Noncommutative Algebraic Geometry in Representation Theory

这个项目，涉及工作的Kobi Kremnizer，致力于应用的方法，从非交换代数几何，研究的代表性理论。第一个目标是研究真实的量子群的表示理论。真实的（代数的）约化群理论的量子化是一个新的重要课题。该项目建议使用量子D-模块来研究量子Harish-Chandra模块，从而使分类成为一项可访问的任务。Harish-Chandra模的经典理论可能会从它的变形的研究中受益。第二个目的是给出Lusztig关于具有正特征的约化李代数的非限制表示的特征的几何证明。证明将使用量子微分算子的量子标志品种在一个根的单位。Lusztig的公式仅在有限的情况下通过非常计算的方法得到证明。该项目建议使用Springer分辨率的几何来证明这些几何的一般情况。该项目还建议通过将非限制量子群在单位根上的表示范畴与仿射李代数在临界水平上的表示范畴连接到施普林格分解来证明这两个范畴的等价性。第三个目标是给出双仿射Hecke代数的一个几何构造，并证明扭曲Harish-Chandra同态的一个量子形式。这应该有助于研究的代表性理论，这个代数在一个根的统一和连接它的代表性理论的三角切雷德尼克代数的积极特征。这个项目是在交汇点表示理论，非交换代数几何，几何的积极特征，无限维几何和共形场论。这将有助于澄清圈空间几何和弦论中观察到的非对易几何之间的密切关系，并将为研究开辟新的途径。它提供的方法在其他情况下也应该是有用的，使非交换几何成为表示论和代数几何中的重要工具。

This project, involving the work of Kobi Kremnizer, is devoted to the application of methods from noncommutative algebraic geometry, to the study of representation theory. The first aim is the study of the representation theory of the real quantum group. The quantization of the theory of real (algebraic) reductive groups is a new and important subject. The project proposes using quantum D-modules to study quantum Harish-Chandra modules thus making the classification an accessible task. The classical theory of Harish-Chandra modules might benefit from this study of its deformations. The second aim is to give a geometric proof of Lusztig`s conjectures concerning characters of nonrestricted representations of reductive Lie algebras in positive characteristic. The proof will use quantum differential operators on the quantum flag variety at a root of unity. Lusztig`s conjectures are only proven for the restricted case by a very computational method. The project proposes to give a proof for the general case of these conjectures using the geometry of the Springer resolution. The project also suggests showing the equivalence of the category of representations of the nonrestricted quantum group at a root of unity and the category of representations of the affine Lie algebra at the critical level by connecting both categories to the Springer resolution. The third aim is to give a geometric construction of the double affine Hecke algebra and to prove a quantum version of the twisted Harish-Chandra homomorphism. This should help study the representation theory of this algebra at a root of unity and connect it to the representation theory of the trigonometric Cherednik algebra in positive characteristic. This project lies in the meeting point of representation theory, noncommutative algebraic geometry, geometry in positive characteristic, infinite dimensional geometry and conformal field theory. It will hopefully help clarify the close relations between loop space geometry and noncommutative geometry that is observed in string theory and will open new pathways for research. The methods it offers to use should be useful in other cases as well, making noncommutative geometry an important tool in representation theory and in algebraic geometry.