorphisms of g, thereby recovering Kac’s classification directly from the geometry of the affine Weyl group, without the use of infinite dimensional Lie algebras. Kac’s classification can be roughly stated as follows. Each symmetry θ of the Dynkin graph D(g) of g extends to a certain kind of automorphism of g, which we again denote by θ, called a pinned automorphism. The pinned automorphisms represent the cosets of G in Aut(g) and the order of any torsion element in Gθ is divisible by the order f of θ. For a given pinned automorphism θ of g, Kac defines a certain vector (b0, b1, . . . , bk) of positive integers. Here k is the number of θ-orbits on the nodes of D(g). Then the G-conjugacy classes of elements in Gθ of order m are parametrized by Kac coordinates. These are vectors (s0, s1, . . . , sk) of nonnegative relatively prime integers si satisfying