Invariant harmonic analysis and Selberg zeta functions (B06 #)

非对易调和分析可以被描述为研究函数空间中的群表示，或者研究由下面空间上的作用所诱导的函数空间中的群表示。一个中心问题是将这些表示分解成不可约分量。这是谱分解的一种形式，因为可约表示的等价类集合可以被认为是群代数的谱。在这个项目中，我们讨论了复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，我们考虑了局部域上的约化线性代数群，主要是实数上的约化线性代数群，以及Adele环上的约化线性代数群。在后一种情况下，可以借助于Arthur-Selberg迹公式来研究自同构形式空间中的表示，该公式将谱数据与基础模空间的几何数据联系起来。我们的目的之一是研究这些具有数论解释的几何数据在新情况下的渐近分布。对于这类应用，我们必须扩展迹公式来测试非紧支集的函数，在编码几何数据的迹公式中的项也包含在非紧群上的分布，这些分布可以被分成依赖于其局部因子的分量。在有关群体上的这些地方分配之间的身份转移对于朗兰兹方案的进展至关重要。朗兰兹最近对如何超越目前的内窥镜检查方法的愿景表明，人们需要确定这些分布的准确傅立叶变换。我们将扩展我们在实际情况中得到的部分结果，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实例。

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.