  We prove several results concerning the intersection cohomology and the
perverse filtration associated with a Lagrangian fibration of an irreducible
symplectic variety. We first show that the perverse numbers only depend on the
deformation equivalence class of the ambient variety. Then we compute the
border of the perverse diamond, which further yields a complete description of
the intersection cohomology of the Lagrangian base and the invariant cohomology
classes of the fibers. Lastly, we identify the perverse and Hodge numbers of
intersection cohomology when the irreducible symplectic variety admits a
symplectic resolution. These results generalize some earlier work by the second
and third authors in the nonsingular case.
