Huygens' principle for the wave equation associated with the trigonometric dunkl-cherednik operators

Let a be an Euclidean vector space of dimension N, and let k = (k _α)_α∈ℛ be a multiplicity function related to a root system ℛ. Let Δ(k) be the trigonometric Dunkl-Cherednik differential-difference Laplacian. For (a, t) ∈ exp(a) × ℝ, denote by u_k(a,t) the solution to the wave equation Δ(k)u_k(a,t) = ∂_ttu_k(a, t), where the initial data are supported inside a ball of radius R about the origin. We prove that u_k has support in the shell {(a, t) ∈ exp(a) × ℝ | |t| - R ≤ ∥ log a∥ ≤ |t| + R} if and only if the root system ℛ is reduced, k_α ∈ ℕ for all α ∈ ℛ, and N is odd starting from 3.