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 enveloping algebras 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.
Our research was devoted to the local Langlands correspondence and geometric representation theory, two areas of central importance in modern algebra that have connections to many other subjects (such as mathematical physics). We used the tools of geometric character theory on unipotent groups to make significant progress in two directions. On the one hand, we found a new method of constructing several special cases of the local Langlands correspondence, which makes its nature much clearer than the existing approaches. On the other hand, we developed general techniques for analyzing a conjectural geometric construction of supercuspidal representations of reductive p-adic groups that was proposed in 1979 by George Lusztig (until recently, it was not known how to compute any interesting examples of the representations coming from his construction). Our research resulted in 4 journal publications and several preprints that were disseminated to the public. The PI also gave a number of seminar talks at US universities. Our work also had an influence on the development of several students at the University of Michigan. In particular, the grant supported three undergraduate students' participation in the "Research Experience for Undergraduates" program; all three students later went on to pursue Ph.D. degrees in mathematics. The PI has also supervised the research of a graduate student, who significantly extended the PI's earlier work on the geometric construction of supercuspidal representations of reductive p-adic groups.
