Collaborative Research: The Geometric Dual Space of a Unipotent Group in Characteristic p

合作研究：特征p中单能群的几何对偶空间

• 批准号: 1001769
• 负责人: Dmitriy Boyarchenko
• 金额: $ 14.25万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001769&HistoricalAwards=false
• 关键词: Collaborative; Research; Geometric; Dual; Space

The principal investigators continue their research in geometric character theory and geometric representation theory of unipotent groups over a field of positive characteristic. They define the "dual space" of such a group as an algebro-geometric object parameterizing L-packets of character sheaves on the group. The principal investigators then develop a conjectural geometric theory of the spectral decomposition of the equivariant derived category with respect to the dual space, which is similar to the classical decomposition of representations as direct integrals of irreducible ones. They also formulate several conjectures on this spectral decomposition in the setting where the orbit method is applicable and discuss certain related quantization problems in the theory of tensor categories. The difference between these quantization problems and the standard ones is that the role of the universal envelopingalgebras is played by group algebras.The proposed project combines several active directions of current research in mathematics and mathematical physics -- geometric representation theory, algebraic geometry, the theory of tensor categories, quantum groups, and conformal field theory. The notion of geometric spectral decomposition will deepen our understanding of the general patterns of geometric representation theory. The research will also lead to new applications of the quantization philosophy in the theory of tensor categories.

主要研究人员继续他们的研究在几何特征理论和几何表示理论的幂幺群在一个领域的积极特征。他们定义这样一个群的“对偶空间”为代数几何对象，参数化该群上特征层的L-包。主要研究人员然后开发了一个代数几何理论的谱分解的等变导出范畴的对偶空间，这是类似于经典的分解表示为直接积分的不可约的。他们还制定了几个approachtures对这一频谱分解的设置中的轨道方法是适用的，并讨论了某些相关的量子化问题的理论张量类别。这些量子化问题和标准问题之间的区别在于，通用的代数的作用是由群algebrais.The拟议的项目结合了当前数学和数学物理研究的几个活跃方向--几何表示论，代数几何，张量范畴理论，量子群和共形场论。几何谱分解的概念将加深我们对几何表示理论一般模式的理解。这一研究也将为量子化哲学在张量范畴理论中的应用提供新的思路。