Arithmetic, Geometry and Representation Theory of Reductive Groups

还原群的算术、几何和表示论

• 批准号: 0653512
• 负责人: Gopal Prasad
• 金额: $ 15.88万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653512&HistoricalAwards=false
• 关键词: Arithmetic; Geometry; Representation; Theory; Reductive

In their two recent papers, Gopal Prasad and Sai-Kee Yeung have constructed all arithmetic "fake" projective spaces and have found some of their geometric properties. Their work has led to many interesting questions related to these spaces and also about some of the singular surfaces with small geometric invarients. Prasad proposes to work on these questions. He has also been working with Andrei Rapinchuk to find the extent a locally symmetric space of finite volume is determined by the set of lengths of its closed geodesics, or its spectrum. Their work led them to define a new relationship between "large" (Zariski-dense or arithmetic) subgroups which they call "weak commensurability". They have shown that weak commesurability of two arithmetic subgroups of a simple Lie group, whose Dynkin diagram does not have symmetries, implies that they are commensurable. They are now investigating the situation when the Dynkin diagram does have a symmetry. Prasad and Rapinchuk have used theorems in transcendental number theory, and a widely believed conjecture due to Schanual, to show that if the quotients of the symmetric space of an absolutely simple real Lie group by two arithmetic subgroups have same set of lengths of closed geodesics, or have the same spectrum, then the two arithmetic subgroups are weakly commensurable. So their results on weakly commensurable arithmetic subgroup can be used. Prasad proposes to obtain analogous results for locally symmetric spaces arising from comples semi-simple Lie groups. In a comepletely different direction, Prasad has associated a natural Levi-subgroup to a given irreducible admissible representation of a reductive p-adic group. He will investigate what role this subgroup plays in the representation theory. Prasad has an ongoing collaboration with Rapinchuk to simplify, unify and complete the results on the congruence subgroup problem. They plan to write a book on this topic in near future.Recent work of Prasad with Sai-Kee Yeung on certain interesting geometric objects known as arithmetic fake projective spaces has led to an explicit construction of all of them and helped to determine many of their geometric properties. Their work has also led to some important questions about related geometric objects. Prasad's recent work with Rapinchuk has deep implications for a particularly important class of geometric structures known as locally symmetric spaces. These spaces arise from symmetric spaces, which as the name suggests, have a lot of symmetries. This work of Prasad and Rapinchuk has introduced a new notion of "weak commensurability" of large subgroups of the group of symmetries and studies its consequences in geometry and group theory.There are still some serious unresolved questions on which they will work.They also plan to write a book on the famous congruence subgroup problem to describe a new unified approach to settle it.

在他们最近的两篇论文中，Gopal Prasad和Sai-Kee Yeung构造了所有算术“假”投影空间，并发现了它们的一些几何性质。他们的工作引出了许多与这些空间相关的有趣问题，也引出了一些具有小几何不变量的奇异曲面。普拉萨德建议着手解决这些问题。他还一直在与Andrei Rapinchuk合作，寻找有限体积的局部对称空间的程度是由其封闭测地线的长度集或其谱决定的。他们的工作使他们定义了“大”（zariski密集或算术）子群之间的一种新关系，他们称之为“弱通约性”。他们证明了一个简单李群的两个算术子群的弱可通约性，其Dynkin图不具有对称性，意味着它们是可通约的。他们现在正在研究当Dynkin图确实具有对称性的情况。Prasad和Rapinchuk利用超越数论中的定理和Schanual的一个猜想，证明了两个算术子群所组成的绝对简单实数李群的对称空间中，如果商具有相同的闭测线长度集，或者具有相同的谱，则这两个算术子群是弱可公的。因此，它们在弱可通约算术子群上的结果是可用的。Prasad提出了由复半单李群产生的局部对称空间的类似结果。在一个完全不同的方向上，Prasad将自然列维子群与一个可约p进群的给定不可约容许表示联系起来。他将研究这一群体在表征理论中所起的作用。Prasad正在与Rapinchuk合作，简化、统一和完成同余子群问题的结果。他们计划在不久的将来写一本关于这个主题的书。Prasad和Sai-Kee Yeung最近在一些有趣的几何对象上的工作，即所谓的算术伪投影空间，导致了它们的明确构造，并帮助确定了它们的许多几何性质。他们的工作也引出了一些有关几何物体的重要问题。Prasad最近与Rapinchuk的合作对一类特别重要的几何结构——局部对称空间有着深远的影响。这些空间来自对称空间，顾名思义，对称空间有很多对称性。Prasad和Rapinchuk的这项工作引入了对称群的大子群的“弱可通约性”的新概念，并研究了其在几何和群论中的影响。仍有一些严重的未解决的问题需要他们来解决。他们还计划写一本关于著名的同余子群问题的书，以描述一种新的统一方法来解决它。