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The flag variety of elliptic Lie algebra and elliptic primitive automorphic forms

椭圆李代数和椭圆本原自守形式的标志簇

基本信息

批准号：
13440021
负责人：
SAITO Kyoji
金额：
$ 5.44万
依托单位：
Kyoto University
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 2003
项目状态：
已结题

项目摘要

The main purpose of the present research program is to describe the period map associated to an integral of a primitive form in terms of Lie theory.I. Subject related to elliptic Lie algebra and elliptic Lie groups.1. The construction of the highest weight representation theory for elliptic algebras: even though elliptic algebras are not Kac-Moody algebras, by replacing the Cartan subalgebra by a Heisenberg algebra, we can construct the infinite dimensional highest weight representation. Since the radical is infinite dimensional and create a non-commutative algebra, we develop a new, concept, called the block algebra.2. Bruhat-Tits decomposition of elliptic Lie groups: owing to the above 1., we know that there are ample representations, and we can introduce the elliptic group by the inverse limit of integrable representations. The normalizer of its maximal torus is an extension by the elliptic Weyl group by the torus. Even though the elliptic Weyl group is not a Coxeter group, we show … More for the elliptic Lie group admits the Bruhat-Tits decomposition.3.The Fourier coefficients of the eta-products attached to the characteristic polynomial of the elliptic Coxeter element are non-negative if and only if the corresponding Weyl group invariant ring admit the flat structure, that is: the cases D^<(1,1)>_4, E^<(1,1)>_6, E^<(1,1)>_7 and E^<(1,1)>_8.II. Subject related to classical finite root systems.1. The flat structure on finite reflection groups (reconstruction of the theory, Hodge filtration, Fourier transformation and the uniformization equation (Gauss-Manin connection), the relation with the Frobenius manifold structure, special solutions by means of periodd integrals of the primitive form for odd dimensional fibration, the the monodromy group in the symplectic group, a conjecture on the period domain, a conjecture on Eisenstein series and on discriminant forms, a conjecture on the power root of the discriminant form).2. Construction of new theory: odd Root systems, Period map of type D_4.3. The relationship between the topology of the complexified orbit space and the semi-algebraic geometry of real orbit space of a finite reflection group (presentation of braid group, K(π,1)-space, twisted real structure, connected components of the complement of twisted real discriminants loci, a relation with the regular eigenvector of the Coxeter element, the characteristic variety, bifurcation set, Linearization theorem). Less

本文的主要目的是用李理论描述与一个原始形式积分相联系的周期映射。与椭圆李代数和椭圆李群有关的学科。椭圆代数的最高权表示理论的构造：即使椭圆代数不是Kac-Moody代数，通过用Heisenberg代数代替Cartan子代数，我们可以构造无限维的最高权表示。由于根式是无限维的，并创建一个非交换代数，我们开发了一个新的概念，称为块代数。椭圆李群的Bruhat-Tits分解：由于上述1.，我们知道有足够的表示，我们可以通过可积表示的逆极限来引入椭圆群。其极大环面的正规化子是椭圆Weyl群对环面的扩张。即使椭圆Weyl群不是Coxeter群，我们证明了 ...更多信息 3.椭圆Coxeter元的特征多项式的η-积的Fourier系数非负当且仅当相应的Weyl群不变环具有平坦结构，即：D^<（1，1）>_4、E^<（1，1）>_6、E^<（1，1）>_7和E^<（1，1）>_8的情形。与经典有限根系有关的学科。有限反射群上的平坦结构（理论的重建，霍奇过滤，傅立叶变换和一致化方程（高斯-马宁连接），与Frobenius流形结构的关系，奇维纤维化的原始形式的周期积分的特殊解，辛群中的单值群，关于周期域的一个猜想，关于Eisenstein级数和判别式的一个猜想，关于判别式的幂根的一个猜想）.新理论的构建：奇根系，D_4.3型周期图。复化轨道空间的拓扑与有限反射群的真实的轨道空间的半代数几何之间的关系（辫子群的表示，K（π，1）-空间，扭曲的真实的结构，扭曲的真实的判别轨迹的补的连通分支，与Coxeter元的正则本征向量的关系，特征簇，分歧集，线性化定理）。少