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.
Barchini et al. (Wed,) studied this question.