Notes on quantization of symplectic vector spaces over finite fields

Details Notes on quantization of symplectic vector spaces over finite fields Notes on quantization of symplectic vector spaces over finite fields

2007-08-05

arxiv.org/abs/0708.0669v3

Physics

Mathematical Physics

Notes from the lecture at the AGAQ conference (Istanbul, June 2006) on our solution to Kazhdan's problem on canonical quantiza

Scientific paper

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.

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