Representations of p-adic Covering Groups and Integrable Systems

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This project will establish new connections between mathematics and physics. On the mathematics side, it explores objects originating from number theory that possess a sophisticated collection of symmetries. On the physics side, it involves lattice models from statistical mechanics. These lattice models are attempts to study the behavior of matter (e.g., its phase transitions from solid to liquid to gas) by determining the energy in each individual atomic interaction. While such a reductionist physical approach may seem daunting, it works surprisingly often and, roughly speaking, the models which succeed in this approach are termed "solvable." In this project solvable lattice models are used to represent special functions from number theory and to demonstrate previously unknown properties of them and the connections between mathematics and physics serve to inform both subjects. The project will provide training opportunities for undergraduate and graduate students that are broadly applicable to a wide range of STEM careers. More precisely, the goal of the project is to provide algebraic structure to the study of matrix coefficients on groups over local fields and their covers. The PI and his collaborators have demonstrated a fundamental connection between metaplectic Whittaker functions on p-adic groups and quantum groups, using the solvable lattice models described above as a bridge between the two theories. The research will demonstrate that this connection extends and generalizes in multiple directions, and a solvable lattice model for Iwahori fixed vectors in a metaplectic Whittaker model is now within reach. Once completed, the project will provide a powerful connection between metaplectic representations and quantum affine superalgebras via the R-matrices used to demonstrate Yang-Baxter equations. Additional topics to be investigated include finding direct links between p-adic representation theory and quantum groups; using Hecke algebra characters to categorize matrix coefficients to better understand unramified calculations in the local theory, which are essential in integral representations of L-functions; and describing enhancements to the theory of Kashiwara crystals for superalgebras and to archimedean matrix coefficients on geometric crystals.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.