1990 Fiscal Year Final Research Report Summary
Automorphic Forms and Number Theory
| Project/Area Number |
01540042
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Kyoto University |
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Principal Investigator |
YAMAUCHI Masatoshi Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (30022651)
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| Co-Investigator(Kenkyū-buntansha) |
NISHIYAMA Kyo Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (70183085)
MATUKI Toshiko Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (20157283)
GYOUJYA Akihiko Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (50116026)
FUJIKI Akira Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (80027383)
SAITOU Hiroshi Kyoto University Yoshida College Assistant Professor, 教養部, 助教授 (20025464)
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| Project Period (FY) |
1989 – 1990
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| Keywords | Prehomogenous Space / L-function / Gauss sum / Representation of quaternions / Hecke operator / Kahler manifold / Representation of Lie groups / flagmanifold |
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| Research Abstract |
This research is mainly concerned with the field of modular forms and related topics. We believe we could obtain many fruitful results. The main results are as follows :
(1) H. Saito proves the functional equations of L functions with respect to the prehomogeneous space of the binary quadratic forms and gives their residues. Furthormore H. Saito defines the gaussian sum with respect to the matrix attached to the quadratic character of the quadratic forms over the finite field and showing this sum is a natural generalization of the classical gaussian sum, he gives some important applications to the prehomogeneous space of symmetric matrices and Siegel modular forms.
(2) H. Saito and M. Yamauchi consider the irreducible representation of the multiplicative group of the quaternion algebra over the local field and give an explicit form of the restrictions to Cartan subgroups and the character of these representations. Owing to this result, they give an application to the trace formula of Hec
… More
ke operators and give many important numerical examples of characteristic polynomials of Hecke operators for the modular groups of higher levels (cowork with H. Hijikata).
(3) A. Fujiki shows that the equivalence classes M of the representations from the fundamental group of the compact Kahler manifold to the complex reductive algebraic groups has the structure of the hyper Kahler space which admits a special C actions. Further in the associated Calabi family he showsthat the general fibre is isomorphic to M and special fibre is isomorphic to the moduli space of the Higgs bundle corresponding to the above representation.
(4) The open orbit of the prehomogenenous space acting the reductive algebraic group is an affine variety if and only if the space is regular. A. Gyoja gives a counter example to the conjecture that the regularity is a sufficient condition for the nonreductive case.
(5) T. Matuki describes the mapping which maps any representations of semisimple Lie groups to the principal series representations by the orbit structure over the flag manifolds. And for the classical simple Lie groups, he gives the symbolical expressions of the orbit structure of the flag manifolds. (cowork with T. Ohshima)
(6) K. Nishiyama defines the harmonic oscillator representation for the ortosymplectic Lie superalgebra and shows that it gives a unitary representation. This is an analogy of the Weil representations of the ordinary symplectic group. He also determines the wave front set of the Weil representations. Less
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