Quantization of symplectic vector spaces over finite fields

Abstract

In this paper, we construct a quantization functor, associating a complex vector space $\cal{H}(V)$ 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.