Invariant harmonic analysis and Selberg zeta functions (B06 #)

不变调和分析和 Selberg zeta 函数 (B06

• 批准号: 28916007
• 金额: --
• 依托单位: Fakultät für Mathematik
• 依托单位国家: 德国
• 项目类别: Collaborative Research Centres
• 财政年份: 2006
• 资助国家: 德国
• 起止时间: 2005-12-31 至 2008-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/28916007?language=en
• 关键词: Invariant; harmonic; analysis; Selberg; zeta

Noncommutative harmonic analysis may be characterised as the study of group representationsinduced in spaces of functions, or their generalisations, by an action on the underlyingspace. A central problem is the decomposition of such representations into irreducible components.This is a version of spectral decomposition, since the set of equivalence classes ofirreducible representations may be thought of as the spectrum of the group algebra. In thisproject, we deal with representations of topological groups in complex Hubert spaces andtheir decoposition as direct sums or integrals.More precisely, we consider reductive linear algebraic groups over local fields, mainlyover the real numbers, and over adele rings. In the latter case, the representations in thespaces of automorphic forms can be studied by means of the Arthur-Selberg trace formula,which relates the spectral data with geometric data of the underlying moduli space. Oneof our aims is to study the asymptotic distribution of those geometric data, which have anumber-theoretic interpretation, in new situations. For this kind of applications, we haveto extend the trace formula to test functions of noncompact support.The terms in the trace formula encoding geometric data also contain distributions onthe adelic group which can be split into components living on its local factors. Transferidentities between those local distributions on related groups are critical for progress in theLanglands programme. Langlands' recent vision on how to go beyond the present methodsof endoscopy suggests that one would need to determine the exact Fourier transforms ofthose distributions. We are going to expand our partial results obtained in the real case.This will also enable us to obtain new instances of the functional equations and determinantformulas for zeta functions of Selberg's type.

非对易调和分析可以被描述为研究函数空间中的群表示，或者它们的推广，通过作用于底层空间。一个中心问题是将这些表示分解成不可约分量，这是谱分解的一个版本，因为不可约表示的等价类的集合可以被认为是群代数的谱。在这个项目中，我们研究复Hubert空间中拓扑群的表示及其作为直和或积分的分解，更确切地说，我们研究局部域上的约化线性代数群，主要是真实的数上的约化线性代数群和Adele环上的约化线性代数群。在后一种情况下，自守形式在空间中的表示可以通过Arthur-Selberg迹公式来研究，该公式将谱数据与底层模空间的几何数据联系起来。本文的目的之一是研究几何数据在新的情况下的渐近分布，这些几何数据具有戊二烯理论解释。对于这类应用，我们必须将迹公式推广到非紧支撑的测试函数，迹公式中编码几何数据的项也包含在阿德利克群上的分布，这些分布可以分裂成分布在其局部因子上的分量。在相关群体的地方分布之间转移身份对于朗兰兹计划的进展至关重要。朗兰兹最近关于如何超越目前的内窥镜检查方法的设想表明，人们需要确定这些分布的精确傅立叶变换。我们将推广我们在真实的情况下得到的部分结果，这也将使我们能够得到Selberg型zeta函数的函数方程和行列式公式的新例子。