Representations of arithmetic groups and their associated zeta functions

本项目旨在通过研究算术群的线性作用来研究数论和群论之间的深层联系。算术群用于几何学、动力学、数论和其他纯数学领域。这个项目将从两个角度研究他们的表现。首先，它将建立相关的zeta函数的属性，以解决有关不可约表示的维数的渐近行为的公开问题。第二，它将探索在新的归纳和限制函子下，群族的表示的演化，类似于物理学中的创造和湮灭算子。该项目将提高澳大利亚在表征理论和群论方面的能力，这是一种强调自然界对称性的数学。

This project aims to investigate deep connections between number theory and group theory by studying linear actions of arithmetic groups. Arithmetic groups are used in geometry, dynamics, number theory and other areas of pure mathematics. This project will study their representations from two perspectives. First, it will establish properties of the associated zeta functions to resolve open problems about the asymptotic behaviour of the dimensions of the irreducible representations. Second, it will explore the evolution of representations across families of groups under new induction and restriction functors, in analogy with creation and annihilation operators in physics. The project will enhance Australia's capacity in representation theory and group theory, the mathematics that underline symmetry in nature.