Sheaves on Affine Flags, Springer Fibers, and Representation Theory

研究者和合作者先前的结果建立了复简单代数群的Steinberg变体上的相干束和Langlands对偶群的仿射旗变体上的反常束之间的关系。由于后者是已知的，通过Kazhdanand Lusztig的工作，与统一根上的量子群的表示有关，因此也与初始特征中的代数群的表示有关，我们的结果对这些理论具有启示意义。他们还指出，在仿射标志簇上的反常束的几何与量子群或李代数在素数特征上的非限制表示之间应该存在一种关系。本项目的主要目标是发展这种关系，从而为非限制表征理论提供一个新的工具。特别是，它被期望为Lusztig最近的数值猜想提供一个证明和概念上的解释。由于德林菲尔德的原因，目前工作的方法和思想部分是基于朗兰兹纲领的几何方法。该项目的另一个目标是调查上述结果对该理论的影响。数学新发展的关键和灵感来源往往在于发现看似不相关的物体或理论之间的平行性（等效性）。表征理论是这类例子的丰富来源：虽然形式上是代数的一个分支，但现代理论严重依赖于几何学科，如拓扑和代数几何。目前的主要目标是为表征理论的一个分支开发几何语言，这是迄今为止它所缺乏的，即所谓的非限制模表征理论。在相应的代数构造中，整数可被特定素数整除的性质是相关的。显然与此相关的几何学处理的是特定的无限大维物体，在数学物理（共形场论）和数论（朗兰兹对偶理论）中也遇到过。这种建构有望为表征理论的上述分支提供新的工具；特别是，它有望证明该领域顶尖专家的某些猜想。

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.