Geometric methods in representation theory of rational Cherednik algebras.

这个建议将调查某些问题的表示论，一个主要的分支代数互动强烈的几何和数学物理。纯数学的目的是对熟悉的概念的本质进行抽象和抽象：在对称性的情况下，这导致了群的定义，即给定对象的对称性的集合。然而，数学定义更加不言自明，以至于群体所描述的基本对象几乎消失了。在这些情况下，重要的是试图恢复这个对象，或者更具体地说，找到所有的对象，其对称性产生了所讨论的群。这就是表征理论背后的动机思想。尽管这个问题看起来很抽象，但表象理论在许多科学领域（如物理学）中至关重要。（例如弦理论/镜像对称），化学（分子振动的研究）和计算机科学，以及是数学的中心。有理切雷德尼克代数，正如由埃廷霍夫和金兹伯格介绍的，涉及到，并建立在辛代数几何的结果，李理论和几何表示理论，代数组合学。特别是，它们已经被用来证明非常困难的结果，如回答辛商奇点的crepant决议的存在性问题，并解决与Mark Haiman的n！猜想这些结果说明了应用非交换代数表示论中存在的技术来解决纯数学相关领域中的难题的能力。这项建议有两个部分。在第一部分中，我计划调查之间的联系，合理的Cherednik代数在t=0仿射李代数在关键的水平，从而提供了一种方法，使用强大的工具已经开发在这一领域（如几何的Opers）的Frenkel，Gaitsgory和其他人获得更好地理解的代表性理论的合理Cherednik代数。这也是很自然的期望，我们的理解合理的切雷德尼克代数将有许多有趣的应用在研究仿射李代数的关键水平。在过去的30年里，表示论领域最著名的成果之一是Beilinson和伯恩斯坦利用局部化的思想证明了Kazhdan-Luzstig猜想。在第二部分的建议，我将探讨的后果理性Cherednik代数在t=1的最近推广柏原和Rouquier的本地化方法。我的目标是使用本地化的方法，介绍强大的几何技术，如理论的反常层，研究的表示理论的理性切雷德尼克代数。这里开发的技能适用于许多其他对象目前感兴趣的表示理论家，如有限W-代数和量子哈密顿约化的量子变种。

This proposal will investigate certain problems in representation theory, a major branch of algebra interacting strongly with geometry and mathematical physics. Pure mathematics aims to abstract and distil the essence of familiar concepts: in the case of symmetries this leads to the definition of a group, the collection of symmetries of a given object. However, the mathematical definition is far more axiomatic, to the point that the underlying object that the group is describing all but disappears. In these cases it is important to try to recover this object, or more specifically to find all objects whose symmetries give rise to the group in question. This is the motivating idea behind representation theory. Despite this seemingly abstract problem, representation theory is crucially important in many areas of science such as physics (e.g. string theory / mirror symmetry), chemistry (study of molecular vibrations) and computer science, as well as being central for mathematics.Rational Cherednik algebras, as introduced by Etingof and Ginzburg relate to, and build upon results in symplectic algebraic geometry, Lie theoretic and geometric representation theory, and algebraic combinatorics. In particular, they have already been used to prove very difficult results such as answering the question of existence of crepant resolutions for symplectic quotient singularities and solving combinatorial conjectures on the properties of certain rings of coinvariants related to Mark Haiman's n!-conjecture. These results illustrate the power of applying the techniques that exist in the representation theory of noncommutative algebras to solving hard problems in related areas of pure mathematics. There are two parts to this proposal. In the first part I plan to investigate the connection between rational Cherednik algebras at t=0 to affine Lie algebras at the critical level, thereby providing a way of using the powerful tools already developed in that area (such as the geometry of Opers) by Frenkel, Gaitsgory and others to gain a much better understanding of the representation theory of rational Cherednik algebras. It is also natural to expect that our understanding of rational Cherednik algebras will have many interesting applications in the study of affine Lie algebras at the critical level. One of the most celebrated results in the field of representation theory in the past 30 years has been the proof by Beilinson and Bernstein of the Kazhdan-Luzstig conjecture using the idea of localization. In the second part of this proposal I will explore the consequences for rational Cherednik algebras at t=1 of a recent generalization by Kashiwara and Rouquier of the localization method. I aim to use the localization method to introduce powerful geometric techniques, such as the theory of perverse sheaves, to the study of the representation theory of rational Cherednik algebras. The skills developed here are applicable to many other objects currently of interest to representation theorists such as finite W-algebras and quantum Hamiltonian reduction of quiver varieties.