河北师范大学学报—

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刊名：河北师范大学学报（自然科学版）Journal of Hebei Normal University (Natural Science)
主办：河北师范大学
ISSN： 1000-5854
CN： 13-1061/N
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酉群上的Siegel Eisenstein级数的常数项原则

宋旭¹

王艺儒²

熊玮³

张哲⁴

(1.河北师范大学数学科学学院，河北石家庄 050024； 2.南京特殊教育师范学院数学科学学院，江苏南京 210038; 3.湖南大学数学学院,湖南长沙 410082; 4.西安电子科技大学数学与统计学院,陕西西安 710126)
DOI： 10.13763/j.cnki.jhebnu.nse.202601011

The principle of constant terms for Siegel Eisenstein series on unitary groups

Song Xu¹,Wang Yiru²,Xiong Wei³,Zhang Zhe⁴

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摘要/Abstract

摘要：

Siegel Eisenstein级数是一种特殊的Eisenstein级数,最早由Siegel在其二次型解析理论的研究引进,其一般形式最早出现在Weil证明的Siegel-Weil公式中,即该Siegel Eisenstein级数在某点处的值等于一个Theta函数的积分.在Siegel-Weil公式的后续发展中,需要研究Siegel Eisenstein级数的解析性质.Eisenstein级数沿着一个抛物子群的常数项是个更简单的研究对象，且包含许多解析性质.通过采用Kudla和Rallis研究数域上辛群的Siegel Eisenstein级数解析性质的方法,本文证明了整体域上拟分裂酉群U(n,n)Siegel Eisenstein级数的常数项原则,即该级数的极点与其沿着一个抛物子群的常数项的极点相同.

Abstract：

Siegel Eisenstein series is a special Eisenstein series first introduced by Siegel in his work on the analytic theory of quadratic forms,and its general form first appeared in the Siegel-Weil formula proved by Weil,i.e.the value of the Siegel Eisenstein series at some point is equal to the integral of a theta function.In later developments of the Siegel-Weil formula,it is necessary to study the analytic properties of the Siegel Eisenstein series.The constant term of the Eisenstein series along a parabolic subgroup is a simpler object to study and embodies many of its analytic properties.In this paper,by following the method of Kudla and Rallis in studying the analytic properties of the Siegel Eisenstein series on symplectic groups over number fields,the authors prove the principle of constant terms for Siegel Eisenstein series on quasi-split unitary groups U(n,n) over global fields,i.e.the poles of the Siegel Eisenstein series are the same as the poles of its constant term along a parabolic subgroup.

关键词

关键词： 整体域；酉群； Siegel Eisenstein级数；抛物子群；常数项

Key words： global fields ; unitary groups ; Siegel Eisenstein series ; parabolic subgroups ; constant terms

参考文献 8

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