Quantization of symplectic vector spaces over finite fields
arXiv:0705.4556
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. 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.
Results obtained: March 2005 (under the direction of Ph.D. advisor Joseph Bernstein, Tel-Aviv, Israel) Keywords: Strong Stone-von Neumann theorem, canonocal Hilbert space, quantization functor, Weil representation