The Minimal Representation of the Conformal Group and Classical Solutions to the Wave Equation

Let n be an integer ≥ 2. We consider the wave operator = −∂ t + Pn i=1 ∂ xi , where (t, x) = (t, x1, . . . , xn) are the canonical coordinates on Minkowski space R 1,n = R× R . Lie’s prolongation algorithm calculates the Lie algebra of infinitesimal symmetries of to be isomorphic to the conformal Lie algebra g = so(2, n+1) plus an infinite dimensional piece reflecting the fact that is linear. In particular, ker = {f ∈ C(R) | f = 0} is a representation of g. This Lie algebra action does not exponentiate to a global action of the conformal group G = SO(2, n+1)0 or any cover group. However, for n odd, it is known that ker contains a nice g-invariant subspace that carries the minimal representation of G. In this paper, we give a uniform realization of the minimal representation of a double cover of G in ker as a positive energy representation H for n even and odd. Using this realization, we obtain an explicit orthonormal basis for H that is well behaved with respect to energy and angular momentum. The lowest positive energy solution is, up to normalization, f(t, x) = 1 p (1− it) + ‖x‖2 n−1 , where √ denotes the principal branch of the square root. Of special note, for n odd, all functions in our basis are rational functions. Finally, using Fourier analysis with respect to this basis, we prove that every classical real-valued solution to the wave equation is the real part of a unique continuous element in the representation H.