Lusztig-Vogan双射组合意义的相关研究

In mathematics, the orbit method (also known as the Kirillov theory) is a branch of Lie groups and representation theory. Its target is to establish a correspondence between irreducible unitary representations of a reductive Lie group and its coadjoint nilpotent orbits. The research method of the orbit method involves many fields, such as harmonic analysis, combinatorics, algebraic geometry and so on. Also since it has important applications in quantum mechanics, then the orbit method has become an active research field. . For complex reductive Lie groups, the Lusztig-Vogan bijection is an important bijection in the orbit method. Its content is to establish a bijection between the set of pairs, each of which consists of a nilpotent orbit as well as an algebraic representation of its isotropy subgroup, and the set of integral dominant weights. Firstly, G. Lusztig conjectured the existence of this bijection using his work on affine Weyl groups, and later from the point of view of Harish-Chandra modules, D. Vogan also constructed such a bijection between these two sets. The research on the combinatorial description of the Lusztig-Vogan bijection and related questions becomes a valuable research topic for us. It will play a quite important role in the research on unitary representations, as well as the relationship between nilpotent orbits and weights. Moreover, this will improve the interdisciplinary developments among combinatorics, representation theory and differential geometry. . In our previous work, we have tried some effective new methods, such as two-step induction, orbit cover induction, diagrammatic method, etc., which provide a feasible research idea for this research topic. In our research plan, we will further extend these combinatorial methods together with geometric methods to the general cases of complex Lie groups in order to make a breakthrough in this research subject. Meanwhile, our research will enrich the tools of the orbit method and increase the vitality of the deep applications of representation theory in geometry and physics.

轨道方法是李群表示理论的分支，目的是建立约化李群的酉表示与幂零轨道的对应，因研究方法涉及调和分析、组合理论以及代数几何等诸多领域，且在量子力学中有重要应用，故成为活跃的研究领域。.在复李群中，Lusztig-Vogan双射由G.Lusztig在研究放射外尔群与D.Vogan借助Harish-Chandra理论分别提出，建立了幂零轨道与其迷向子群的代数表示组成的集合与支配整权集合间的一一对应。该双射组合意义的相关研究，对研究酉表示、轨道与权的关系、探索组合理论与表示理论及微分几何间的跨学科领域发展等都有很大促进作用，从而是一有价值的研究课题。.在前期研究中，我们尝试一些有效的新方法，如：二步诱导、轨道覆盖诱导、图形化方法等，为该课题提供可行的研究思路，我们将进一步结合这些组合和几何方法，完成该项目的研究目标。同时这也将丰富轨道方法的研究工具，也为表示理论在几何与物理等学科的深入应用增加活力。

在复李群中，Lusztig-Vogan双射建立了幂零轨道与其迷向子群的代数表示组成的集合与支配整权集合间的一一对应。在本项目中，我们主要关注该双射的组合意义及相关的几何和理论物理等课题的研究。.在本项目中，我们重点研究了以下几个问题：.（1）利用轨道覆盖诱导得到了关于诱导表示结构的相关研究成果，在一定程度上简化了Lusztig-Vogan双射的研究，且拓宽了几何工具在表示理论中的应用。.（2）利用了图形化方法这一组合工具得到了李超代数上具有辛结构的对称对的分类，李超代数的辛对称对在几何及理论物理中有着重要的应用。.（3）利用图形化方法得到了李超代数及李超代数的实形式的外自同构群的结构的描述，这在无限维代数及可积系统等都有着重要应用。.取得的成果如下：.[1] 利用轨道覆盖诱导得到了关于诱导表示结构的相关研究成果，该部分结构已整理成论文，目前处于投稿中。.[2] 得到了具有辛结构的李超代数的对称对的分类，该部分成果发表在Ann. Mat. Pura Appl.上。.[3]得到了李超代数及李超代数的实形式对应的外自同构群的结构的描述，该部分成果发表在J. Algebra与Algebr. Represent. Theory上。

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