Zeta functions on weakly spherical homogeneous spaces

弱球形均匀空间上的 Zeta 函数

基本信息

批准号：
09440024
负责人：
SATOU Fumihiro
金额：
$ 3.01万
依托单位：
Rikkyo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440024/
关键词：
zeta function functional equation weakly spherical homogeneous space local density automorphic form Koecher-Maass zeta function Eisenstein series prehomogeneous vector spaces 二次形式 p-進球関数 Koecher-Maass級数 Siegel保型形式 Jacobi Eisenstein

项目摘要

1.Sato, the head investigator, studied the weakly spherical homogeneous spaces (WSHS for short) obtained from the prehomogeneous vector spaces (PV for short) (SpinィイD210ィエD2 × GLィイD23ィエD2, half-spin 【cross product】 ΛィイD21ィエD2), (SLィイD25ィエD2 × GLィイD23ィエD2, ΛィイD22ィエD2 【cross product】 ΛィイD21ィエD2) and proved the functional equations satisfied by Eisenstein series attached to the spaces, which is a joint work with T. Kimura of Tsukuba Univ. and his students. As an application we can calculate explicitly the Fourier transforms of the complex powers of relative invariants of the PV's above. This shows that the theory of WSHS is very fruitful, since the structure of these two PV's is very complicated and even the method of micro local calculus can not be applied to them. We also made several attempts to generalize the theory to WSHS of reductive groups other than the general linear group. The result is still unsatisfactory ; however several suggestive partial results have been obtained.2.Hiron … More aka succeeded in calculating the explicit formula for spherical functions of spherical homogeneous spaces over p-adic fields in a rather general setting. Using the formula, she constructed the theory of spherical functions of the space of hermitian forms over unramified quadratic extensions of the base p-adic field and gave an explicit formula for local densities of hermitian forms. Moreover, jointly with Sato, she calculated local densities of quadratic forms over nondyadic p-adic fields explicitly in the most general setting ; the results can be extended to the space of hermitian forms over ramified quadratic extensions.3.Arakawa studied mainly Koecher-Maass zeta functions attached to Siegel modular forms and obtained some results including the following : (a) a generalization of Koecher-Maass zeta functions to Jacobi forms, (b) an explicit expression of the Koecher-Maass zeta function attached to the Nagaoka Eisenstein series of weight 1 of degree 2, which is obtained from p-adic Siegel Eisenstein series. Less

1.Sato, the head investigator, studiod the weakly spherical homogeneous spaces (WSHS for short) obtained from the prehomogeneous vector spaces (PV for short) (Spini D210E D2 × GLII D23E D2, half-spin [cross product] Λi D21E D2), (SLI D25E D2 × GLII D23E D2, Λi D22E D2 [跨产品]λiD21E D2），并证明了与空间相连的Eisenstein系列满足的功能方程，这是与Tsukuba Univ的T. kimura的联合作品。和他的学生。作为应用程序，我们可以明确计算上述PV相对不变的复杂功率的傅立叶变换。这表明WSH的理论非常富有成果，因为这两种PV的结构非常复杂，甚至微观局部计算的方法也无法应用于它们。我们还进行了几次尝试，将理论推广到除通用线性群以外的还原群体的WSH中。结果仍然不令人满意。然而，已经获得了几个暗示性的部分结果。2.Hiron…又名更多的又名成功的公式在相当通用的环境中在P-ADIC领域上的球形均匀空间的球形函数进行了明确的公式。她使用该公式，在基本padic领域的未受到的二次扩展上构建了赫尔米利亚形式空间的球形函数理论，并给出了针对Hermitian形式的局部密度的明确公式。此外，她与佐藤（Sato）共同计算了在最一般的环境中明确的非横态P-Adic田地上二次形式的局部密度。 the results can be extended to the space of hermitian forms over ramified quadratic extensions.3.Arakawa studioded mainly Koecher-Maass zeta functions attached to Siegel modular forms and obtained some results including the following: (a) a generalization of Koecher-Maass zeta functions to Jacobi forms, (b) an explicit expression of the Koecher-Maass zeta function attached to the Nagaoka Eisenstein系列的重量为2，该系列的重量1，从P-Adic Siegel Eisenstein系列获得。较少的