  Fix an integral semisimple element $\lambda$ in the Lie algebra
$\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the
centralizer of $\lambda$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$
eigenspace of $\mathrm{ad}(\lambda)$ in $\mathfrak{g}$. Under a natural
hypothesis (which is always satisfied for classical subgroups of
$\mathrm{GL}(n)$), we embed the closure of each $L$ orbit on $\mathfrak{g}(-1)$
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.
