Unique Functionals and Quantum Groups

Stanford University, Stanford CA

Investigators

Abstract

This is a research project at the interface of number theory, representation theory, and theoretical physics. Recent work of the investigator and collaborators has shown that certain matrices, known as R-matrices, that arise in theoretical physics in the study of statistical mechanics and phase transitions also arise in the study of a class of functions, known as Whittaker functions, that encode arithmetic information when these functions are viewed representation-theoretically, that is, when inherent algebraic symmetries are taken into account. This discovery opens the door to a rich collection of questions that will be explored in this project. It is anticipated that in the process, important new connections between number theory and mathematical physics will be developed. In more detail, Whittaker functions on p-adic groups are a fundamental tool in automorphic forms. Whittaker functions on metaplectic covers of these groups are less well-understood, but the corresponding matrices on intertwining operators have been computed. The investigator and collaborators have shown that these matrices for the n-fold mataplectic cover of GL(n) agree with the R-matrices for a quantum group, which is a Drinfield twist of the quantized enveloping algebra of the affinized Lie superalgebra gl(1/n). The effect of the Drinfield twisting is to introduce Gauss sums into the R-matrix, and therefore to make it "number-theoretic." This clarifies a great deal, but also presents new questions that will be explored in this project. A separate but related project that will be pursued is an on-going study of connections between unique models and representations of Hecke algebras.