Representations of Real Lie Groups, Symmetry Breaking, and Automorphic Forms

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The investigator will continue long-term research into the study of symmetries and some of their applications. Symmetry is a mathematical concept with direct applications in other sciences such as chemistry, physics, biology, and engineering. The study of algebra, of which this research is a part, is in many ways the study of symmetries. This project aims to advance understanding of important algebraic structures, with potential applications to number theory and theoretical physics.This research project is concerned with symmetry breaking, that is, obtaining representations of a smaller Lie group from the representations of a large reductive semi-simple Lie group. The representations of the large group are usually infinite dimensional, and the representations of smaller groups are related to quotients of the original representation of the large group. Symmetry breaking of representations is far from well understood, and so the constructions and the understanding of examples are very important. This research has applications to number theory and possibly to theoretical physics. Many of the techniques used in these constructions are analytic, and thus some new, interesting, and challenging problems in analysis arise and need to be solved.