Symplectic Twistor Operator on $$\mathbb R ^{2n}$$ and the Segal–Shale–Weil Representation

Abstract

The aim of our article is the study of solution space of the symplectic twistor operator $$T_s$$ in symplectic spin geometry on standard symplectic space $$(\mathbb R ^{2n},\omega )$$ , which is the symplectic analogue of the twistor operator in (pseudo) Riemannian spin geometry. In particular, we observe a substantial difference between the case $$n=1$$ of real dimension $$2$$ and the case of $$\mathbb R ^{2n}, n>1$$ . For $$n>1$$ , the solution space of $$T_s$$ is isomorphic to the Segal–Shale–Weil representation.