Geometric Methods in the Representation Theory of Affine Hecke Algebras and Quantum Groups

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9732805 Lusztig This award supports the continued study of periodic W-graphs attached to affine Hecke algebras and the investigation of their possible connections with unrestricted representations of semisimple Lie algebras in positive characteristic. The principal investigator will also study canonical bases of quantized enveloping algebras from the point of view of perverse sheaves. He will continue the study of unipotent representations of semisimple p-adic groups, as well as character sheaves on (possibly disconnected) reductive groups. Representation theory of Lie algebras and Lie groups, initially developed at the turn of the century, studies the structure of symmetries and their realizations. This theory encompasses many areas in mathematics and has fundamental applications to theoretical physics. In the past twenty years, the study of representation theory of a special class of affine Lie algebras and Lie groups led to many new unexpected discoveries in mathematics and theoretical physics, as well as to a synthesis of many established areas in both disciplines. In particular, it yielded a pure mathematical description of a simplest quantum field theory -- a prototype theory of fundamental interactions. It is expected that representation theory of new classes of infinite-dimensional Lie algebras and groups might substantially deepen our understanding of mathematics and theoretical physics during the next decade.