Notes on quantization of symplectic vector spaces over finite fields

TL;DR

This paper constructs a canonical quantization framework for symplectic vector spaces over finite fields, leading to a model of the Weil representation and addressing a question about canonical Hilbert spaces for coadjoint orbits.

Contribution

It introduces a quantization functor for symplectic vector spaces over finite fields and proves a stronger Stone-von Neumann theorem for the Heisenberg group over F_q.

Findings

Established a canonical model for the Weil representation

Proved a stronger form of the Stone-von Neumann theorem over finite fields

Provided a canonical Hilbert space for coadjoint orbits of unipotent groups

Abstract

In these notes we construct a quantization functor, associating an Hilbert space H(V) to a finite dimensional symplectic vector space V over a finite field F_q. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp(V). The main technical result is a proof of a stronger form of the Stone-von Neumann theorem for the Heisenberg group over 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 F_q.