Criteria for rational smoothness of some symmetric orbit closures

Abstract

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 $\iot = \{θ(w^{-1})w\mid w\in W\}$ 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 $\iot$. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on $\iot$ is rank symmetric. In the special case $K=\Sp_{2n}(\C)$, $G=\SL_{2n}(\C)$, 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.