Weyl group orbits on Kac–Moody root systems*

Abstract

Let be a Dynkin diagram and let be the simple roots of the corresponding Kac–Moody root system. Let denote the Cartan subalgebra, let W denote the Weyl group and let Δ denote the set of all roots. The action of W on , and hence on Δ, is the discretization of the action of the Kac–Moody algebra. Understanding the orbit structure of W on Δ is crucial for many physical applications. We show that for , the simple roots αi and αj are in the same W–orbit if and only if vertices i and j in the Dynkin diagram corresponding to αi and αj are connected by a path consisting only of single edges. We introduce the notion of ‘the Cayley graph of the Weyl group action on real roots’ whose connected components are in one-to-one correspondence with the disjoint orbits of W. For a symmetric hyperbolic generalized Cartan matrix A of rank we prove that any two real roots of the same length lie in the same W–orbit. We show that if the generalized Cartan matrix A contains zeros, then there are simple roots that are stabilized by simple root reflections in W, that is, W does not act simply transitively on real roots. We give sufficient conditions in terms of the generalized Cartan matrix A (equivalently ) for W to stabilize a real root. Using symmetry properties of the imaginary light cone in the hyperbolic case, we deduce that the number of W–orbits on imaginary roots on a hyperboloid of fixed radius is bounded above by the number of root lattice points on the hyperboloid that intersect the closure of the fundamental region for W.