Special Functions of p-adic Algebraic Groups and Quantum Groups

This is a project to develop connections between number theory and physics. A modern paradigm in number theory uses highly symmetric functions to answer the most fundamental questions about solutions of equations in several variables. Quite surprisingly, these same symmetries arise in physics, particularly statistical mechanics, where one seeks to determine global behavior of molecules based on local interactions between particles. The PI, collaborators, and students, will explain and explore further mathematical consequences of this connection. The project will provide research training opportunities for both undergraduate and graduate students. More precisely, the bridge between number theory and statistical mechanics alluded to above is the theory of quantum groups and most of the specific projects pursued will use the representation theory of quantum group modules. To make connections with special functions in number theory, particularly matrix coefficients of algebraic groups over local fields, one needs new results on quantum group modules. The PI and collaborators will use quantum affine Lie superalgebra modules to produce lattice models with the required symmetry used in the study of matrix coefficients for metaplectic groups. In reverse, by expressing new classes of special functions from representation theory as partition functions of solvable lattice models, one obtains conjectural invariants of multi-parameter quantum groups. The primary scientific goals include deeper insight from quantum groups into various aspects of the Langlands program. 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.