Analytic and combinatorial aspects of representation theory

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Representation theory is an area of mathematics that interweaves ideas from several disciplines, including algebra, analysis, combinatorics, and mathematical physics, to study realizations (also known as representations) of abstract mathematical objects that parametrize symmetries. In many problems that emerge in dynamical systems and physics, one encounters continuous groups of symmetries, which are called Lie groups. Lie groups are objects that belong to differential geometry, but the study of their representations relies substantially on certain algebraic objects called Lie algebras. Representation theory of Lie groups and Lie algebras plays a distinguished role in harmonic analysis, number theory, algebraic combinatorics, and theoretical physics. ******In my research over the next five years, I aim to answer questions in representation theory of Lie groups and Lie algebras, and a generalization of Lie algebras which are called Lie superalgebras. These questions are closely related to the theory of symmetric polynomials. Symmetric polynomials, such as Jack and Macdonald polynomials and their deformations, occur frequently in representation theory. This has lead to remarkable interactions between representation theory and algebraic combinatorics, and the insight provided by each of these branches of mathematics has enriched the other one substantially. The goal of my proposed research is to shed more light on these interactions by focusing on the case of Lie superalgebras, and to establish new connections with other algebraic objects such as quantum groups. The passage from Lie algebras to Lie superalgebras results in many new challenges and technical difficulties, and tackling them requires novel ideas.******I also plan to continue my study of unitary representations of Lie groups using tools from analysis. These representations are realized on Hilbert spaces, and typically the underlying Lie algebra does not act on the entire representation space. In particular, canonical dense subspaces which carry an action of the Lie algebra, such as the space of smooth vectors, play a crucial role in the theory. In my research I will study representations of both finite and infinite dimensional Lie groups. The finite dimensional case includes real and p-adic semisimple Lie groups, and therefore the impact of my research will be in the theory of automorphic forms. In the infinite dimensional case, the groups under investigation include loop groups and the Virasoro group. One of the most important classes of representations of the latter groups is the class of unitary representations of positive energy. Therefore in the infinite dimensional case the impact of my research will be in mathematical physics.