Geometric representation theory and moduli spaces of bundles

这一提议属于几何表示理论领域。本研究主要由4个主题组成，其统一的主题是代数曲线和代数曲面上的丛的模空间的表示理论与几何之间的关系。*第一部分是关于局部域上仿射Kac-Moody群的Hecke代数(这个课题是由A.Braverman和D.Kazhdan发起的，我们打算进一步研究一些问题。具体地说，我们计划发展一种类似于双仿射Hecke代数的Kazhdan-Lusztig多项式的理论)。*在第二部分中，我们提出了一个新的局部L-函数的统一几何定义，用于表示具有正特征的局部域上的约化群G(附在对偶群的任意有限维表示上)。这个定义沿用了Godement和Jacquet的经典模式，但相关Schwartz空间的定义要复杂得多，而且它是基于G的循环群上的某些无限维空间的倒叶的几何。我们的计划是为了证明我们的构造是定义良好的，并且它与局部朗兰兹对应的一些已知结果(例如Lusztig对幂等表示的分类)是相容的。*在第三部分中，我们提出了所谓的三维N=4超对称规范理论的库仑分支的数学定义及其量子化。该定义基于相应规范群的仿射Grassmanian几何。我们计划将我们的构造应用于由Braden，Licata，ProudFoot和Webster提出的所谓的“辛对偶”的猜想范畴版本的证明。*在第四部分中，我们建议给出有限维单李代数和仿射李代数的Kazhdan-Lusztig猜想的一个新的证明(部分地，涵盖一些新的情况--例如临界级上仿射李代数的表示)。所提出的证明是基于所谓的Zastava空间(在有限维情况下)和Uhlenbeck丛空间(在仿射情况下)的几何。

The proposal lies in the field of geometric representation theory. The proposed research essentially consists of 4 subjects; the unifying theme for all of them is the relation between representation theory and geometry of moduli space of bundles on algebraic curves and algebraic surfaces. ***The first part has to do with Hecke algebras for affine Kac-Moody groups over local fields (this subject has been initiated by A.Braverman and D.Kazhdan; we propose to study some further questions. In particular, we plan to develop an analog of the theory of Kazhdan-Lusztig polynomials for double affine Hecke algebras).***In the second part we propose a new uniform geometric definition of local L-functions for representations of a reductive group G over local fields of positive characteristic (attached to an arbitrary finite-dimensional representation of the Langlands dual group) . The definition follows the classical pattern of Godement and Jacquet, but the definition of the relevant Schwartz space is much more involved and it is based on the geometry of perverse sheaves of certain infinite-dimensional spaces attached to the loop group of G. The plan is to show that our construction is well-defined and that it is compatible with some known results on the local Langlands correspondence (such as Lusztig's classificiation of unipotent representations).***In the 3rd part we propose a mathematical definition of the so called Coulomb branch of 3-dimensional N=4 super-symmetric gauge theories and its quantization. The definition is based on the geometry of the affine Grassmannian of the corresponding gauge group. We plan to apply our construction to the proof of the conjectural categorical version of the so called "symplectic duality" due to Braden, Licata, Proudfoot and Webster.***In the 4th part we propose to give a new proof of the Kazhdan-Lusztig conjecture for simple finite-dimensional and affine Lie algebras (in partucular, covering some new cases - such as the case of representations of affine Lie algebras on critical level). The proposed proof is based on the geometry of the so called Zastava spaces (in the finite-dimensional case) and Uhlenbeck spaces of bundles (in the affine case).********