Representations of Quantum Groups, Special Functions, and Geometry

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KIRILLOV This research is concerned with the representation theory of quantum groups and its relation with combinatorics and mathematical physics. The principal investigator will work on two problems. The first problem concerns a geometric realization of Lusztig's canonical basis, using the identification of modules over a quantum group with the homology of a certain local system on the configuration space, which is due to Schechtman and Varchenko. Such a realization would reveal a new deep relation between combinatorics and the geometric structures of mathematical physics. The second continues the study of special functions appearing in representation theory and related combinatorial identities. In particular, the principal investigator will work on the generalization of the inner product identities for Macdonald's polynomials to affine root systems, and on on further properties of this inner product, such as positivity and its relation with the inner product on the space of conformal blocks in conformal field theory. This study is in the general area of representation theory of Lie algebras and related objects, and is important both for mathematics and physics.