**1. Introduction**

The wave problem consists of the wave equation and some initial data,

$$
\Delta u(\mathbf{x}, t) = \partial\_{\mathcal{U}} u(\mathbf{x}, t), \quad u(\mathbf{x}, 0) = f(\mathbf{x}), \\
\partial\_{\mathcal{U}} u(\mathbf{x}, 0) = \mathbf{g}(\mathbf{x}), \qquad \text{for } \mathbf{x} \in \mathbb{R}^n \text{ and } t \in \mathbb{R}.
$$

This problem is certainly one of the most interesting problems of mathematical physics. Standard techniques involving the Fourier transform show that there are two distributions *P*1 and *P*2 on R*n* ˆ R such that *u* " *P*1 ˚ *f* ` *P*2 ˚ *g*. Here ˚ represents the Euclidean convolution product. The distributions *P*1 and *P*2 are called propagators.

One of the most celebrated features of the wave equation is Huygens' principle: When the dimension *n* is odd and starting from 3, the propagators are supported entirely on spherical shells. This is the reason why in our three-dimensional word, transmission of signals is possible and we can hear each other. A two-dimensional world would be drastically different from this point of view.

The problem of classifying all second order differential operators which obey Huygens' principle is known as the Hadamard problem [1]. This problem has received a good deal of attention and the literature is extensive (see, for instance, [2–13]). Nevertheless, this problem is still far from being fully solved.

In this paper we will consider a deformed wave equation where the Laplacian Δ is replaced by a certain differential-difference operator. We will prove the non-existence of Huygens' principle for the deformed wave equation for all *n* ě 1. The main tool is the representation theory of the Lie algebra slp2, Rq.

More precisely, we will consider the deformed wave equation <sup>2</sup>}*x*}Δ*k<sup>u</sup>*p*<sup>x</sup>*, *<sup>t</sup>*q"B*ttu*p*<sup>x</sup>*, *t*q with compactly supported initial data p*f* , *g*q. Here Δ*k* is the differential-difference Dunkl Laplacian (see (2)), where *k* is a multiplicity function for the Dunkl operators. The operator }*x*}Δ*k* appeared in [14] and played a crucial rule in the study of the so-called p*k*, <sup>1</sup>q-generalized Fourier transform. When *k* " 0, the deformed wave equation becomes <sup>2</sup>}*x*}Δ*u*p*<sup>x</sup>*, *<sup>t</sup>*q"B*ttu*p*<sup>x</sup>*, *t*q and the p0, <sup>1</sup>q-generalized Fourier transform reduces to a Hankel type transform on R*<sup>n</sup>*. We refer the reader to [14] for a detailed study on the generalized Fourier transform.

We begin with a straightforward treatment of the Cauchy problem for the deformed wave equation by means of the p*k*, <sup>1</sup>q-generalized Fourier transform, and derive the existence of propagators *Pk*,<sup>1</sup> and *Pk*,2, in terms of which, the Cauchy problem is solved. Huygens' principle for the deformed wave equation is that *Pk*,<sup>1</sup> and *Pk*,<sup>2</sup> are supported entirely on the set tp*<sup>x</sup>*, *t*q P R*n* ˆ R : }*x*} ´ 1 2 *t* 2 " 0u. It is not a simple task to study the support property from the precise form of the propagators. However, subtler dilatation properties of the propagators allow us to show that Huygens' principle holds true if, and only if, *Pk*,<sup>1</sup> and *Pk*,<sup>2</sup> generate a finite dimensional representation of the Lie algebra slp2, Rq. It is here that the construction of a representation of slp2, Rq plays a crucial role. This construction was inspired by [14]. A closer investigation shows that *Pk*,<sup>1</sup> and *Pk*,<sup>2</sup> cannot generate finite dimensional representations of slp2, Rq, and therefore, Huygens' principle does not hold for the deformed wave equation for any *n* ě 1 and any multiplicity function *k*. The strategy uses proof by contradiction. It is noteworthy mentioning that the case *k* " 0 is already new.

It would be interesting to understand the interpretation(s) of the non-existence of Huygens' principle for the deformed wave equation from the physics point of view. It would also be fascinating to ask whether Huygens's principle holds for other seminal Dunkl-type equations such as the Dunkl–Dirac equation (see [15] for more details about the Dunkl–Dirac operator). For the Euclidean Dirac equation, this problem has been investigated in [16].
