Total positivity, finite reflection groups, and a formula of Harish-Chandra

Let W be a finite reflection (or Coxeter) group and K: R² → R. We define the concept of total positivity for the function K with respect to the group W. For the case in which W = G_n, the group of permutations on n symbols, this notion reduces to the classical formulation of total positivity. We prove a basic composition formula for this generalization of total positivity, and in the case in which W is the Weyl group for a compact connected Lie group we apply an integral formula of Harish-Chandra (Amer. J. Math.79 (1957), 87-120) to construct examples of totally positive functions. In particular, the function K(x, y)= e^xy, (x, y) ∈ ℝ², is totally positive with respect to any Weyl group W. As an application of these results, we derive an FKG-type correlation inequality in the case in which W is the Weyl group of SO(5).