Lie Groups and Their Discrete Subgroups

首席研究员(约翰·米尔森)提出了三条研究主线，全部在约化代数群和几何的一般框架内。在第一条也是最重要的一条主线中，巴黎大学的Nicolas Bergeron和Colette Moeglin和PI将试图将他们的结果推广到酉群SU(p，q)，他们的结果是在与正交群相关的简单算术类型的局部对称空间中的完全测地圈所承载的类的Poincare对偶So(p，q)跨越低次和特殊精化Hoge型的上同调群，参见“关于与正交群相关的算术流形的Hodge类型定理”。此外，PI还建议将他与Jens Funke(来自达勒姆大学)关于与正交群相关的特殊上同调类的边值的早期合作工作(包括五篇论文)推广到酉型情形。第二条主线讨论了复射影m空间上n阶点的射影不变量环R以及相应模空间的等变辛几何及其环面退化。上面提到的所有已完成工作都得到了国家科学基金会的DMS-0907446赠款的支持。第三条主线是试图将PI以前与Michael Kapovich(以及部分与Thomas Haines，Shrawan Kumar和Bernhard Leeb)一起处理还原李代数中的广义三角不等式和相关(饱和)问题的工作推广到Kac Moody李代数。这个项目是国际和平组织早些时候与Prakash Belkale、Thomas Haines、Michael Kapovich和Shrawan Kumar一起获得的联邦共和国政府拨款DMS-0554254的主题。这个问题看起来很难，但有一个测试例子可以表明早期的理论是否会推广。这个例子就是仿射SL(2)。PI提出了三条研究主线，全部在约化代数群和几何的一般框架内。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·阿瑟关于塞尔伯格迹公式的工作，部分基于罗伯特·朗兰兹的思想)之间的显著和意想不到的相互作用。这项工作应该适用于数论，就像史蒂文·库德拉在2002年国际数学家大会演讲中所描述的那样。提案的第二部分和第三部分的动机部分是因为它们与许多研究问题有关，这些问题可以追溯到19世纪末不变量理论的开始。PI关于第三部分的早期工作涉及表示论中的基本问题，例如，在许多学科中广泛使用的张量积分解和分支公式。目前的提案概述了将这项工作扩展到其他环境。所有上述项目都是与来自美国或国外的其他数学家合作进行的。在过去的四年里，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)。

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).