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，即群及其商在一个域上作为矩阵群的实现，例如复数。即使对于特殊的群类，如“半简单”算术群，也很难找出所有可能的表示。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。利用数论的工具和狄利克雷生成函数，我们可以对这类群的渐近分布进行编码。随后，我们可以应用分析、几何和模型理论等技术来探索这种分布的特征。由此产生的ζ函数表示构成了著名的黎曼ζ函数的深远推广。后者在数论中起着核心作用，因为它编码有关素数分布的信息。近年来，对算术群和p进李群的表示zeta函数的研究取得了一些有趣的进展。该项目的目的是以不同的方式改进zeta函数的原始定义，以便将新兴的理论置于更广泛的基础上。在一个方向上，我们将研究数字域的表示，而不是复数。在另一个方向上，我们将把ζ函数与每个允许的无限维表示联系起来。为此，需要开发在其他环境中也有相关应用的新方法。该项目由具体目标组成，构成了扩展渐近群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.