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Representation theoretic research of spherical functions on p-adic homogeneous spaces

p进齐次空间上球函数的表示理论研究

基本信息

批准号：
15540022
负责人：
KATO Shin-ichi
金额：
$ 1.92万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540022/
关键词：
p-adic fields reductive groups admissible representations spherical functions symmetric spaces spherical homogenous spaces distinguished representations Weyl groups 認容表現部分表現定理尖点表現放物型部分群 p-進群 Wey1群ルート系 Hecke環 Macdonald公式

项目摘要

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 … More 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

S.Kato，首席研究员，研究了p-adic域上对称空间上的球面函数，与K. Takano一起。利用对称空间在极大紧子群下的轨道分解（Cartan分解，其一般公式仍为几何形式），得到了用对称空间的Weyl群（小Weyl群）上的和表示环面上的值的球函数的Macdonald型公式.问题有明确的公式，球函数一般仍然存在。然而，对于包括辛群的二次基变换在内的几个例子，我们有这样的公式。作为对称空间研究的一个副产品，我们得到了关于对称空间表示（更确切地说，关于约化群的对称子群的可容许表示）的一个表示理论结果：我们成功地建立了Jacquet的子表示的一个相对版本（=对称空间版本） ...更多信息设P是一个p进约化群G的不可约容许表示V，则至少存在一个抛物线P和一个不可约尖点W，使得在Cartan分解的假设下，V可以嵌入到W的导出表示中。也就是说，通过定义相对尖点性的概念，我们证明了对称空间的任何不可约表示都可以嵌入到与由σ-分裂抛物子群和其Levi子群的不可约区别表示组成的对相关联的诱导表示中。这个结果可以看作是将p-adic群上的调和分析推广到对称空间上的第一步。利用“相对尖点性”的概念建立p进群和/或其他群上的对称空间的表示理论是很有趣的。其他研究者也得到了一些关于自守表示和自守形式的结果（H.Saito和A.Murase），以及关于真实的李群的结构理论和表示理论的结果（T.Matsuki和K.Nishiyama）。少