Invariant harmonic analysis and Selberg zeta functions (B06 #)

非交换调和分析可以被描述为研究在函数空间或其推广空间中通过作用于底层空间而归纳出的群表示。一个中心问题是将这些表示分解成不可约的组件。这是谱分解的一个版本，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，我们处理复Hubert空间中拓扑群的表示及其作为直接和或积分的分解。更准确地说，我们考虑局部域上的约化线性代数群，主要是实数上的，以及阿黛尔环上的。在后一种情况下，可以利用Arthur-Selberg迹公式研究自同构形式空间中的表示，该公式将谱数据与底层模空间的几何数据联系起来。我们的目标之一是研究在新情况下具有数论解释的几何数据的渐近分布。对于这类应用，我们必须将跟踪公式扩展到非紧凑支撑函数的测试。编码几何数据的轨迹公式中的项也包含在曲线群上的分布，这些分布可以被分解成依赖于其局部因子的分量。在相关群体的这些地方分布之间转移身份对兰兰项目的进展至关重要。Langlands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里叶变换。我们将在实际情况下展开我们得到的部分结果。这也将使我们能够获得Selberg型zeta函数的泛函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.