Criteria for rational smoothness of some symmetric orbit closures

Let G be a connected reductive linear algebraic group over C with an involution θ. Denote by K the subgroup of fixed points. In certain cases, the K-orbits in the flag variety G/B are indexed by the twisted identities ι={θ(w−1)w|w∈W} in the Weyl group W. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a “Bruhat graph” whose vertices form a subset of ι. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on ι is rank symmetric.In the special case K=Sp2n(C), G=SL2n(C), we strengthen our criterion by showing that only the degree of a single vertex, the “bottom one”, needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.