Notes on Canonical Quantization of Symplectic Vector Spaces over Finite Fields

Abstract

In these notes we construct a quantization functor, associating a Hilbert space \(\mathcal{H}(V )\) to a finite dimensional symplectic vector space V over a finite field \({\mathbb{F}}_{q}\). As a result, we obtain a canonical model for the Weil representation of the symplectic group SpV . The main technical result is a proof of a stronger form of the Stone–von Neumann theorem for the Heisenberg group over \({\mathbb{F}}_{q}\). Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical Hilbert space attached to a coadjoint orbit of a general unipotent group over \({\mathbb{F}}_{q}\).