Criteria for rational smoothness of some symmetric orbit closures

Abstract

Let G be a connected reductive linear algebraic group over 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 = \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 . Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on is rank symmetric. In the special case K=2n(), G=2n(), 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.