ar X iv : 0 71 0 . 14 25 v 1 [ m at h . D G ] 7 O ct 2 00 7 Classification of 1 st order symplectic spinor operators over contact projective geometries

We give a classification of 1 st order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations , which are derived from the so called higher symplectic, harmonic or generalized Kostant spinor modules. Higher symplectic spinor modules are arising from the Segal-Shale-Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one 1 st order invariant differential operator up to a complex multiple and give an equivalence condition for the existence of such an operator. Contact projective analogues of the well known Dirac, twistor and Rarita-Schwinger operators appearing in Riemannian geometry are special examples of these operators.