关于AIII型的广义Steinberg理论
On generalized Steinberg theory for type AIII
AI总结：
针对AIII型对称对的双旗簇$K$对角作用轨道，本文给出轨道的部分置换对参数化等描述，并通过部分置换组合算法显式刻画对称化与异值Steinberg映射，推广了$p=q=r$特殊情形下的已有结果。
摘要：
给定AIII型对称对$(G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_{p}(\mathbb{C})\times \mathrm{GL}_{q}(\mathbb{C}))$，我们研究$K$在双旗簇$\mathfrak{X}=\mathrm{Grass}(\mathbb{C}^{p+q},r)\times K/B_K$上的对角作用，其中第一个分量是$G$的Grassmann簇，第二个分量是$K$的全旗簇。该作用的轨道数目有限，我们的第一项结果是对这些轨道的描述：包括参数化、维数、闭包关系和覆盖关系。具体而言，这些轨道由特定的部分置换对进行参数化。\n$\mathfrak{X}$中的每个轨道都对应一个余法丛。与参考文献[5]和[6]中的做法类似，我们利用与该作用相关的矩映射，定义了所谓的对称化Steinberg映射和异值Steinberg映射，分别将每个余法丛（即每个轨道）映射到$K$的李代数中的幂零轨道，以及该李代数的Cartan补中的幂零轨道。我们的主要结果是通过部分置换上的组合算法对这些Steinberg映射进行显式描述，该算法推广了经典的置换Robinson–Schensted过程。这是对文献[5]中结果的全面推广，文献[5]中我们假设$p=q=r$且仅考虑特殊形式的轨道。
英文摘要：
Given a symmetric pair $(G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_{p}(\mathbb{C})\times \mathrm{GL}_{q}(\mathbb{C}))$ of type AIII, we consider the diagonal action of $K$ on the double flag variety $\mathfrak{X}=\mathrm{Grass}(\mathbb{C}^{p+q},r)\times K/B_K$ whose first factor is a Grassmann variety for $G$ and whose second factor is a full flag variety of $K$. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization, dimensions, closure relations, and cover relations. Specifically, the orbits are parametrized by certain pairs of partial permutations. Each orbit in $\mathfrak{X}$ gives rise to a conormal bundle. As in the references [5] and [6], by using the moment map associated to the action, we define a so-called symmetrized Steinberg map, respectively an exotic Steinberg map, which assigns to each such conormal bundle (thus to each orbit) a nilpotent orbit in the Lie algebra of $K$, respectively in the Cartan complement of that Lie algebra. Our main result is an explicit description of these Steinberg maps in terms of combinatorial algorithms on partial permutations, extending the classical Robinson--Schensted procedure on permutations. This is a thorough generalization of the results in [5], where we supposed $p=q=r$ and considered orbits of special forms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
