Sheaves on Affine Flags, Springer Fibers, and Representation Theory

仿射旗上的滑轮、施普林格纤维和表示论

• 批准号: 0341076
• 负责人: Roman Bezrukavnikov
• 金额: $ 1.52万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-09-30 至 2004-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0341076&HistoricalAwards=false
• 关键词: Sheaves; Affine; Flags; Springer; Fibers

Previous results of the investigator and collaborators establisha relation between coherent sheaves on a Steinberg variety of a complexsimple algebraic group, and perverse sheaves on the affine flag variety ofthe Langlands dual group. Since the latter are known, by the work of Kazhdanand Lusztig, to be related to representations of quantum groups at a rootof unity, and hence also to representations of algebraic groups in primecharacteristic, our results have implications for that theories. They also indicate that there should be a relation between the geometryof perverse sheaves on affine flag varieties, and non-restricted representations of quantum groups or Lie algebras in prime characteristics.The principal goal of the present project is to develop this kind of relationship, thus providing a new tool for the theory of non-restricted representations. It is expected, in particular, to yield a proof and aconceptual explanation for recent numerical conjectures by Lusztig. The methods and ideology of the present work are partly based on the geometricapproach to the Langlands program, due to Drinfeld. Another goal of the projectis to investigate consequences of the above mentioned results to that theory. A key to new developments, and a source of inspiration in mathematics often lie in discovery of parallelism (equivalence) betweenseemingly unrelated objects or theories. Representation theory is a richsource of examples of that kind: though formally being a branch of algebra,the modern theory relies heavily on geometric disciplines, such as topology and algebraic geometry. The principle goal of the present projectis to develop geometric language for a branch of representation theorywhere it has been lacking so far, namely the so called theory of non-restrictedmodular representations. In the corresponding algebraic constructions divisibilityproperties of integral numbers by a particular prime number are relevant.The geometry apparently related to this deals with particular infinitedimensional objects, encountered also in mathematical physics(conformal field theory), and in number theory (Langlands duality theory).This construction is expected to provide new tools for the above mentionedbranch of representation theory; in particular, it is expected to yield aproof of certain conjectures by leading experts in the field.

研究者和合作者以前的结果建立了复单代数群的Steinberg簇上的相干层和Langlands对偶群的仿射旗簇上的反常层之间的关系。由于后者是已知的，由KazhdanandLusztig的工作，是有关的表示量子群在一个rootofunity，因此也表示代数群的primecharteristic，我们的结果有影响的理论。它们还表明仿射旗簇上的反常层的几何与量子群或李代数的素特征标的非限制表示之间应该存在着某种联系，本课题的主要目的就是发展这种联系，从而为非限制表示理论提供一种新的工具。预计，特别是，产生一个证明和概念解释最近的数值计算Lusztig。目前工作的方法和思想部分是基于几何方法的朗兰兹计划，由于德林费尔德。该项目的另一个目标是研究上述结果对该理论的影响。数学新发展的关键和灵感的源泉往往在于发现看似无关的对象或理论之间的平行（等价）。表示论是这类例子的丰富来源：虽然形式上是代数的一个分支，但现代理论在很大程度上依赖于几何学科，如拓扑学和代数几何。本项目的主要目标是为表示论的一个分支，即所谓的非限制模表示理论，发展几何语言。在相应的代数结构中，整数被某一特定素数整除的性质是相关的。与此相关的几何学显然是处理在数学物理中也遇到的特定的无限维对象（共形场论），在数论中（朗兰兹对偶理论）。这种构造有望为上述表示论分支提供新的工具;特别是，预计该领域的主要专家将对某些成果进行验证。