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

• 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
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440011/
• 关键词: prehomogeneous vector space; spherical function; automorphic form; Eisenstein series; 表現論; ゼータ関数; Siegel保型形式; 球等質空間; アイゼンシュタイシ級数; L関数

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

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

项目成果

T.Ibukiyama: "On "easy" zeta functions"Sugaku Expositions. 14. 191-204 (2001)

T.Ibukiyama：“论“简单”zeta 函数”朱乐阐述。

F.Sato: "Local densities of representations of quadratic forms over p-adic integers : the non-dyadic case"J.Number Theory. 83. 106-136 (2000)

F.Sato：“p 进整数上二次形式表示的局部密度：非二进情况”J.数论。

Y.Yamada: "Segre threefold and $N=3$ reflection equation"Phys.Lett.A. 298. 350-360 (2002)

Y.Yamada：“Segre 三重和 $N=3$ 反射方程”Phys.Lett.A。

Akihiko Gyoja: "Characteristic cycles of certain characger sheaves"Indagationes Mathematicae. 12. 329-335 (2001)

Akihiko Gyoja：“某些字符轮的特征周期”Indagationes Mathematicae。

Yumiko Hironaka: "A remark on Kitaoka's power series attached to local densities"Comment. Math. Univ. St. Pauli. 50. 141-146 (2001)

Yumiko Hironaka：“关于北冈幂级数与局部密度相关的评论”评论。