Combinatorics of the Affine Hecke Algebra and Module Categories

该提案由4个部分组成。在第一部分中，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代数是通常Hecke代数的一个合适的极限。它的表示理论与Hecke代数本身的表示理论密切相关。这里的目的之一是证明Lusztig用初等K-理论项描述渐近Hecke代数的猜想。在第二部分中，研究者研究了么半群范畴上的模范畴。这门学科与现代物理学密切相关，现代物理学中的模范类出现在边界共形场理论的背景下。在第三部分中，研究者和合作者研究了仿射Weyl群中的特殊对合。特别地，在许多情况下，他们对典型区分对合进行了大量的显式计算。在第四部分中，研究者和他的同事研究了双重仿射Hecke代数。本文的目的是用某些无限维代数簇的Kazhdan-Lusztig型组合学来刻画这种代数簇的交上同调.在这个方案中，研究者研究了表示论的各种问题.表示论是数学的一部分，它研究对称性可以用来解决具体物理或技术问题的所有可能的方法。许多物理和技术系统在某些变换下不会改变(例如，太阳的引力场只取决于与太阳的距离，因此它在围绕太阳的空间旋转时不会改变)。这种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对这类系统的研究。因此，表示理论在物理学(其中连续对称性是最基本的概念之一)、化学(特别是在量子化学中用于计算分子内部化学力)、计算机科学(例如，傅立叶分析，可以被认为是表示理论最简单的情况，是所有计算技术中最广泛使用的技术之一)以及数学本身的数论(其中表示理论是朗兰兹计划的重要部分)中有许多应用也就不足为奇了。表示论的中心研究对象之一是仿射Hecke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。这一建议主要致力于仿射Hecke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.