Relating real and p-adic Kazhdan-Lusztig polynomials

Fix an integral semisimple element [Formula: see text] in the Lie algebra [Formula: see text] of a complex reductive algebraic group [Formula: see text]. Let [Formula: see text] denote the centralizer of [Formula: see text] in [Formula: see text] and let [Formula: see text] denote the [Formula: see text]-eigenspace of [Formula: see text] in [Formula: see text]. Under a natural hypothesis (which is always satisfied for classical groups), we embed the closure of each [Formula: see text] orbit on [Formula: see text] into the closure of an orbit of a symmetric subgroup [Formula: see text] containing [Formula: see text] on a partial flag variety for [Formula: see text]. We use this to relate the local intersection homology of the latter orbit closures to the former orbit closures. This, in turn, relates multiplicity matrices for split real and [Formula: see text]-adic groups.