Matrix Coefficients of Covering Groups, Quantum Groups, and Lie Superalgebras

The study of symmetry is of fundamental scientific importance. In particular, many physical theories of the universe and of elementary particles are described by collections of symmetries called Lie groups. Understanding the spaces on which these symmetries may act provides important insights into such physical theories. This project will study surprising connections between number theory, quantum groups, and mathematical physics that provide a new way of understanding such spaces.

More specifically, the investigator and his students and collaborators will explore matrix coefficients of representations of p-adic groups and their arithmetic covers, known as metaplectic groups. Matrix coefficients allow one to extract numerical invariants from representations. They play a key role in both the construction of automorphic L-functions and the proofs of their analytic properties, and also in the determination of scattering amplitudes in string theory. The project will develop new relations between matrix coefficients, quantum groups, and statistical mechanics, and new methods to study matrix coefficients using Hecke algebras.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.