The Symplectic Fueter-Sce Theorem

TL;DR

This paper introduces a symplectic analogue of the Fueter theorem, enabling the construction of special polynomial solutions for the symplectic Dirac operator, which is invariant under the symplectic Lie algebra.

Contribution

It develops a novel symplectic version of the Fueter theorem, expanding the theory of invariant differential operators in symplectic geometry.

Findings

Constructed polynomial solutions for the symplectic Dirac operator.

Established the symplectic Fueter theorem as an analogue to the classical version.

Enhanced understanding of symplectic invariant differential operators.

Abstract

In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator $D_s$, which is defined as the first-order $\mathfrak{sp}(2n)$-invariant differential operator acting on functions on ${\mathbb R}^{2n}$ taking values in the metaplectic spinor representation.