Notes on quantization of symplectic vector spaces over finite fields

2007/08/05 by Shamgar Gurevich, Ronny Hadani, Gurevich, Shamgar +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Representation Theory (math.RT) #math-ph #math.MP #math.RT

paper · pdf · doi:10.48550/arxiv.0708.0669

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

openalex publication_date 2007/08/05 · arxiv created 2009/04/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Heisenberg Realizations, Eigenfunctions and Proof of the Kurlberg-Rudnick Supremum Conjecture
• Weil Representation, Deligne Sheaf, and Proof of the Kurlberg-Rudnick Conjecture
• The Two Dimensional Hannay-Berry Model

Related

• Quantization of bounded symplectic domains associated with compact Lie groups
• Weil representations over finite fields and Shintani lift
• The Weil Representation in Characteristic Two
• Regular characters of GL_n(O) and Weil representations over finite fields
• The categorical Weil representation