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

本研究的主要目的是用李理论来描述与本原形式的积分有关的周期映射。1.与椭圆李代数和椭圆李群有关的课题。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，我们也可以构造无限维的最高权表示。由于根是无限维的，并且产生了一个非交换代数，我们提出了一个新的概念，称为块代数。椭圆李群的Bruhat-Tits分解：由于上面的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。尽管椭圆Weyl群不是Coxeter群，但我们证明了…3.椭圆Coxeter元特征多项式上的ETA-积的傅立叶系数为非负当且仅当相应的Weyl群不变环具有平坦结构，即：D^&lt；(1，1)&gt；_4，E^&lt；(1，1)&gt；_6，E^&lt；(1，1)&gt；_7和E^&lt；(1，1)&gt；有限反射群上的平面结构(理论的重建、Hodge滤波、傅立叶变换和均匀化方程(Gauss-Manin连接)、与Frobenius流形结构的关系、奇维纤维的本原形式的周奇异积分的特解、辛群中的单调群、周期域上的猜想、关于Eisenstein级数和判别形式的猜想、关于判别形式的方根的猜想)。新理论的构建：奇根系统，D_4.3型周期映射。有限反射群的复化轨道空间的拓扑与实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，扭实结构，扭实判别轨迹补的连通分支，与Coxeter元的正则特征向量的关系，特征簇，分支集，线性化定理)。较少

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