Noncommutative Algebraic Geometry in Representation Theory

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This project, involving the work of Kobi Kremnizer, is devoted to the application of methods from noncommutative algebraic geometry, to the study of representation theory. The first aim is the study of the representation theory of the real quantum group. The quantization of the theory of real (algebraic) reductive groups is a new and important subject. The project proposes using quantum D-modules to study quantum Harish-Chandra modules thus making the classification an accessible task. The classical theory of Harish-Chandra modules might benefit from this study of its deformations. The second aim is to give a geometric proof of Lusztig`s conjectures concerning characters of nonrestricted representations of reductive Lie algebras in positive characteristic. The proof will use quantum differential operators on the quantum flag variety at a root of unity. Lusztig`s conjectures are only proven for the restricted case by a very computational method. The project proposes to give a proof for the general case of these conjectures using the geometry of the Springer resolution. The project also suggests showing the equivalence of the category of representations of the nonrestricted quantum group at a root of unity and the category of representations of the affine Lie algebra at the critical level by connecting both categories to the Springer resolution. The third aim is to give a geometric construction of the double affine Hecke algebra and to prove a quantum version of the twisted Harish-Chandra homomorphism. This should help study the representation theory of this algebra at a root of unity and connect it to the representation theory of the trigonometric Cherednik algebra in positive characteristic. This project lies in the meeting point of representation theory, noncommutative algebraic geometry, geometry in positive characteristic, infinite dimensional geometry and conformal field theory. It will hopefully help clarify the close relations between loop space geometry and noncommutative geometry that is observed in string theory and will open new pathways for research. The methods it offers to use should be useful in other cases as well, making noncommutative geometry an important tool in representation theory and in algebraic geometry.