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)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 I D 24 I D 2，E I D 26 I D 2，E I D 27 I D 2，E I D 28 I D 2)，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 ii 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：Less

(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