Ellipticity of the symplectic twistor complex

Svatopluk Krýsl

Address: Charles University of Prague, Sokolovská 83, Praha 8, Czech Republic

E-mail: krysl@karlin.mff.cuni.cz

Abstract: For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain infinite rank vector bundles defined over this manifold. The differential operators are symplectic analogues of the twistor operators known from Riemannian or Lorentzian spin geometry. It is known that the mentioned sequences form complexes if the symplectic connection is of Ricci type. In this paper, we prove that certain parts of these complexes are elliptic.

AMSclassification: primary 22E46; secondary 53C07, 53C80, 58J05.

Keywords: Fedosov manifolds, Segal-Shale-Weil representation, Kostant’s spinors, elliptic complexes.