Geometric Representation theory of reductive group

约简群的几何表示论

• 批准号: 11440011
• 负责人: OCHIAI Hiroyuki
• 金额: $ 5.18万
• 依托单位: Tokyo Institute of Technology (2001)Kyushu University (1999-2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440011/
• 关键词: semisimple; highest weight; degree; theta correspondence; nilpotent orbit; associated variety; prehomogeneous; Kazhdan-Lusztig; 半単純リー郡; テータ対応; 最高ウエイト表現; Giambelliの公式; 重複度自由; ベキ零軌道; Selberg 積分; Kazhdan-Lusztig予想; ガンマ因子; 保型形式

The Bernstein degree, associated cycles and isotropy representation for a unitary highest weight representation of reductive Lie groups are important family of geometric invariants. For the symplectic groups, Ochiai with K. Nishiyama determines the Bernstein degree. For orthogonal and unitary groups, Nishiyama, Ochiai and K. Taniguchi determines the Bernstein degree and associated cycles. The general case including exceptional groups are done by Ochiai and Shohei Kato. This results are generalized to describe the degree of spherical orbits of a family of multiplicity-free representations by Ochiai and Kato. This is a new application of the Selberg-type integral.A representation is considered as a quantization of a function space. Oshima introduces the notion of homogenized enveloping algebra, in order to deal with universal enveloping algebras and coordinate rings simultaneously. Using the Capelli operator, which is a quantization of minors, Oshima describes the annihilator ideals of generalized Verma modules of scalar type. Related to this work, Ochiai describes the difference of the centers of the enveloping algebras and the rings of invariant differential operators. Based on the work above, Oshima with Shimeno characterize the image of the Poisson transform associated to non-minimal parabolic.Wakimoto discuss the modular invariance of characters of various affine algebras. Kaneko investigates the modular function *j from the geometry of singular moduli and supersingular elliptic curves over finite fields. Ochiai also discuss the quasi-modularity of the generating functions of an elliptic curve with a specified type of ramification indeces. Konno considers the unipotent representations of reductive groups over local fields, especially gives the description of CAP representations, which is a special subclass of unipotent representation, for low-rank groups.

约化李群的酉最高权表示的伯恩斯坦度、相伴圈和迷向表示是一类重要的几何不变量。对于辛群，Ochiai与K.西山确定了伯恩斯坦度。对于正交群和酉群，Nishiyama，Ochiai和K.谷口确定了伯恩斯坦度和相关的循环。一般情况下，包括特殊群体是由落合和Shohei加藤。这个结果被推广到描述由Ochiai和Kato提出的一族多重性自由表示的球面轨道的度。这是Selberg型积分的一个新应用，一个表示被认为是函数空间的一个量子化. Oshima为了同时处理泛包络代数和坐标环，引入了齐次包络代数的概念。使用Capelli算子，这是一个量子化的未成年人，大岛描述了零化子理想的广义Verma模的标量型。与此相关的工作，落合介绍了差异的中心包络代数和环不变微分算子。在此基础上，Oshima和Shimeno刻画了非极小抛物型的Poisson变换的象，Wakimoto讨论了各种仿射代数特征标的模不变性。Kaneko从有限域上奇异模和超奇异椭圆曲线的几何学出发研究了模函数 *j。Ochiai还讨论了具有特定类型分歧指数的椭圆曲线的生成函数的拟模性。Konno研究了局部域上约化群的幂幺表示，特别是对低秩群给出了幂幺表示的一个特殊子类CAP表示的刻画.

项目成果

N.Shimeno: "An analogue of Hardy's theorem on the Poincare disk"Bull.of Okayama Univ. of Science. 36A. 7-10 (2001)

N.Shimeno：“庞加莱圆盘上哈代定理的模拟”冈山大学的 Bull.。

H. Ochiai: "Non-commutative harmonic, oscillators and Fuchsian ordinary differential equations, Kyushu Univ. preprint series 1998-18"Comm. Math. Phys.. 217, no. 2. 357-373 (2001)

H. Ochiai：“非交换谐波、振子和 Fuchsian 常微分方程，九州大学预印本系列 1998-18”

Shohei Kato and H. Ochiai: "The degree of orbits of multiplicity-free actions (with Shohei Kato)"Asterisque. 273. 139-158 (2001)

Shohei Kato 和 H. Ochiai：“多重自由作用的轨道度（与 Shohei Kato）”星号。

S.-J.Cheng and V.G.Kac,M.Wakimoto: "Extensions of Neveu-Schwarz conformal modules"Jour.Math.Phys.. 41. 2271-2294 (2000)

S.-J.Cheng 和 V.G.Kac,M.Wakimoto：“Neveu-Schwarz 共形模的扩展”Jour.Math.Phys.. 41. 2271-2294 (2000)

M.Kaneko, N.Todaka: "Hypergeometric modular forms and supersingular elliptic curves"CRM Proceedings and Lecture Notes. 30. 79-83 (2002)

M.Kaneko、N.Todaka：“超几何模形式和超奇异椭圆曲线”CRM 论文集和讲义。