Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 22 (2012), No. 2, 301--360Copyright Heldermann Verlag 2012 The Minimal Representation of the Conformal Group and Classical Solutions to the Wave Equation Markus Hunziker Dept. of Mathematics, Baylor University, Waco, TX 76798-7328, U.S.A. Markus_Hunziker@baylor.edu Mark R. Sepanski Dept. of Mathematics, Baylor University, Waco, TX 76798-7328, U.S.A. Mark_Sepanski@baylor.edu Ronald J. Stanke Dept. of Mathematics, Baylor University, Waco, TX 76798-7328, U.S.A. Ronald_Stanke@baylor.edu [Abstract-pdf] \def\R{\mathbb{R}} Using an idea of Dirac, we give a geometric construction of a unitary lowest weight representation ${\cal H}^{+}$ and a unitary highest weight representation ${\cal H}^{-}$ of a double cover of the conformal group SO$(2,n+1)_{0}$ for every $n\geq 2$. The smooth vectors in ${\cal H}^{+}$ and ${\cal H}^{-}$ consist of complex-valued solutions to the wave equation $\Box f=0$ on Minkowski space $\R^{1,n}=\R\times \R^{n}$ and the invariant product is the usual Klein-Gordon product. We then give explicit orthonormal bases for the spaces ${\cal H}^{+}$ and ${\cal H}^{-}$ consisting of weight vectors; when $n$ is odd, our bases consist of rational functions. Furthermore, we show that if $\Phi, \Psi\in {\cal S}(\R^{1,n})$ are real-valued Schwartz functions and $u\in {\cal C}^{\infty}(\R^{1,n})$ is the (real-valued) solution to the Cauchy problem $\Box u=0$, $u(0,x)=\Phi(x)$, $\partial_tu(0,x)=\Psi(x)$, then there exists a unique real-valued $v\in {\cal C}^{\infty}(\R^{1,n})$ such that $u+iv\in {\cal H}^{+}$ and $u-iv\in{\cal H}^{-}$. Keywords: Conformal group, minimal representation, wave equation, classical solutions, Cauchy problem. MSC: 22E45, 22E70, 35A09, 35A30, 58J70 [ Fulltext-pdf  (529  KB)] for subscribers only.