| Co-Investigator(Kenkyū-buntansha) |
TAKAHASHI Atsushi Research Institute for Mathematical Sciences, Research Associates, 数理解析研究所, 助手 (50314290)
MORI Shigehumi Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (00093328)
KASHIWARA Masaki Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (60027381)
TERAO Hiroaki Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90119058)
OKA Matsuo Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40011697)
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| Research Abstract |
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
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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
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