New Unifying Structures in Lie Theory and the Cubic Dirac Operator

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A whole array of questions in different mathematical areas seemto be converge to questions having to do with the set of alcovesin a fixed Weyl chamber of a semisimple Lie group. The areasinclude Kac-Moody theory, homology of loop groups, quantumcohomology, the Verlinde algebra, MacDonald identities, Schubertcalculus, ideals in the Borel subalgebra, symmetric space theoryand the Cartan-Weyl representation theory of compact Liegroups. In effect our proposal is to sort out what is going on tounify what seems to us to be unifiable in the subjects listedabove.Group theory, and especially Lie group theory, lies at the heartof mathematics and the application of mathematics to problems inthe real world. Included in the latter are applications to bothclassical and quantum mechanics, control theory, string theory,chemistry and crystallography. Lie group theory is extremelyintricate and the extent to which it is applicable depends highlyon a knowledge of its intricacies. The proposed project expectsto discover highly exciting new structures in the subject andwould unify many existing structures. The effect that would be togreatly increase our understanding and our ability to use thesepowerful structures. This award is jointly funded by the programsin Geometric Analysis and Algebra, Number Theory, & Combinatorics.