Around Langlands duality for representations of affine Kac-Moody groups

仿射 Kac-Moody 群表示的朗兰兹对偶性

• 批准号: 1200807
• 负责人: Alexander Braverman
• 金额: $ 30.04万
• 依托单位: Brown University
• 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: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200807&HistoricalAwards=false
• 关键词: Around; Langlands; duality; representations; affine

In the current proposal the PI proposes to study several problems related to algebraic groups over 2-dimensional local and global fields, their representations and related geometric problems. More specifically the PI intends to continue his study of Hecke algebras of affine Kac-Moody groups over a local non-archomedian field. He plans to apply those ideas to the theory of affine Eisenstein series; this should produce applications to the theory of (usual) automorphic L-functions. In the second part of his project the PI suggests to attack several (mathematically well-posed) problems related to 4-dimensional gauge theory. Both subjects can be reformulated in terms of various questions about G-bundles on algebraic surfaces. The PI believes that above questions may also be connected to the (not yet formulated) 2-dimensional geometric Langlands duality.The proposed research lies on the border of such fields as number theory, algebraic geometry, representation theory and mathematical physics; succesful implementation of the project might shed some new light on the connection between these fields. For example, number theory is perhaps one of the oldest mathematical subjects and one of its most important parts is called the Langlands program. Recently it has been realized that geometric aspects of the Langlands program have many connections with modern mathematical physics (such as 4-dimensional quantum field theory). The proposed research project should confirm the existence of such links as well broaden and generalize them.

在目前的提案中，PI建议研究与二维局部和全局域上的代数群有关的几个问题，它们的表示和相关的几何问题。更具体地说，PI打算继续他的研究仿射Kac-Moody群的Hecke代数在局部非古古场。他打算把这些想法应用到仿射爱森斯坦级数的理论中；这将产生对（通常）自同构l函数理论的应用。在他的项目的第二部分，PI建议攻击几个与四维规范理论相关的（数学上良好的）问题。这两个主题都可以用代数曲面上g束的各种问题来重新表述。PI认为上述问题也可能与（尚未公式化的）二维几何朗兰兹对偶有关。本文的研究处于数论、代数几何、表示理论和数学物理等领域的边缘；该项目的成功实施可能会使人们对这些领域之间的联系有一些新的认识。例如，数论可能是最古老的数学学科之一，其最重要的部分之一被称为朗兰兹纲领。最近，人们认识到朗兰兹程序的几何方面与现代数学物理学（如四维量子场论）有许多联系。拟议的研究项目应确认这种联系的存在，并扩大和概括它们。