2005 Fiscal Year Final Research Report Summary
Representation theoretic research of spherical functions on p-adic homogeneous spaces
| Project/Area Number |
15540022
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kyoto University |
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Principal Investigator |
KATO Shin-ichi Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90114438)
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| Co-Investigator(Kenkyū-buntansha) |
SAITO Hiroshi Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20025464)
MATSUKI Toshihiko Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20157283)
NISHIYAMA Kyo Kyoto University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70183085)
MURASE Atsushi Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (40157772)
TAKANO Keiji Akashi College of Technology, Associate Professor, 一般科目, 助教授 (40332043)
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| Project Period (FY) |
2003 – 2005
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| Keywords | p-adic fields / reductive groups / admissible representations / spherical functions / symmetric spaces / spherical homogenous spaces / distinguished representations / Weyl groups |
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| Research Abstract |
S.Kato, the Head investigator, studied the spherical functions on symmetric spaces over p-adic fields, together with K.Takano. By using orbit decomposition of symmetric spaces under maximal compact subgroups (Cartan decomposition, general formula of which is still in conjectural form), we obtained a Macdonald-type formula for spherical functions which expressed the value on tori by a sum over the Weyl groups of symmetric spaces (the little Weyl groups). The problem to have explicit formulas for spherical functions in general remained. However, for several examples including quadratic base change of symplectic groups, we had such formulas. As a byproduct of our study of symmetric spaces, we obtained a representation theoretical result about the representations of symmetric spaces (more precisely, about distinguished admissible representations for symmetric subgroups of reductive groups) : We succeeded in establishing a relative version (=symmetric space version) of Jacquet's subrepresen
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tation theorem which asserts that for any irreducible admissible representation V of a p-adic reductive group G, there exists at least one parabolic P and one irreducible cuspidal W such that V may be embedded into the induced representation of W from P under the assumption of the Cartan decomposions. Namely, by defining the notion of relative cuspidality, we showed that any irreducible representation of a symmetric space can be embedded in a induced representation associated with a pair consisting of a sigma-split parabolic subgroup and an irreducible distinguished representation of its Levi subgroup. This result can be viewed as a first step to generalize the harmonic analysis on p-adic groups to that on symmetric spaces. It is interesting to build representation theory of symmetric spaces on p-adic groups and/or other groups over various fields by using the notion of "relative cuspidality".
Other investigators also obtained several results on automorphic representations and automorphic forms (H.Saito and A.Murase ), and on structure theory and representation theory of real Lie groups (T.Matsuki and K.Nishiyama). Less
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