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.