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，群及其子群作为域上的矩阵群的实现，例如复数。即使对于特殊类的群，如“半单”算术群，也很难算出所有可能的表示。 在适当的技术意义上，我们甚至不可能对所有的不可约表象给出一个概观。使用工具从数论，某些狄利克雷生成函数，我们能够编码的渐近分布的表示，这样的群体。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。由此产生的表示zeta函数构成了著名的黎曼zeta函数的广泛推广。后者在数论中起着核心作用，因为它编码了有关素数分布的信息。近年来，算术群和p-adic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完善表征zeta函数的原始定义，以便将新兴理论置于更广泛的基础上。 在一个方向上，我们将研究数域上的表示，而不是复数。在另一个方向上，我们将把一个zeta函数与每一个可容许的无限维表示相关联。为此目的，需要开发在其他情况下也具有相关应用的新方法。该项目包括具体的目标，形成一个长期战略的一部分，以扩展渐近群理论。

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.