A study on relations between the theory of prehomogeneous vector spaces, the theory of group representations and the theory of automorphic forms

预齐次向量空间理论、群表示理论和自守形式理论之间的关系研究

基本信息

批准号：
12440011
负责人：
SATO Fumihiro
金额：
$ 6.91万
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：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

项目摘要

In this research project, we investigated zeta functions of prehomogeneous vector spaces from the view points(1)relations between functional equations and representations of general linear groups,(2)relations to automorphic L-functions,(3)generalization of the theory to non-regular prehomogeneous vector spaces.(1)We showed that the functional equations of zeta functions are closely related to intertwining operators between degenerate principal series representations of general linear groups, and, using the relation, we obtained an integral expression of Eulerian type of the gamma matrices of functional equations. This enables us to identify the variable change in functional equations as an action of an element in the Weyl group of a general linear group, and to decompose functional equations into a product of more elementary functional equations. There exists a similar results for p-adic local zeta functions. As an application of p-adic theory, we investigated the Fourier coefficients … More of Elsenstein series of Sp(n) and GL(n) and the theory of spherical transforms on certain spherical homogeneous spaces.(2)We identified the Koecher-Maass series of real analytic Siegel Eisenstein series with a zeta function associated with a certain prehomogeneous vector space on which the Siegel parabolic subgroup of SO(n, n) acts. It is quite probable that this result can be extended to other classical groups. A considerable progress has been made in explicit calculation of zeta functions. We obtained an explicit expression of zeta functions in terms of the Riemann zeta function and the Mellin transforms of the Cohen Eisenstein series for more than 70 percent of irreducible regular reduced prehomogeneous vector spaces.(3)For non-regular prehomogeneous vector spaces, we developed a general theory of integral representations and the functional equation of the zeta integrals, which is a formal generalization of the theory for regular prehomogeneous vector spaces. We also gave the first example of explicit functional equations for non-regular spaces. Less

在该研究项目中，我们从观点（1）功能方程与一般线性组的功能方程和表示之间的关系之间的Zeta Zeta功能，（2）与自动形态L功能的关系，（3）理论将理论的概括性化为非规范的前载体空间之间的ZERINES函数，我们表明了Zeta的相互作用。一般线性基团的主序列表示，并使用这种关系，我们获得了功能方程的伽马级材料的欧拉类型的积分表达。这使我们能够识别功能方程的变化是通用线性组的Weyl组中元素的动作，并将功能方程分解为更基本功能方程的产物。 P-ADIC局部Zeta函数也存在类似的结果。 As an application of p-adic theory, we investigated the Fourier Coefficients … More of Elsenstein series of Sp(n) and GL(n) and the theory of spherical transforms on certain spherical homogeneous spaces.(2)We identified the Koecher-Maass series of real analytic Siegel Eisenstein series with a zeta function associated with a certain prehomogeneous vector space on which the Siegel parabolic SO（N，N）行为的亚组。可以将此结果扩展到其他古典群体是很有问题的。在明确计算Zeta功能方面取得了很大进展。 We obtained an explicit expression of zeta functions in terms of the Riemann zeta function and the Mellin transforms of the Cohen Eisenstein series for more than 70 percent of irreducible regular reduced prehomogeneous vector spaces.(3)For non-regular prehomogeneous vector spaces, we developed a general theory of integral representations and the functional equation of the zeta integrals, which is a Formal generalization定期均匀前载体空间的理论。我们还为非规范空间提供了明确的功能方程的第一个示例。较少的