A Weyl group generating function that ought to be better known

Abstract

Let W be a finite Weyl group with root system Φ, simple roots α 1,…, α n , exponents e 1,…, e n , and index of connection ƒ. Let b 1,…, b n denote the simple root coordinates for the sum of all positive roots, and for w ϵ W let ℓ( w) denote the length and D( w) = { i: w −1 α i < 0} the descent set of w. By analyzing the structure of the corresponding affine Weyl group, we prove that ∑ wϵWq σ(w)−ℓ(w) = ƒ · Π n i = 1 (1 − q bi) (1 − q ei) , where σ( w) = ∑ iϵD( w) bi .