A curious identity and the volume of the root spherical simplex
C. De Concini, C. Procesi
Take a maximal parabolic subgroup of an affine Weyl group. Take the product of its exponents, and divide by the product of its degrees. Now, sum that over all maximal parabolic subgroups; you get 1. Now that's an elegant equation.
And further, acconding to E. Vinberg, this also tells us about the ratio of the volumes of the Weyl chamber and its intersection with the unit sphere. I'm excited to see where this goes.