Symplectic Twistor Operator on $R^2n$ 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 $(\mR^{2n},\omega)$, 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 $\mR^{2n}$, $n>1$.

For $n>1$, the solution space of $T_s$ is isomorphic to the Segal-Shale-Weil representation.

Keywords

• Symplectic twistor operator
• Symplectic Dirac operator
• Metaplectic Howe duality.