Geometric representation theory and moduli spaces of bundles

几何表示理论和丛的模空间

• 批准号: RGPIN-2016-05542
• 负责人: Braverman, Alexander
• 金额: $ 2.4万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615760
• 关键词: Geometric; representation; theory; moduli; spaces

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).

该提案属于几何表示理论领域。拟议的研究主要包括 4 个主题；它们的统一主题是表示论与代数曲线和代数曲面上丛的模空间几何之间的关系。 第一部分与局部域上的仿射 Kac-Moody 群的 Hecke 代数有关（该课题由 A.Braverman 和 D.Kazhdan 发起；我们建议研究一些进一步的问题。特别是，我们计划开发双仿射 Hecke 代数的 Kazhdan-Lusztig 多项式理论的模拟）。 在第二部分中，我们提出了局部 L 函数的新统一几何定义，用于表示正特征局部域上的约简群 G（附加到朗兰兹对偶群的任意有限维表示）。该定义遵循 Godement 和 Jacquet 的经典模式，但相关 Schwartz 空间的定义要复杂得多，并且它基于附加到 G 环群的某些无限维空间的反常滑轮的几何形状。该计划是为了表明我们的构造是明确定义的，并且它与局部 Langlands 对应关系的一些已知结果兼容（例如 Lusztig 的分类 单能表示）。 在第三部分中，我们提出了所谓的3维N=4超对称规范理论库仑分支的数学定义及其量子化。该定义基于相应规范组的仿射格拉斯曼函数的几何形状。我们计划将我们的构造应用于证明由 Braden、Licata、Proudfoot 和 Webster 提出的所谓“辛对偶性”的猜想绝对版本。 在第四部分中，我们建议给出简单有限维仿射李代数的 Kazhdan-Lusztig 猜想的新证明（特别是涵盖一些新情况 - 例如仿射李代数在临界水平上的表示情况）。所提出的证明基于所谓的 Zastava 空间（在有限维情况下）和束的 Uhlenbeck 空间（在仿射情况下）的几何形状。