## Journal of Symplectic Geometry

- J. Symplectic Geom.
- Volume 7, Number 4 (2009), 475-502.

### Quantization of symplectic vector spaces over finite fields

Shamgar Gurevich and Ronny Hadani

#### 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.

#### Article information

**Source**

J. Symplectic Geom., Volume 7, Number 4 (2009), 475-502.

**Dates**

First available in Project Euclid: 22 October 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.jsg/1256219055

**Mathematical Reviews number (MathSciNet)**

MR2552002

**Zentralblatt MATH identifier**

1220.53094

#### Citation

Gurevich, Shamgar; Hadani, Ronny. Quantization of symplectic vector spaces over finite fields. J. Symplectic Geom. 7 (2009), no. 4, 475--502. https://projecteuclid.org/euclid.jsg/1256219055