Sheaves on Affine Flags, Springer Fibers, and Representation Theory

研究人员和合作者以前的结果建立了复单代数群的Steinberg簇上的凝聚层和朗兰兹对偶群的仿射FLAG簇上的逆层之间的关系。由于Kazhdan和Lusztig的工作已知后者与单位根上的量子群的表示有关，因此也与原始特征中的代数群的表示有关，我们的结果对这些理论有意义。它们还指出，仿射旗簇上的倒叶的几何与量子群或素数特征的李代数的非限制表示之间应该存在某种关系，本课题的主要目的是发展这种关系，从而为非限制表示理论提供一种新的工具。特别是，它有望对Lusztig最近提出的数值猜想给出证明和概念上的解释。由于Drinfeld，本工作的方法和思想部分是基于对朗兰兹程序的几何方法。该项目的另一个目标是调查上述结果对该理论的影响。新发展的关键和数学灵感的源泉往往在于发现看似不相关的对象或理论之间的平行(等价性)。表示论是这类例子的丰富来源：尽管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理论的一个分支发展几何语言，这是迄今为止所缺乏的，即所谓的非限制模表示理论。在相应的代数结构中，整数被特定素数整除的性质是相关的。显然，与此相关的几何学涉及在数学物理(保形场论)和数论(朗兰兹对偶理论)中也遇到的特殊的无限维物体。这种结构有望为上述表示论的分支提供新的工具；特别是，它有望给该领域的主要专家对某些猜想的证明。

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.