Collaborative Research: The Geometric Dual Space of a Unipotent Group in Characteristic p

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The principal investigators continue their research in geometric character theory and geometric representation theory of unipotent groups over a field of positive characteristic. They define the "dual space" of such a group as an algebro-geometric object parameterizing L-packets of character sheaves on the group. The principal investigators then develop a conjectural geometric theory of the spectral decomposition of the equivariant derived category with respect to the dual space, which is similar to the classical decomposition of representations as direct integrals of irreducible ones. They also formulate several conjectures on this spectral decomposition in the setting where the orbit method is applicable and discuss certain related quantization problems in the theory of tensor categories. The difference between these quantization problems and the standard ones is that the role of the universal envelopingalgebras is played by group algebras.The proposed project combines several active directions of current research in mathematics and mathematical physics -- geometric representation theory, algebraic geometry, the theory of tensor categories, quantum groups, and conformal field theory. The notion of geometric spectral decomposition will deepen our understanding of the general patterns of geometric representation theory. The research will also lead to new applications of the quantization philosophy in the theory of tensor categories.