Fundamental polytope for the Weyl group acting on a maximal torus of a compact Lie group

Abstract

We provide a fundamental domain for the action of the finite Weyl group on a maximal torus of a compact Lie group of the corresponding type. The general situation is reduced to the adjoint case and, from the perspective of root data, this problem can be rephrased by asking for a fundamental polytope for the action of the extended affine Weyl group on the (dual) toral subalgebra. We solve the problem in this second form. Using the theory of minuscule weights, we obtain a description of this fundamental polytope as a convex hull of explicit vertices, and as an intersection of closed half-spaces. The latter description was first obtained by Komrakov and Premet in 1984 but, as the present work is independent of that of Komrakov-Premet, we give a new self-contained proof of it. We also derive some consequences on the structure of automorphism groups of extended Dynkin diagrams.