Pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the flag variety

We give a pattern avoidance criterion to classify the orbits of Sp(p,C)×Sp(q,C) (resp. GL(n,C)) on the flag variety of type Cp+q (resp. Dn) with rationally smooth closure. We show that all such orbit closures fiber (with smooth fiber) over a smaller flag variety, and hence are in fact smooth. In addition we prove that the classification is insensitive to isogeny.