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We give a pattern avoidance criterion to classify the orbits of Sp(p,C) xSp(q,C) (resp. GL(n,C)) on the flag variety of type C_{p+q} (resp. D_n) withrationally smooth closure. We show that all such orbit closures fiber (withsmooth fiber) over a smaller flag variety, and hence are in fact smooth. Inaddition we prove that the classification is insensitive to isogeny.

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