Project/Area Number  09440031 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
Geometry

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
SAITO Kyoji RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (20012445)

CoInvestigator(Kenkyūbuntansha) 
古田 幹雄 京都大学, 数理解析研究所, 助教授 (50181459)
NAKAYAMA Noboru RIMS, KYOTO UNIVERSITY ASSOCIATE PROFESSOR, 数理解析研究所, 助教授 (10189079)
KASHIWARA Masaki RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (60027381)
MIYAOKA Yoichi RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (50101077)
MORI Shigefumi RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (00093328)
MIWA Tetsuji RIMS, KYOTO UNIVERSITY PROFESSOR, 数理解析研究所, 教授 (10027386)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥9,500,000 (Direct Cost : ¥9,500,000)
Fiscal Year 1999 : ¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1998 : ¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 1997 : ¥3,400,000 (Direct Cost : ¥3,400,000)

Keywords  primitive forms / period maps / elliptic singularities / elliptic root systems / elliptic Lie algebra / elliptic Lfunctions / regular systems of weights / flat structure / 原始形式 / weight系 / 楕円ルート系 / 楕円特異点 / 楕円ワイル群 / 楕円リー環 / 楕円L函数 / 平坦構造 / 平坦座標 / 楕円アルティン群 / 原始保型形式 / エータ積 / Weight系 
Research Abstract 
(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 Lfunctions [4]. The elliptic Lfunction (the Mellin transform of the etaproduct associated to the characteristic polynomial of the elliptic Coxeter element) is either an Artin Lfunction or a difference of Artin Lfunctions. As a consequence, if the Galois groups is abelian
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(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 etaproduct is noncuspidal and all Fourier coefficients are nonnegative. 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 selfduality 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 Lfunction 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
