Torsion automorphisms of simple Lie algebras

Abstract

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 [11], 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 Elie Cartan [5] 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. The desire for such a treatment arose from the work of Benedict Gross and myself on adjoint gamma factors of discrete Langlands parameters, which involves the characteristic polynomials of torsion automorphisms. Jean-Pierre Serre pointed us to Cartan’s paper, which led to the reformulation of Kac’s classification presented here. Gross’ insights, examples, predictions and requests have helped form this paper, through many discussions and his careful reading of an earlier version. In particular, Gross suggested that the inner case be treated in detail, before studying general torsion automorphisms. Throughout, I make frequent use Kostant’s theory of the principal PGL2, along with conjugacy results of Segal and Steinberg. I include some facts about centers and component groups of centralizers that may not have appeared in the literature, and the last section gives a twisted analogue (Prop. 4.1) of a result of Kostant on principal elements. These complements are used in [8].