Combinatorics of the Affine Hecke Algebra and Module Categories

本建议书由四部分组成。在第一部分，研究者和他的同事研究了G. Lusztig引入的渐近仿射Hecke代数。当参数趋于零时，渐近Hecke代数是一般Hecke代数的一个合适的极限。它的表征理论与赫克代数本身的表征理论密切相关。本文的目的之一是证明用初等k论术语描述渐近Hecke代数的Lusztig猜想。第二部分研究了模类与一元类的关系。这门学科与现代物理学密切相关，其中模块范畴出现在边界共形场论的背景下。在第三部分中，研究者和合作者研究了仿射Weyl组中不同的牵涉。特别是，他们在许多情况下对规范区分的对合进行了广泛的明确计算。第四部分研究了双仿射赫克代数。本文的目的是用无限维代数变体的Kazhdan-Lusztig型组合来描述该代数变体的交上同调。在这个建议中，研究者研究表征理论的各种问题。表征理论是数学的一部分，它研究所有可能的方法，其中对称性可以用于解决具体的物理或技术问题。许多物理和技术系统在某些转换下不会改变（例如，太阳的引力场仅取决于与太阳的距离，因此它在围绕太阳旋转的空间下不会改变）。这样的变换称为系统的对称性。在许多情况下，对称性可以用来简化这类系统的研究。因此，毫不奇怪，表征理论在物理学（连续对称是最基本的概念之一）、化学（特别是在量子化学中，它被用于计算分子内部的化学力）、计算机科学（例如，傅立叶分析，可以被认为是表征理论的最简单的例子，是所有计算技术中使用最广泛的一种）和数学本身中有许多应用。在数论（其中表征理论是朗兰兹纲领的重要组成部分）。表征理论研究的中心对象之一是仿射赫克代数，因为许多看似无关的问题的答案都编码在这个代数的结构中。本文主要致力于仿射赫克代数的研究。

The proposal consists of 4 parts. In the first part the investigator and his colleagues study the asymptotic affine Hecke algebra introduced by G. Lusztig. The asymptotic Hecke algebra is a suitable limit of the usual Hecke algebra as parameter tends to zero. Its representation theory is closely related with representation theory of the Hecke algebra itself. One of the aims here is a proof of Lusztig's Conjecture describing the asymptotic Hecke algebra in the elementary K-theoretic terms. In the second part the investigator studies module categories over monoidal categories. This subject is closely related with modern physics where module categories appear in the context of the Boundary Conformal Field Theory. In the third part the investigator and collaborators study distinguished involutions in the affine Weyl group. In particular they make extensive explicit calculations of canonical distinguished involutions in number of cases. In the fourth part the investigator and his colleagues study the Double Affine Hecke Algebra. The aim here is to describe Intersection Cohomology of certain infinite dimensional algebraic varieties in terms of Kazhdan-Lusztig type combinatorics of this algebra.In this proposal the investigator studies various questions of Representation Theory. Representation Theory is a part of mathematics that studies all possible ways in which symmetry can be used for solving concrete physical or technical problems. Many physical and technical systems do not change under some transformations (e.g. the gravitational field of the Sun depends only on the distance from the Sun and so it does not change under rotating of the space around the Sun). Such transformations are called symmetries of the system. In many cases symmetries can be used to simplify the study of such systems. So it is not surprising that Representation Theory has many applications in physics (where continuous symmetry is one of the most fundamental concepts), chemistry (especially in quantum chemistry where it is used in computations of chemical forces inside molecules), computer science (for example, Fourier analysis, which can be considered as a simplest case of Representation Theory, is one of the most widely used of all calculation techniques), and inside of mathematics itself, in number theory (where Representation Theory is an essential part of the Langlands program). One of the central objects of study in Representation Theory is the affine Hecke algebra, because answers to many seemingly unrelated questions are encoded in the structure of this algebra. This proposal is mainly devoted to the study of the affine Hecke algebra.