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This paper studies the Lie symmetries of the equationGenerically the symmetry group is sl(2, R). In particular, we show the local action of the symmetry group extends to a global representation of SL(2, R) on an appropriate subspace of smooth solutions. In fact, every principal series is realized in this way. Moreover, this subspace is naturally described in terms of sections of an appropriate line bundle on which the given differential operator is intimately related to the Casimir element.

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We realize the oscillator representation of the metaplectic group Mp .n/ in the space of solutions to a system of Schrödingier type equations on R n ¢ Sym.n; R/. Our realization has particularly simple intertwining maps to the realizations given by Kashiwara and Vergne.