Reduced holonomy and Hodge theory

We develop Hodge theory for a Riemannian manifold $(M,g)$ with a background closed 3-form, H. Precisely, we prove that if the metric connections with torsion $\pm H$ have holonomy groups $G_{\pm }$, then the $d^H$-Laplacian preserves the irreducible representations of the Lie algebras of the holonomy groups on the space of forms.