Symplectic Spinor Valued Forms and Invariant Operators Acting between Them

While the spinor twisted de Rham sequence for orthogonal spin structures is well understood from the point of view of representation theory (see, e.g., Delanghe, Sommen, Souček [4]), its symplectic analogue seems to be untouched till present days. In Riemannian geometry, a decomposition of spinor twisted de Rham sequence (i.e., exterior differential forms with values in basic spinor bundles) into invariant parts is well known. Suppose a principal connection on the frame bundle of orthogonal repers (of the tangent bundle) is given. It induces in a canonical way a covariant derivative on differential forms with values in the basic spinor bundles. In this case, it is known, which parts of the covariant derivatives acting between the spinor bundle valued forms are zero if we restrict it to an invariant part of the sequence. Namely, the covariant derivative maps each invariant part only in at most three invariant parts sitting in the next gradation (some degeneracies on the ends of the sequence could be systematically described). In symplectic geometry, the first question which naturally arises is, what are the spinors for a symplectic Lie algebra. This question was successfully answered by Bertram Kostant in [14]. He offered a candidate for symplectic spinors. We will call these spinors basic symplectic spinors and denote their underlying vector spaces S+ and S−. They are analogous to the ordinary orthogonal spinors in at least two following ways.