Huygens' principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators

Abstract

Let a be an Euclidean vector space ofdimension N, and let k = (kα)α∈R be a multiplicity function related to a root system R. Let ∆(k )b e the trigonometric Dunkl-Cherednik differential-difference Laplacian. For (a, t) ∈ exp(a) × R, denote by uk(a, t) the solution to the wave equation ∆(k)uk(a, t )= ∂ttuk(a, t), where the initial data are supported inside a ball ofradius R about the origin. We prove that uk has support in the shell {(a, t) ∈ exp(a) × R || t |− R ≤ � log a �≤| t| + R} ifand only ifthe root system R is reduced, kα ∈ N for all α ∈ R, and N is odd starting from 3.