Lie Groups and Their Discrete Subgroups

李群及其离散子群

• 批准号: 1206999
• 负责人: John Millson
• 金额: $ 25.95万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206999&HistoricalAwards=false
• 关键词: Lie; Groups; Their; Discrete; Subgroups

The Principal Investigator (John Millson) proposes three main lines of research all within the general framework of reductive algebraic groups and geometry. In the first and most important main line, Nicolas Bergeron and Colette Moeglin (both of the University of Paris) and the PI will try to generalize to the unitary groups SU(p,q) their result that the Poincare duals of classes carried by totally geodesic cycles in the locally symmetric spaces of simple arithmetic type associated to the orthogonal groups SO(p,q) span the cohomology groups of low degree and special refined Hodge type, see "Hodge type theorems for arithmetic manifolds associated to orthogonal groups." Also, the PI proposes generalizing to the unitary case his earlier joint work (comprising five papers) with Jens Funke (from Durham University) on the boundary values of special cohomology classes associated to orthogonal groups. The second main line deals with the ring R of projective invariants of n ordered points on complex projective m space and the equivariant symplectic geometry of the corresponding moduli spaces and their toric degenerations. All of the completed work referred to above was supported by the NSF grant DMS-0907446. The third main line is an attempt to generalize to Kac Moody Lie algebras the PI's previous work with Michael Kapovich (and in parts with Thomas Haines, Shrawan Kumar, and Bernhard Leeb) dealing with the generalized triangle inequalities and related (saturation) problems from reductive Lie algebras. This project was the subject of the PI's earlier FRG grant DMS-0554254 with Prakash Belkale, Thomas Haines, Michael Kapovich and Shrawan Kumar. The problem looks difficult but there is a test example that will indicate whether the earlier theory will generalize. That example is affine SL(2). The PI proposes three main lines of research all within the general framework of reductive algebraic groups and geometry. The first part of the proposal deals with a remarkable and unexpected interaction between geometry, analysis and two different areas of representation theory (the oscillator/Weil representation and the work of James Arthur on the Selberg trace formula based in part on ideas of Robert Langlands). This work should have applications to number theory along the lines described by Steven Kudla in his 2002 International Congress of Mathematicians talk. The second and third parts of the proposal are motivated in part because they are related to much studied problems going back to the beginning of invariant theory in the late nineteenth century. The earlier work of the PI on the third part deals with basic problems in representation theory, e.g. decomposing tensor products and branching formulas which are much used in a number of disciplines. The current proposal outlines an extension of this work to other settings. All the above projects are in collaboration with other mathematicians from within the USA or abroad. In the last four years, the PI has had collaborations with ten mathematicians continuing a history of extensive collaboration (over fifty joint papers).

首席研究员（约翰·米尔森）提出了三个主要研究方向，全部在还原代数群和几何的总体框架内。 在第一个也是最重要的主线中，Nicolas Bergeron 和 Colette Moeglin（均来自巴黎大学）和 PI 将尝试将他们的结果推广到酉群 SU(p,q)，即与正交群 SO(p,q) 相关的简单算术类型的局部对称空间中的完全测地线循环所承载的类的庞加莱对偶跨越低次和特殊精炼的上同调群 Hodge 类型，请参阅“与正交群相关的算术流形的 Hodge 类型定理”。 此外，PI 建议将他早期与 Jens Funke（来自杜伦大学）关于与正交群相关的特殊上同调类的边界值的联合工作（包括五篇论文）推广到酉情况。 第二条主线讨论复射影 m 空间上 n 个有序点的射影不变量的环 R 以及相应模空间的等变辛几何及其环面简并。上述所有已完成的工作均得到 NSF 拨款 DMS-0907446 的支持。 第三条主线是尝试将 PI 之前与 Michael Kapovich（以及部分与 Thomas Haines、Shrawan Kumar 和 Bernhard Leeb）合作处理广义三角不等式和还原李代数相关（饱和）问题的工作推广到 Kac Moody 李代数。该项目是 PI 早期与 Prakash Belkale、Thomas Haines、Michael Kapovich 和 Shrawan Kumar 共同获得的 FRG 拨款 DMS-0554254 的主题。 这个问题看起来很困难，但有一个测试示例可以表明早期的理论是否可以推广。该示例是仿射 SL(2)。 PI 提出了三个主要研究方向，均在还原代数群和几何的总体框架内。该提案的第一部分涉及几何、分析和表示理论的两个不同领域（振子/韦尔表示和詹姆斯·阿瑟在部分基于罗伯特·朗兰兹的思想的塞尔伯格迹公式上的工作）之间的显着且意想不到的相互作用。这项工作应该按照 Steven Kudla 在 2002 年国际数学家大会演讲中描述的思路应用于数论。该提案的第二部分和第三部分的部分动机是因为它们与可追溯到十九世纪末不变理论之初的许多研究问题相关。 PI 的早期工作第三部分涉及表示论中的基本问题，例如分解张量积和分支公式在许多学科中广泛使用。当前的提案概述了这项工作到其他设置的扩展。 所有上述项目都是与美国或国外的其他数学家合作的。在过去四年中，PI 与十位数学家进行了合作，延续了广泛合作的历史（超过 50 篇联合论文）。