Geometric methods in representation theory

主要研究人员：George Lusztig提案编号：0243345机构：麻省理工学院摘要表示论中的几何方法李型群的表示论是数学的核心部分。它涉及通过用矩阵形式表示具有对称性的系统来理解它们。表示论中最困难的领域之一是p-ady域上的群问题，它与数论有着很强的联系。研究这些群的主要工具之一是使用仿射Hecke代数。G.Lusztig建议继续研究具有不等参数的仿射Hecke代数，特别是建立其标准基的几何解释。并证明了相应的渐近Hecke代数的存在性。这将给p-ady域上的群的表示理论提供新的信息。在有限域被某些有限环代替的情况下，进一步研究了Deligne-Lusztig理论的类比。这同样应该应用于p-ady域上的群的表示理论。从斜层的角度进一步研究了量子化包络代数的典范基。并建议继续研究约化群上的特征标层。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性的思想。表示论最古老的应用之一是傅里叶级数理论，广泛应用于工程和应用科学。最近，表象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的研究涉及应用代数拓扑学(通过代数研究形状)和代数几何(研究方程的几何)的方法来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.