Mathematical Sciences: Automorphic Representations of Semisimple Lie Groups

首席研究员计划研究与（半单李群）表示理论有关的数学分析问题。后者的一个合适的例子是一个球体的旋转群。像这样的群很重要，因为它们作为对称群出现在数学的许多领域（例如几何、微分方程、代数数论、数学物理）。表征理论允许人们利用对称性来解决问题。更具体地说，这个项目将研究来自自同构形式的空间（不变解析函数）在群模一个离散子群的商上的表示。目的是根据离散群的代数性质和它在周围李群中的位置信息，得到这些自同构表示中已知不可约表示的多重性的显式公式。

The principal investigator plans to work on problems in mathematical analysis concerning the representation theory of (semisimple) Lie groups. A suitable example of the latter is the group of rotations of a sphere. Groups like this are important because they occur in many areas of mathematics (e.g. geometry, differential equations, algebraic number theory, mathematical physics) as groups of symmetries. Representation theory allows one to take advantage of symmetries in solving problems. More specifically, this project will study representations that come from spaces of automorphic forms (invariant analytic functions) on the quotient of the group modulo a discrete subgroup. The aim is to obtain explicit formulas for the multiplicities of known irreducible representations in these automorphic representations, based on algebraic properties of the discrete group and information about how it sits in the ambient Lie group.