Quantum Affine Algebras: BGG reciprocity, Macdonald Polynomials, Schur postivity

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The proposal is on the interplay between the representation theory of affine algebras, its standard maximal parabolic subalgebra, namely the current algebra, and the quantum affine algebra associated to a simple Lie algebra. It focuses on the study of families of infinite-dimensional and level zero representations for each of these algebras and develops connections with Macdonald polynomials, Demazure characters and Schur positivity. One of the goals is to establish a Bernstein-Gelfand-Gelfand type duality principle for these categories and to investigate the combinatorial and homological consequences of having such a duality. Another goal of this proposal is to develop a connection between the homological properties of the category of infinite-dimensional representations of the quantum affine algebra and the tensor structure of this category. The existence of such a connection is unexpected and somewhat mysterious and is suggested by some recent work of the PI. It exists only at the quantum level and a deeper understanding of this should have a substantial impact on the study of this category. It should also also yield connections with recent work of others on cluster algebras and categorification.The study of affine Lie algebras and their quantum analogs have long had remarkable connections to a number of different fields including string theory, conformal field theory, topological field theory, infinite dimensional geometry and mathematical physics. Many of the themes of the project are motivated by questions arising in solvable lattice models. Representations of affine Lie algebras and the standard maximal parabolic subalgebras will capture important physical information. The project will also provide a representation theoretic framework in which to understand various combinatorial problems.