Positive product-formula measure for Dunkl–Bessel functions

Establish the existence, for every finite root system R ⊂ ℝ^N and every nonnegative multiplicity function θ ∈ θ(R), of a nonnegative probability measure μ^{R(θ)}_{a_1,a_2} on ℝ^N such that for all a_1,a_2 ∈ ℝ^N and all x ∈ ℂ^N, J^{R(θ)}_{a_1}(x) J^{R(θ)}_{a_2}(x) = ∫_{ℝ^N} J^{R(θ)}_{a}(x) dμ^{R(θ)}_{a_1,a_2}(a).

Background

This conjecture asks for a positive product formula for the symmetric Dunkl–Bessel functions J^{R(θ)}_a(x) associated with a finite root system R and a nonnegative multiplicity θ. A signed measure representation of this product is known (Trimeche), but positivity is not established in general.

Rösler proved such a product formula in the radially symmetric setting, while for the nonsymmetric eigenfunctions the analogous statement is known to be false. The paper uses this conjecture as an assumption to deduce weak convergence results to (rectangular) free convolutions, highlighting the significance of a positive measure representation.