A Curious Identity and the Volume of the Root Spherical Simplex

In this note we shall consider a finite root system R spanning an euclidean space E of dimension l (for all the facts about root systems which we are going to use in this note we refer to [1]). l is called the rank of R. We shall choose once and for all a set of positive roots R+ and in R+ the set of simple roots ∆ = {α1, . . . , αl}. We shall also denote by W the Weyl group of R i.e. the finite group generated by the reflections with respect to the hyperplanes orthogonal to the roots in R. Given such a root α ∈ R, we shall denote by sα ∈ W the reflection with respect to the hyperplane orthogonal to α. Set si = sαi for each i = 1, . . . , l and call S = {s1, . . . , sl} the set of simple reflections. One know that S generates W and that the pair (W,S) is a Coxeter group. We know that the ring of regular functions on E, invariant under the action of W , is a polynomial ring generated by homogenous elements of degrees d1 ≤ d2 ≤ · · · ≤ dl. The di’s are called the degrees. We shall also consider the sequence of exponents, d1 − 1, d2 − 1, . . . , dl − 1. Recall that