Relating Real and P-adic Kazhdan-Lusztig Polynomials

Fix an integral semisimple element λ in the Lie algebra g of a complex reductive algebraic group G. Let L denote the centralizer of λ in G and let g(-1) denote the $-1$ eigenspace of ad(λ) in g. Under a natural hypothesis (which is always satisfied for classical subgroups of GL(n)), we embed the closure of each L orbit on g(-1) into the closure of an orbit of a symmetric subgroup K containing L on a partial flag variety for G. We use this to relate the local intersection homology of the later orbit closures to the former orbit closures. This, in turn, relates multiplicity matrices for split real and p-adic groups. We also describe relationships between "microlocal packets'' of representations of these groups.