Some invariant connections on symplectic reductive homogeneous spaces

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