Metaplectic automorphic forms and matrix coefficients

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This research project will explore topics in number theory and representation theory with connections to geometry, algebraic combinatorics, and statistical mechanics. Over the past several decades it has become clear that some of the deepest questions and conjectures in number theory, most notably those connected with the Langlands program, have powerful analogs in geometry and physics. However, the mechanism behind this relationship and associated conjectures remains largely mysterious. This project aims to explore possible sources of the connections by broadening the class of objects under consideration and using the winnowed set of techniques that apply to this larger class.In particular, many of the projects proposed center around the investigation of matrix coefficients for p-adic algebraic groups and their arithmetic covers. These matrix coefficients play a key role in the construction of automorphic L-functions. Their explicit computation in the context of metaplectic covers leads to surprising connections with geometry of Schubert varieties, to various specializations of Macdonald polynomials, and to quantum groups via both canonical bases and lattice models. These will be further developed in the proposed work and a framework for classifying matrix coefficients on algebraic groups as intertwining operators for Hecke algebra modules will be pursued. New distribution results for arithmetic functions will be another byproduct of these investigations.