Some invariant connections on symplectic reductive homogeneous spaces

TL;DR

This paper investigates invariant symplectic connections on symplectic reductive homogeneous spaces, identifies a unique natural connection, and relates it to known structures like Wallach flag manifolds, providing explicit curvature properties.

Contribution

It introduces a family of invariant connections, identifies a unique symplectic connection within this family, and connects it to known geometric structures such as Wallach flag manifolds.

Findings

The connection $ abla^{0,1}$ is flat iff the space is locally a symplectic Lie group.

The unique symplectic connection $ abla^ extbf{s}$ exists at $a=b=1/3$ within the family.

The Wallach flag manifold's invariant connection coincides with $ abla^ extbf{s}$ and is Ricci-parallel.

Abstract

A symplectic reductive homogeneous space is a pair $(G/H,\Omega)$, where $G/H$ is a reductive homogeneous $G$-space and $\Omega$ is a $G$-invariant symplectic form on it. The main examples include symplectic Lie groups, symplectic symmetric spaces, and flag manifolds. This paper focuses on the existence of a natural symplectic connection on $(G/H,\Omega)$. First, we introduce a family $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$ of $G$-invariant connection on $G/H$, and establish that $\nabla^{0,1}$ is flat if and only if $(G/H,\Omega)$ is locally a symplectic Lie group. Next, we show that among all $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$, there exists a unique symplectic connection, denoted by $\nabla^\mathbf{s}$, corresponding to $a=b=\tfrac{1}{3}$, a fact that seems to have previously gone unnoticed. We then compute its curvature and Ricci curvature tensors. Finally, we demonstrate that the $\operatorname{SU}(3)$-invariant preferred symplectic connection of the Wallach flag manifold $\operatorname{SU}(3)/\mathbb{T}^2$ (from Cahen-Gutt-Rawnsley) coincides with the natural symplectic connection $\nabla^\mathbf{s}$, which is furthermore Ricci-parallel.