The wave equation for Dunkl operators

Let k = (kα)αεℝ, be a positive-real valued multiplicity function related to a root system ℝ, and Δk be the Dunkl-Laplacian operator. For (x, t) ε ℝN, × ℝ, denote by uk(x, t) the solution to the deformed wave equation Δkuk,(x, t) = δttuk(x, t), where the initial data belong to the Schwartz space on ℝN. We prove that for k ⩾ 0 and N ⩾ l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N − 3)/2 + Σαεℝ+kα 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 uk(x, t) is contained in the conical shell {(x, t), ε ℝN × ℝ | |t| − R ⩽ ‖x‖ ⩽ |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 uk is, for large |t|, partitioned into equal potential and kinetic parts.