Geometric methods in representation theory of rational Cherednik algebras.

这项建议将研究表示论中的某些问题，表示论是代数的一个主要分支，与几何和数学物理有很强的相互作用。纯数学的目的是抽象和提炼熟悉概念的本质：在对称性的情况下，这导致了组的定义，即给定对象的对称性的集合。然而，数学上的定义要公理得多，以至于小组所描述的底层对象几乎消失了。在这些情况下，重要的是尝试恢复这个对象，或者更具体地说，找到其对称性导致问题群的所有对象。这就是表象理论背后的激励思想。尽管有这个看似抽象的问题，但表示理论在许多科学领域都是至关重要的，例如物理(如弦理论/镜像对称性)、化学(研究分子振动)和计算机科学，以及数学的中心。Etingof和Ginzburg引入的有理Cherednik代数与辛代数几何、李理论和几何表示理论以及代数组合学的结果相关，并建立在这些结果的基础上。特别地，它们已经被用来证明非常困难的结果，如回答辛商奇点的预解的存在性问题，以及解决与Mark Haiman的n！猜想有关的某些余不变环的性质的组合猜想。这些结果说明了应用非对易代数表示理论中存在的技巧来解决纯数学相关领域中的困难问题的能力。这项提议包括两个部分。在第一部分中，我计划研究t=0的有理Cherednik代数与临界水平的仿射李代数之间的联系，从而提供一种方法来使用Frenkel和Gaitsgory等人在那个领域已经开发的强大工具(如OPERS几何)来更好地理解有理Cherednik代数的表示理论。我们对有理Cherednik代数的理解在仿射李代数的研究中将有许多有趣的应用，这也是自然的。在过去的30年里，表象理论领域最著名的成果之一是Beilinson和Bernstein用局部化的思想证明了Kazhdan-Luzstig猜想。在这个建议的第二部分，我将探索Kashiwara和Rouquier最近推广的局部化方法对t=1的有理Cherednik代数的结果。我的目的是用局部化的方法将强大的几何技巧，如倒叶理论，引入到有理Cherednik代数的表示理论的研究中。这里发展的技巧也适用于目前表示理论家感兴趣的许多其他对象，例如有限W-代数和箭图簇的量子哈密尔顿约化。

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.