Let $G$ be a simple algebraic group and $P$ a parabolic subgroup of $G$ with abelian unipotent radical $P^u$, and let $B$ be a Borel subgroup of $G$ contained in P. Let $\mathfrak{p}^u$ be the Lie algebra of $P^u$ and let $L$ be a Levi factor of $P$, then $L$ is a Hermitian symmetric subgroup of $G$ and $B$ acts with finitely many orbits both on $\mathfrak{p}^u$ and on $G/L$. In this paper we study the Bruhat order of the $B$-orbits in $\mathfrak{p}^u$ and in $G/L$, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.

中文翻译：

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Hermitian 对称簇和阿贝尔 nilradicals 的 Bruhat 阶数

令$G$是一个简单代数群，$P$是$G$带有阿贝尔单能根$P^u$的抛物子群，令$B$是包含在P中的$G$的Borel子群。令$\ mathfrak{p}^u$ 是 $P^u$ 的李代数，令 $L$ 是 $P$ 的 Levi 因子，则 $L$ 是 $G$ 的 Hermitian 对称子群，$B$ 与$\mathfrak{p}^u$ 和 $G/L$ 上的有限多个轨道。在本文中，我们研究了 $\mathfrak{p}^u$ 和 $G/L$ 中 $B$-轨道的布鲁哈特阶，分别证明了 Panyushev 的猜想和 Richardson 和 Ryan 的猜想。