  Motivated by relating the representation theory of the split real and
$p$-adic forms of a connected reductive algebraic group $G$, we describe a
subset of $2^r$ orbits on the complex flag variety for a certain symmetric
subgroup. (Here $r$ is the semisimple rank of $G$.) This set of orbits has the
property that, while the closure of individual orbits are generally singular,
they are always smooth along other orbits in the set. This, in turn, implies
consequences for the representation theory of the split real group.
