Towards Langlands duality for affine Kac-Moody groups

在当前的计划中，PI建议将所谓的Langlands对偶的几个方面与仿射Kac-Moody群进行类比。更具体地说，PI建议研究仿射Kac-Moody群的球面和岩堀Hecke代数以及它们的几何版本（该计划的某些部分已经在PI与D.Kazhdan和M.Finkelberg的联合工作中开发）。PI还建议研究仿射Kac-Moody群的类似自守形式，以及推广有限维约化群的几何Langlands对应的一些结构。拟议的研究处于数论、表示论、代数几何和数学物理等领域的边缘;该项目的成功实施可能会对这些领域之间的联系产生一些新的影响。例如，数论可能是最古老的数学学科之一，它最重要的部分之一被称为朗兰兹纲领。最近人们意识到朗兰兹纲领的几何方面与代数几何和现代数学物理（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，并扩大和推广这种联系。

In the current proposal the PI proposes to develop an analog of several aspects of the so-called Langlands duality to affine Kac-Moody groups.More specifically, the PI proposes to study the spherical and Iwahori Hecke algebras for affine Kac-Moody groups as well as their geometric versions (some parts of that program have already been developed in the PI's joint works with D.Kazhdan and M.Finkelberg). The PI also proposes to study an analog automorphic forms for affine Kac-Moody groups as well as some structures generalizing the so-called geometric Langlands correspondence for finite-dimensional reductive groups.The proposed research lies on the border of such fields as number theory, representation theory, algebraic geometry and mathematical physics; successful implementation of the project might shed some new light on the connections 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 algebraic geometry as well as 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.