Primitive Forms and Period Maps

（1） Elliptic case。i） Presentation of the elliptic Weyl group [1] ： The elliptic Weyl group （generated by reflections for elliptic root） and their central extensions is presented by generators and relations in terms of elliptic （dynkin）dragram。ii） Presentation of the elliptic Lie algebra [2]， [3]。elliptic algebra is introduced as the isomorphism class of the three different constructions ： a） the algebra generated by the“vacumes”of Bosonic representations corresponding to elliptic roots， b） the algebra generated by Chevalley basis for the elliptic diagram and defined by relations generalizing the Serre relations， and c） an amalgamation algebra of an affine algebra and a Heisenberg algebra。iii） Elliptic L-functions [4]。有弹性L-函数（与有弹性Coxeter元素相关的以太产品的特征多态性的梅林变换）要么是Artin L-函数，要么是Artin L-函数的差异。作为一个后果，如果Galois集团是阿拉伯人 ... More （the types D_D24_D2， E_D26_D2， E_D27_D2）， the elliptic eta-product is non-cuspidal and all Fourier coefficients are non-negative。iv） Introduction of elliptic Artin Group [5] ： H. Yamada表示，作为补充的椭圆形鉴别的基本群体，可以在椭圆形图表的术语中进行改革。通过对关系进行简单的修改，一个可以定义一个有积极关系的椭圆形艺术组。v） Classification of unstable Principal bundles over an elliptic curve in is given in connection with roop group （Helmke and Slodowy）。（2） Regular systems of weights and their duality [6]， [7]， [8]。The duality described in terms of regular systems of weManagement in the present work Management ins in purely arithmetic way the strange duality of Arnold and the self-duality of AイイD2， DイD2， EイD2。这与弦乐理论中的某些二元性是相等的（A.）Takahashi）。目前工作中L-函数的概念是部分解决在上面工作中（1） iii）。（3）经典有限根系统案例。i）多氢离子与高氧计反射超平面的双重[9] ：这是一个精确的高氧计组的鉴别，它在轨道空间上连接了两个结构：拓扑（模糊组， K（π， 1）-属性等）和平坦的结构。ii） Geometry of finite reflection groups [10]， [11] ： The flat structure for finite reflection group was not perly written in the literature。我写了一个关于这个主题的讲座，包括许多新的计算，为一个未来的想法，当它需要一个研究的时间图为奇数尺寸的纳米纤维。Less（低）

(1) Elliptic case. i) Presentation of the elliptic Weyl group [1] : The elliptic Weyl group (generated by reflections for elliptic root) and their central extensions is presented by generators and relations in terms of elliptic (dynkin) diragram. ii) Presentation of the elliptic Lie algebra [2], [3]. The elliptic algebra is introduced as the isomorphism class of the three different constructions : a) the algebra generated by the "vacumes" of Bosonic representations corresponding to elliptic roots, b) the algebra generated by Chevalley basis for the elliptic diagram and defined by relations generalizing the Serre relations, and c) an amalgamation algebra of an affine algebra and a Heisenberg algebra. iii) Elliptic L-functions [4]. The elliptic L-function (the Mellin transform of the eta-product associated to the characteristic polynomial of the elliptic Coxeter element) is either an Artin L-function or a difference of Artin L-functions. As a consequence, if the Galois groups is abelian … More (the types DィイD24ィエD2, EィイD26ィエD2, EィイD27ィエD2, EィイD28ィエD2), the elliptic eta-product is non-cuspidal and all Fourier coefficients are non-negative. iv) Introduction of elliptic Artin group [5] : H. Yamada has shown that the fundamental group of the complement of the elliptic discriminates can be reformulated in terms of the elliptic diagram. By a slite modification of the relations, one can define an elliptic Artin group with positive relations. v) Classification of unstable principal bundles over an elliptic curve in is given in connection with roop group (Helmke and Slodowy).(2) Regular systems of weights and their duality [6], [7], [8]. The duality described in terms of regular systems of weights in the present work explains in purely arithmetic way the strange duality of Arnold and the self-duality of AィイD2ιィエD2, DィイD2ιィエD2, EィイD2ιィエD2. It is equivalent to certain duality in string theory (A. Takahashi). The conjecture on the L-function posed in the present work is partialy solved in the above work (1) iii).(3) Classical finite root system case. i) The polyhedron dual to the Coxeter reflection hyperplanes [9] : This is a real geometric the discriminate for a finite Coxeter group, which connects two structures on the orbit space : the topology (braid groups, K(π, 1)-property, etc.) and the flat structure. ii) Geometry of finite reflection groups [10], [11] : The flat structure for finite reflection group was not properly written in the literature. I made a lecture note on the subject including many new calculations, for a future poupose when it will be necessary when one studies the period maps for odd dimensional Milnor fibers. Less