Representation theory of finite W-algebras

最近，世界著名数学家对有限W-代数及其表示理论产生了极大的兴趣。这在很大程度上是由于连接到表示理论和几何的有限维约化李代数。有限W-代数和它们的仿射对应物在数学物理中也引起了很大的兴趣，这是因为它们在共形场论中作为非线性对称代数出现。本研究的目的是取得重大进展的表示理论的有限W-代数，特别是解决的重要问题的分类有限维简单modules.In最近的联合工作Brundan，Kleshchev和笔者，已经制定了一个最高的重量理论的有限W-代数。这导致了一个自然的策略来解决有限维简单模块的分类问题，类似于复杂的约化李代数的情况。本建议的主要目的是成功地实施这一战略。最近的发展领域的有限W-代数和不同领域的数学饲料在其表示理论将使我们能够采用各种技术来实现这一目标。特别是，我们打算利用有限W-代数的三个等价定义之间的相互作用：惠特克模型定义，通过量子哈密顿约化的定义，并通过Fedosov约化的定义。除了我们的主要目标，还有一些其他重要的问题，我们将调查的表示理论的有限W-代数。

Recently there has been a great deal of interest in finite W-algebras and their representation theory from world leading mathematicians. This is largely due to connections to the representation theory and geometry of finite dimensional reductive Lie algebras. Finite W-algebras and their affine counterparts have also attracted much interest in mathematical physics motivated by their occurrence as non-linear symmetry algebras in conformal field theory. The purpose of this research is to make significant advances in the representation theory of finite W-algebras and in particular to resolve the important problem of classifying finite dimensional simple modules.In recent joint work Brundan, Kleshchev and the author, have developed a highest weight theory for finite W-algebras. This leads to a natural strategy for tackling the problem of classifying finite dimensional simple modules, in analogy with the situation for complex reductive Lie algebras. The primary aim of this proposal is to successfully follow this strategy. The recent developments in the area of finite W-algebras and the diverse areas of mathematics feeding in to their representation theory will allow us to employ a variety of techniques in achieving this goal. In particular, we intend to exploit the interplay between the three equivalent definitions of a finite W-algebra: the Whittaker model definition, the definition via quantum Hamiltonian reduction, and the definition via Fedosov reduction. In addition to our main objective, there are a number of other important problems concerning the representation theory of finite W-algebras that we will investigate.