Demazure Flags, Hypergeometric Series, and Quantum Affine Algebras

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The study of Lie groups and Lie algebras has a long and distinguished history going back to the mid-nineteenth century. It had its roots in the idea that the geometry of space is determined by its group of symmetries. As our understanding of space has evolved, the study of Lie groups and Lie algebras (Lie theory) has become a central part of modern mathematics. The subject has also always benefited by interactions with physics and these led to two of the major developments in the last century; the definition and study of Kac-Moody algebras and quantum groups. These objects turned out to have many connections to physics, combinatorics and number theory. One of the early successes of their study was the fact that many classical identities established by the famous Indian mathematician Srinivasa Ramanujan (1887-1920) could be recovered and interpreted in the language of the representation theory of Kac-Moody algebras. Quantum groups are deformations of Kac-Moody algebras and their study has led to important connections with knot theory and mathematical physics. One of the goals of this project it to exploit the synergy between the study of Kac-Moody algebras and their quantum analogs to study questions in representation theory and to relate them to various series of functions which appear in the work of Ramanujan.This project focuses on the study of the connections between Demazure modules in an affine Kac-Moody Lie algebra and the theory of hypergeometric series, quantum affine algebras, generalized Q-systems and cluster algebras. In the simplest cases, Demazure modules are indexed by a pair of positive integers one of which is called the level of the module. It can be shown that a module of a fixed level admits a flag where the successive quotients are all higher level Demazure modules. This information can be encoded in the form of a generating series and a goal of the project is to relate them to q-hypergeometric series and to study their modularity properties. The study of Demazure modules is also important in the theory of quantum affine Kac-Moody algebras. Another goal of this project is to establish character formulae for prime representations of quantum affine algebras. The latter are known to be related to cluster algebras and the proposal will study the combinatorial problems arising from these many connections.