Representation zeta functions associated to arithmetic groups and compact analytic groups

群论中的一个重要学科是研究群的线性表示，即群及其商在域上的矩阵群的实现，例如复数。即使对于特殊类别的群，如“半单”算术群，也很难计算出所有可能的表示。通常，在适当的技术意义上，甚至不可能给出所有不可约表示的概述。利用数论中的工具，某些Dirichlet母函数，我们能够编码这些群的表示的渐近分布。随后，我们可以应用分析、几何和模型理论的技术来探索这种分布的特征。由此得到的表示Zeta函数构成了著名的Riemann Zeta函数的推广。后者在数论中起着核心作用，因为它编码关于素数分布的信息。近年来，算术群和p-进李群的表示Zeta函数的研究取得了令人感兴趣的进展。该项目的目的是以不同的方式改进表示Zeta函数的原始定义，以便将新兴理论放在更广泛的基础上。在一个方向上，我们将研究数字域上的表示法，而不是复数。在另一个方向上，我们将Zeta函数与每个可接受的无限维表示相关联。为此，人们需要开发新的方法，这些方法将在其他情况下也有相关的应用。该项目由具体目标组成，构成了扩展渐近群理论的长期战略的一部分。

An important discipline within group theory is concerned with the study of linear representations of groups, i.e., realizations of groups and their quotients as matrix groups over a field, such as the complex numbers. Even for special classes of groups, such as `semisimple' arithmetic groups, it is difficult to work out all possible representations. Often it is not even possible - in a suitable technical sense - to give an overview over all irreducible representations. Using tools from number theory, certain Dirichlet generating functions, we are able to encode the asymptotic distribution of representations for such groups. Subsequently we can apply techniques from analysis, geometry and model theory to explore features of this distribution. The resulting representation zeta functions constitute farreaching generalizations of the famous Riemann zeta function. The latter plays a central role in number theory as it encodes information about the distribution of prime numbers. In recent years interesting advances have be made in the study of representation zeta functions of arithmetic groups and p-adic Lie groups. The aim of the project is to refine the original definition of a representation zeta function in different ways in order to put the emerging theory on a much wider basis. In one direction we will study representations over number fields instead of the complex numbers. In another direction we will associate a zeta function to each admissible infinite dimensional representation. For this purpose one needs to develop new methods that will have relevant applications also in other contexts. The project consists of concrete aims forming part of a long term strategy to extend asymptotic group theory.