Canonical intertwiners for representations of unipotent groups, generalized Weil representations, and their geometric analogues

A. Weil defined a remarkable projective representation of the symplectic group over a local field by studying the intertwiners between representations of the Heisenberg group. If the ground field is a finite field or the complex numbers, then it is known that the Weil representation can be linearized; in other words, it comes from a true representation. Otherwise, the Weil representation comes from a true representation of a two-sheeted cover of the symplectic group known as the metaplectic group. In my Master's Thesis, I give a proof of this fact using symplectic K-theory, as developed by A. Suslin. I also develop explicit formulas of the linearized Weil representation over a finite field. In my future research, I plan to study intertwiners between representations of unipotent groups. Using these intertwiners, one can define certain projective representations which can be considered as generalizations of the Weil representation. It is expected that these projective representations can be linearized over a finite field. Recently, S. Gorevich and R. Hadani have given a geometric construction of the Weil representation following P. Deligne. Following V. Drinfeld, I have outlined a program for geometrizing the generalized Weil representations.

One dimensional character sheaves, true commutator and stacky abelianization

Character sheaves for reductive groups and unipotent groups have been defined by G. Lusztig and V. Drinfeld, respectively. In contrast to usual character theory, there exists one-dimensional character sheaves whose restrictions to the commutator subgroup is not trivial. In my PhD thesis, I construct a new commutator for an algebraic group called the true commutator. It satisfies the property that every one-dimensional character sheaf splits over the true commutator. The quotient stack of the group by the new commutator has a universal property similar to the usual abelianization of groups.