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Huygens’ principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators. (English) Zbl 1088.39018
Summary: Let $${\mathfrak a}$$ be an Euclidean vector space of dimension $$N$$, and let $$k= (k_\alpha)_{\alpha\in{\mathcal R}}$$ be a multiplicity function related to a root system $${\mathcal R}$$. Let $$\Delta(k)$$ be the trigonometric Dunkl-Cherednik differential-difference Laplacian [cf. I. Cherednik, Invent. Math. 106, No. 2, 411–432 (1991; Zbl 0725.20012)]. For $$(a,t)\in \exp({\mathfrak a})\times\mathbb R$$, denote by $$u_k(a,t)$$ the solution to the wave equation $$\Delta(k) u_k(a,t)= \partial_{tt}u_k(a,t)$$, 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 $$\{(a,t)\in \exp({\mathfrak a})\times\mathbb R\mid |t|-R\leq \|\log a\|\leq |t|+R\}$$ if and only if the root system $${\mathcal R}$$ is reduced, $$k_\alpha\in \mathbb N$$ for all $$\alpha\in{\mathcal R}$$, and $$N$$ is odd starting from 3.

##### MSC:
 39A70 Difference operators 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 58J45 Hyperbolic equations on manifolds 35L05 Wave equation 35R10 Partial functional-differential equations
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