Quantization of symplectic vector spaces over finite fields

Abstract

In this paper, we construct a quantization functor, associating a complex vector space H(V ) to a finite-dimensional symplectic vector space V over a finite field of odd characteristic.As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp(V ).The main new technical result is a proof of a stronger form of the Stone-von Neumann property for the Heisenberg group H(V ).Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical vector space attached to a coadjoint orbit of a general unipotent group over finite field.