
Journal of Lie Theory 22 (2012), No. 2, 301360 Copyright 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 767987328, U.S.A. Markus_Hunziker@baylor.edu Mark R. Sepanski Dept. of Mathematics, Baylor University, Waco, TX 767987328, U.S.A. Mark_Sepanski@baylor.edu Ronald J. Stanke Dept. of Mathematics, Baylor University, Waco, TX 767987328, U.S.A. Ronald_Stanke@baylor.edu [Abstractpdf] \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 complexvalued 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 KleinGordon 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 realvalued Schwartz functions and $u\in {\cal C}^{\infty}(\R^{1,n})$ is the (realvalued) solution to the Cauchy problem $\Box u=0$, $u(0,x)=\Phi(x)$, $\partial_tu(0,x)=\Psi(x)$, then there exists a unique realvalued $v\in {\cal C}^{\infty}(\R^{1,n})$ such that $u+iv\in {\cal H}^{+}$ and $uiv\in{\cal H}^{}$. Keywords: Conformal group, minimal representation, wave equation, classical solutions, Cauchy problem. MSC: 22E45, 22E70, 35A09, 35A30, 58J70 [ Fulltextpdf (529 KB)] for subscribers only. 