Criteria for rational smoothness of some symmetric orbit closures

TL;DR

This paper provides a criterion for determining the rational smoothness of certain symmetric orbit closures in flag varieties, extending classical results and simplifying the process in specific cases.

Contribution

It generalizes classical smoothness criteria to symmetric orbit closures using Bruhat graphs and introduces a simplified test for the case of $K=Sp_{2n}(C)$ and $G=SL_{2n}(C)."

Findings

Rational smoothness can be checked via degrees in a Bruhat graph.

Orbit closure smoothness corresponds to rank symmetry in Bruhat order intervals.

In specific cases, only the degree of the bottom vertex needs to be examined.

Abstract

Let $G$ be a connected reductive linear algebraic group over $\C$ with an involution $\theta$. 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 $\iot = \{\theta(w^{-1})w\mid w\in W\}$ 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.