Geometric representation theory and moduli spaces of bundles

该建议属于几何表示理论领域。拟议的研究基本上由4个主题，他们所有的统一主题是代数曲线和代数曲面上的丛的模空间的表示理论和几何之间的关系。 第一部分是关于局部域上仿射Kac-Moody群的Hecke代数（这一课题是由A.Braverman和D.Kazhdan发起的;我们打算研究一些进一步的问题。特别是，我们计划开发一个模拟的理论Kazhdan-Lusztig多项式的双仿射Hecke代数）。 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的表示的约化群G在局部域的正特征（附加到任意有限维表示的Langlands对偶群）。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，它是基于某些无限维空间的反常层的几何，这些空间附着在G的loop群上。该计划是要表明，我们的建设是明确的，它是兼容的一些已知的结果，当地朗兰兹对应（如Lusztig的分类幂幺表示）。 第三部分给出了三维N=4超对称规范理论库仑分支的数学定义及其量子化。这个定义是基于相应规范群的仿射格拉斯曼几何。我们计划将我们的构造应用于Braden，Licata，Proudfoot和韦伯斯特提出的所谓“辛对偶”的范畴化版本的证明。 在第四部分中，我们对简单有限维仿射李代数给出了Kazhdan-Lusztig猜想的一个新的证明（特别是覆盖了一些新的情形，如仿射李代数在临界水平上的表示）。建议的证明是基于几何的所谓Zastava空间（在有限维的情况下）和乌伦贝克空间的丛（在仿射的情况下）。

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).