Representation Theory, Quantum Groups, and Canonical Bases

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The main theme of this proposal is to investigate the area lying at the crossroads of the representation theory of Lie groups, quantum groups, cluster algebras, and noncommutative algebraic geometry. A new approach to the study of Lusztig's canonical bases and Kashiwara's crystal bases is proposed, based on quantum cluster algebras and geometric crystals as developed in recent papers of the proposer. New information resulting from this study will be applied to computing the multiplicities for the symmetric powers of representations of reductive groups, and constructing new totally positive varieties. The results of this study will be applied for solving problems emerging in the representations of discrete subgroups of reductive algebraic groups and Cherednik algebras as well as for explication and elaboration of related combinatorial and geometric structures including the ``geometric lifting'' of crystal bases as a new tool in understanding the local Langlands correspondence. Representation theory is one of the most dynamically developing fields of modern Mathematics. It has a large impact in other fields of Mathematics and numerous applications in other Natural Sciences. The concepts of canonical and crystal bases are of great importance for the representation theory: a mere establishing of existence of such bases has helped in solving classical enumeration problems like computing multiplicities of irreducible representations or decomposing tensor products of irreducible representations. A new class of canonical bases discovered by the proposer is expected to settle an old problem of decomposing symmetric powers of representations. Therefore, any information on canonical or crystal bases would be very beneficial for the representation theory. It has been revealed in the works of G. Lusztig and the subsequent works of the proposer that there are algebro-geometric counterparts for the purely discrete canonical and crystal bases: totally positive varieties, geometric crystals, and cluster algebras. Understanding the relationship between these structures underlying the canonical bases is one of main priorities of this proposal. This relationship has proved to be a useful tool in the study of Langlands correspondence -- the most mysterious and inspiring correspondence between Algebra and Geometry of the 20th century Mathematics.