某些对称轨道闭包的有理光滑性判据
Criteria for rational smoothness of some symmetric orbit closures
摘要：
设$G$是$\C$上的连通约化线性代数群，其对合为$θ$ .用$K$表示不动点的子群。在某些情况下，标志变种$G/B$中的$K$ -轨道由外尔群$W$中的扭曲恒等式$\iot = \{θ(w^{-1})w\mid w\in W\}$索引。在此假设下，我们建立了Schubert簇轨道闭包的有理光滑性准则，推广了Carrell和Peterson的经典结果。也就是说，轨道闭合在给定点是否有理光滑，可以通过检查"布鲁哈特图“中的度来确定，布鲁哈特图的顶点形成$\iot$的子集。此外，轨道闭包处处有理光滑当且仅当它在$\iot$上的Bruhat序中的对应区间是秩对称的。在特殊情况下，$K=\Sp_{2n}(\C)$，$G=\SL_{2n}(\C)$，我们加强我们的标准表明，只有一个单一的顶点，“底部的一个”，需要检查。这推广了Deodhar的$A$型舒伯特品种的结果。
英文摘要：
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.
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