The wave equation for Dunkl operators

Let k = (k_α)_αεℝ, be a positive-real valued multiplicity function related to a root system ℝ, and Δ_k be the Dunkl-Laplacian operator. For (x, t) ε ℝ^N, × ℝ, denote by u_k(x, t) the solution to the deformed wave equation Δ_ku_k,(x, t) = δ_ttu_k(x, t), where the initial data belong to the Schwartz space on ℝ^N. We prove that for k {greater than or slanted equal to} 0 and N {greater than or slanted equal to} l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N - 3)/2 + Σ_αεℝ+k_α ε ℕ. Here ℝ⁺ ⊂ ℝ is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of u_k(x, t) is contained in the conical shell {(x, t), ε ℝ^N × ℝ | |t| - R {less-than or slanted equal to} {norm of matrix}x{norm of matrix} {less-than or slanted equal to} |t| + R}. Our approach uses the representation theory of the group SL(2, ℝ), and Paley-Wiener theory for the Dunkl transform. Also, we show that the (t-independent) energy functional of u_k is, for large |t|, partitioned into equal potential and kinetic parts.