Spineurs symplectiques purs et indice de Maslov de plans lagrangiens positifs

Abstract

As is well known, each point of the closed generalized unit-disk X can be associated to a holomorphically induced representation of the Heisenberg group. First canonical intertwining operators are constructed between pairs of such representations. Next, after having introduced suitable definitions, it is noted that the classical correspondence between group extensions and 2-cocycles also makes sense when applied to transformation spaces. As an example of transformation space extension, the manifold of pure symplectic spinors is described. It is the analogue of the manifold of pure spinors when the spin representation of the Clifford algebra is replaced by the Stone-Von Neumann representation of the Heisenberg group. Then, the associated 2-cocycle m 2 is worked out, which is a T -valued function on X × X × X, and the composition law of the canonical intertwining operators is given. Lifting m 2, an R -valued 2-cocycle m is constructed whose restriction to the Shilov boundary of X takes integer values and coincides with the ordinary Maslov index. For this reason, it is called the generalized Maslov index. Finally, using these results, explicit realizations of the metaplectic group, its Shale-Weil representation, and the universal covering of the symplectic group are given.