Symmetry breaking for real reductive groups and applications to automorphic forms
Final Report Abstract
Symmetries are mathematically described in terms of symmetry groups and their representations. Each abstract symmetry group can have several different representations, i.e. there exist different incarnations of the same abstract symmetry type. For instance, the symmetries of quantum mechanical systems in physics are described by unitary representations of symmetry groups. Representation theory is the mathematical discipline which seeks to understand representations of symmetry groups. One of the fundamental problems in this area are is decompose representations into their smallest building blocks and to classify these building blocks. In this project, the aspect ot symmetry breaking was investigated, i.e. the reduction of symmetries. Mathematically this corresponds to the restriction of a representation of a symmetry group to a subgroup, hence reducing the symmetries by only considering a those coming from a smaller symmetry group. Here we focused on two particular types of symmetries, those coming from rank one semisimple Lie groups and from the general linear groups. In both cases we explicitly constructed so-called symmetry breaking operators which encode the symmetries of the smaller symmetry group inside the representation of the larger symmetry group. In the former situation we were even able to classify all such symmetry breaking operators in certain situations. Moreover, these operators were used to explicitly decompose some of the building blocks of the representation theory of the larger symmetry group into building blocks of the smaller symmetry group. Such decompositions are referred to as branching laws. The key new idea in this project was to make significant use of the fact that such symmetry breaking operators often come in families depending on certain parameters. The dependence on these parameters is crucial and was one of the main objects of research. Once this dependence is well-understood, one can reduce the study of branching laws to a single representation in this family and then simply vary the parameters. A direct application of our construction and classification of symmetry breaking operators to analytic number theory was given. Here the symmetry groups describe symmetries of so-called automorphic functions which encode number theoretic information. Our previously gained understanding of the symmetry reduction allowed us to obtain some new estimates for automorphic functions when comparing them to automorphic functions belonging to smaller symmetry groups.
Publications
- Symmetry breaking operators for line bundles over real projective spaces, J. Lie Theory 29 (2019), no. 2, 511–558
Jan Frahm and Clemens Weiske
(See online at https://doi.org/10.48550/arXiv.1712.06344) - Symmetry breaking operators for real reductive groups of rank one, J. Funct. Anal. 279 (2020), no. 5, 108568, 70
Jan Frahm and Clemens Weiske
(See online at https://doi.org/10.1016/j.jfa.2020.108568)
