Geometric methods in representation theory

项目负责人：George Lusztig项目编号：0243345机构：麻省理工学院摘要表示理论中的几何方法李型群的表示理论是数学的核心部分。它关注的是通过用矩阵形式表示对称系统来理解它们。表示论中最困难的领域之一是p进域上的群，它与数论有很强的联系。研究这些群的主要工具之一是使用仿射赫克代数。G. Lusztig建议继续研究具有不等参数的仿射Hecke代数，特别是建立其正则基的几何解释。并给出了相应的渐近Hecke代数的存在性。这将为p进域上群的表示理论提供新的信息。在有限域被有限环代替的情况下，进一步研究了delign - lusztig理论的类似性。这同样适用于p进域上群的表示理论。从反常束的角度进一步研究了量子化包络代数的正则基。并提出继续研究约化群上的特征束。这些课题的进展有望应用于数学和理论物理的各个部分。群表示理论试图通过更易于计算的矩阵来研究对称的思想。傅立叶级数是表征理论最古老的应用之一，广泛应用于工程和应用科学。最近，表征理论的思想被用于化学（晶体研究）和物理学（基本粒子理论）。G. Lusztig的研究涉及代数拓扑（通过代数研究形状）和代数几何（方程的几何研究）方法的应用，以获得其他方法无法获得的关于群表示的新结果。

Principal Investigator: George Lusztig Proposal Number: 0243345Institution: Massachusetts Institute of TechnologyABSTRACTGeometric methods in representation theoryRepresentation theory of groups of Lie type is a central part of mathematics. It is concerned with understanding systems with symmetry by representing them in matrix form. One of the most difficult areas of representation theory is that of groups over p-adic fields, which has strong connections with number theory. One of the main tools in the study of these groups is the use of affine Hecke algebras. G. Lusztig proposes to continue the study of affine Hecke algebras with unequal parameters and in particular to establish a geometric interpretation for their canonical basis. Also it is proposed to establish the existence of the corresponding asymptotic Hecke algebras. This should give new information on the representation theory of groups over p-adic fields. It is also proposed to further investigate the analogue of the Deligne-Lusztig theory in the case where finite fields are replaced by certain finite rings. This again should have applications to the representation theory of groups over p-adic fields. It is proposed to further investigate the canonical bases of quantized enveloping algebras from the point of view of perverse sheaves. It is also proposed to continue the study of character sheaves on reductive groups. Progress in these topics is expected to have applications to various parts of mathematics and theoretical physics.The theory of group representations attempts to study the idea of symmetry by means of matrices, which are more amenable to computation. One of the oldest applications of representation theory is the theory of Fourier series, widely used in engineering and applied science. More recently, ideas from representation theory have been used in chemistry (study of crystals) and physics (theory of elementary particles). G. Lusztig's research is concerned with applications of methods of algebraic topology (study of shapes by means of algebra) and algebraic geometry (geometric study of equations) to obtain new results on group representations which could not be obtained by other methods.