  The purpose of this paper is to establish several new results about the Hodge
theory of Lagrangian fibrations on (not necessarily compact) holomorphic
symplectic manifolds. Let $M$ be a holomorphic symplectic manifold of dimension
$2n$ that is K\"ahler but not necessarily compact, and let $\pi \colon M \to B$
be a Lagrangian fibration. We establish a relationship between the bundle of
holomorphic $(n+i)$-forms on $M$ and the $i$-th perverse sheaf $P_i$ in the
decomposition theorem for $\pi$. This is formulated using Saito's theory of
Hodge modules and the BGG correspondence (between graded modules over the
symmetric and exterior algebra). Along the way, we prove a relative Hard
Lefschetz theorem for the action by the symplectic form; we prove two recent
conjectures by Maulik, Shen, and Yin; we give a short proof for Matsushita's
theorem (about higher direct images of the structure sheaf); and we show,
without using hyperk\"ahler metrics, that every Lagrangian fibration gives rise
to an action by the Lie algebra $\mathfrak{sl}_3(\mathbb{C})$ (in the
noncompact case) or $\mathfrak{sl}_4(\mathbb{C})$ (in the compact case).
