Symplectic twistor operator on ${\mathbb R}^{2n}$ and the Segal-Shale-Weil representation

TL;DR

This paper investigates the symplectic twistor operator on standard symplectic space, revealing that for dimensions greater than two, its solution space aligns with the Segal-Shale-Weil representation, highlighting a key difference from the two-dimensional case.

Contribution

It characterizes the solution space of the symplectic twistor operator and establishes its isomorphism with the Segal-Shale-Weil representation for higher dimensions.

Findings

Solution space for n>1 is isomorphic to the Segal-Shale-Weil representation.

Significant difference observed between the n=1 case and higher dimensions.

The study advances understanding of symplectic spin geometry and twistor operators.

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.