Representation theory of Lie algebras and quantum groups

李代数和量子群的表示论

• 批准号: 13440010
• 负责人: TANISAKI Toshiyuki
• 金额: $ 6.98万
• 依托单位: Osaka City University (2002)Hiroshima University (2001)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440010/
• 关键词: Infimte dimensiand Lie Igobras; quantum groups; algobrarc groups; Highestweight modulas; 最高ウェイト加群; カッツ・ムーディ・リー代数

1. Flag manifolds for quantum groups Tanisaki investigaled represecrtation theory of quantum groups using the flag manifolds of quanlum groups as non-commutative schemes. Especially, he has considered about the D-modyle theory and established a version of Beillnson-Bernstein correspondence proving a part of the conjecture by Lunts-Roscnberg2. Finite dimensional representations of quantum sffine algebras Kashiwara investigated the crystal bases and gave a condition for the tensor product of fundamental modules to be irreducible. Nakajima indroduced a crystal structure on the set of certain monomials and gave a new proof for the fact that the standard modulesare tensor product of fundamental modulcs. He also gave a proof of Lusztig's conjecture about the cell structure of quantum affine algebras3. Green functions assoclated to complex refiection groups Shoji constructed a new type of Macdonald polynomials as a two parameter version of Hall-Littlewood polynomial associated to complex refection groups4. Characters of finite Chevalley groups Shoji investigated about the delermination of some scalars which are the remaining part in Lusztig's program determining the characters, and has decided them for the special lincar groups5. Elliptic Lic algebras and assclated Artin groups and Hecke algebras Saito constructed a representation of exccptional elliptic Lie algebsa using the method of vertex operator algebras and computed its characters. He also get a result about a relation between the elilptic Hecke algebras and the double affino Hecke algebrasSphcrical homogeneous spaces over p-adic flekis Kato gave a dimension formula for spherical functions for some cases and constructed a general theory of spherical functions for symmetric spaces

1. Tanisaki用量子群的标志流形作为非交换格式研究了量子群的表示理论。特别是，他对d模理论进行了思考，建立了一个版本的Beillnson-Bernstein对应，证明了lunts - roscnberg猜想的一部分。Kashiwara研究了晶体基，给出了基本模张量积不可约的一个条件。Nakajima在单项式集合上引入了一种晶体结构，并给出了标准模是基本模的张量积的新证明。他还证明了Lusztig关于量子仿射代数的细胞结构的猜想。Shoji构造了一类新的Macdonald多项式，作为复反射群相关的Hall-Littlewood多项式的双参数版本4。有限Chevalley群的特征Shoji研究了一些标量的确定，这些标量是Lusztig确定特征的方案中剩下的部分，并为特殊线性群确定了它们。椭圆李代数及其相关的Artin群和Hecke代数Saito用顶点算子代数的方法构造了一个例外椭圆李代数的表示，并计算了其性质。他还得到了椭圆Hecke代数与双仿射Hecke代数之间的关系。加托给出了一些情况下球函数的维数公式，并构造了对称空间的球函数的一般理论

项目成果

Y.Morita: "The Radon transform on an exceptional flag manifoid"Hiroshima Math. J. 32(1). 7-15 (2002)

Y.Morita：“氡气变换在一个特殊的旗帜歧管上”广岛数学。

T.Shoji: "Green functions associated to complex reflection groups,ll."J. Algebra. 258(2). 563-598 (2002)

T.Shoji：“与复杂反射群相关的绿色函数，ll。”J。

T.Shoji: "Green functions associated to complex reflection groups, II"J. Algebra. 258(2). 563-598 (2002)

T.Shoji：“与复杂反射群相关的绿色函数，II”J。

C.Marastoni: "Radon transforms for quasi-equivariant D-modules on generalized flag manifolds"Differential geometry and its applications. 18(2). 147-176 (2003)

C.Marastoni：“广义旗流形上准等变 D 模的 Radon 变换”微分几何及其应用。

谷崎俊之: "リー代数と量子群"共立出版. 267 (2002)

Toshiyuki Tanizaki：“李代数和量子群”Kyoritsu Shuppan 267 (2002)。