Project/Area Number  10640020 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
KATO Shinichi Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (90114438)

CoInvestigator(Kenkyūbuntansha) 
吉野 雄二 京都大学, 総合人間学部, 助教授 (00135302)
YAMAUCHI Masatoshi Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (30022651)
NISHIYAMA Kyo Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (70183085)
MATSUKI Toshihiko Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (20157283)
SAITO Hiroshi Kyoto Univ., Graduate School of Human and Environmental studies, Professor, 大学院・人間・環境学研究科, 教授 (20025464)
TAKASAKI Kanehisa Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (40171433)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1999 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1998 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  Representation theory / Algebraic group / Spherical function / Special function / Symmetric space / Spherical homogeneous space / Orbit / 表現論 / 代数群 / 球関数 / 特殊関数 / 対称空間 / 球等質空間 / 軌道 / 特殊関係 / リー群 / ヘッケ環 / ワイル群 / 軌道分解 
Research Abstract 
Special functions (spherical functions) on algebraic groups play an important role in number theory, especially in the study of automorphic forms. In most cases, these spherical functions are related to spherical homogeneous spaces, such as symmetric spaces. In this research project, Kato (head investigator) studied spherical functions on spherical homogeneous spaces of reductive groups over nonarchimedean local fields from a representation theoretic view point. The purpose of this research is twofold : (1) To understand special functions such as zonal spherical functions or Whittaker functions in a uniform manner from the view point as above. (2) To obtain properties of these functions, including the uniqueness and explicit formulas, for important cases which arise in number theory. As for (1), we studied an orbit decomposition of spherical homogeneous spaces first. Then applying this, we obtained a general formula for spherical functions (at least in the case of symmetric spaces) t
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ogether with a method to compute the coefficients in this formula explicitly. As for (2), we got the uniqueness and an explicit formula for e.g. a symmetric space corresponding to quadratic base change by using the above mentioned method. This research is still under way. Other investigators obtained several results related to representation theory and spherical homogeneous spaces as follows. Saito studied zeta functions of prehomogeneous vector spaces, which is closely related to (spherical functions of) spherical homogeneous spaces, and showed the convergence and explicit formulas (in terms of local orbital zeta functions) in general. Matsuki investigated Weyl groups and Jordan decompositions arising from symmetric spaces. Nishiyama studied multiplicity free actions, which is a characteristic property of spherical homogeneous spaces, and the relation between theta correspondences and nilpotent orbits. Other investigators, Takasaki, Yamauchi et al. carried out researches on mathematical physics, automorphic forms and so on. Less
