[Paper Review] Quantization of symplectic vector spaces over finite fields

Shamgar Gurevich, Ronny Hadani|ArXiv.org|May 31, 2007

Advanced Algebra and Geometry5 citations

TL;DR

This paper constructs a canonical quantization functor from symplectic vector spaces over finite fields of odd characteristic to complex vector spaces, yielding a natural realization of the Weil representation of the symplectic group. The key contribution is a proof of a strong Stone–von Neumann theorem for the Heisenberg group over finite fields, establishing a canonical vector space model for each coadjoint orbit via oriented Lagrangian subspaces.

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.

Motivation & Objective

• To construct a canonical quantization functor assigning a complex vector space to each finite-dimensional symplectic vector space over a finite field of odd characteristic.
• To provide a canonical linear model for the Weil representation of the symplectic group $Sp(V)$, avoiding the need for projective representations and subsequent linearization.
• To establish a strong form of the Stone–von Neumann theorem for the Heisenberg group over finite fields, resolving a question posed by Kazhdan on canonical vector spaces for coadjoint orbits of unipotent groups.
• To demonstrate that the quantization functor is monoidal, compatible with duality, and respects symplectic reduction via natural isomorphisms.
• To construct a canonical vector in $\mathcal{H}(V)$ associated to each oriented Lagrangian subspace $L^\circ = (L, o_L)$, via the invariant subspace $\mathcal{H}^L(V) \simeq \mathbb{C}$.

Proposed method

• The quantization functor $\mathcal{H}: \mathsf{Symp} \to \mathsf{Vect}$ is constructed using representation-theoretic methods, assigning to each symplectic space $V$ a complex vector space $\mathcal{H}(V)$.
• The construction relies on a strong version of the Stone–von Neumann theorem for the Heisenberg group $H(V)$, ensuring a unique irreducible representation with a fixed central character.
• Two proofs are provided: one using elementary linear algebra and another using $\ell$-adic perverse Weil sheaves, which are shown to realize the strong Stone–von Neumann property.
• The sheaf-theoretic proof involves constructing an $\ell$-adic perverse sheaf $\mathcal{K}$ on the space of oriented Lagrangians, and proving its convolution isomorphism via Fourier transform techniques.
• The compatibility with symplectic reduction is established via the isomorphism $\mathcal{H}^I(V) \simeq \mathcal{H}(I^\perp / I)$, where $I$ is an oriented isotropic subspace.
• The canonical vector associated to an oriented Lagrangian $L^\circ$ arises from the isomorphism $\mathcal{H}^L(V) \simeq \mathcal{H}(0) = \mathbb{C}$, yielding a well-defined vector $v_{L^\circ} \in \mathcal{H}(V)$.

Experimental results

Research questions

• RQ1 Can a canonical complex vector space be functorially associated to a symplectic vector space over a finite field of odd characteristic?
• RQ2 Does a strong form of the Stone–von Neumann theorem hold for the Heisenberg group over finite fields, ensuring uniqueness and naturality of the representation?
• RQ3 Can the Weil representation be realized as a canonical linear representation, rather than a projective one, via functorial quantization?
• RQ4 Is there a natural vector in $\mathcal{H}(V)$ associated to each oriented Lagrangian subspace $L^\circ = (L, o_L)$?
• RQ5 How does the quantization functor interact with symplectic reduction and duality?

Key findings

• The paper constructs a monoidal quantization functor $\mathcal{H}: \mathsf{Symp} \to \mathsf{Vect}$ that assigns to each symplectic vector space $V$ a complex vector space $\mathcal{H}(V)$, providing a canonical model for the Weil representation of $Sp(V)$.
• The strong Stone–von Neumann theorem for the Heisenberg group over $\mathbb{F}_q$ is proven, ensuring a unique irreducible representation with a fixed central character, which underpins the construction.
• For any oriented isotropic subspace $I^\circ = (I, o_I)$, there is a natural isomorphism $\mathcal{H}^I(V) \simeq \mathcal{H}(I^\perp / I)$, showing compatibility with symplectic reduction.
• When $I = L$ is a Lagrangian subspace, the invariant subspace $\mathcal{H}^L(V)$ is isomorphic to $\mathbb{C}$, yielding a canonical vector $v_{L^\circ} \in \mathcal{H}(V)$ for each oriented Lagrangian $L^\circ$.
• The functor $\mathcal{H}$ is compatible with duality, meaning $\mathcal{H}(V^*) \simeq \mathcal{H}(V)^*$ in a natural way.
• The sheaf-theoretic proof uses an $\ell$-adic perverse Weil sheaf $\mathcal{K}$ on the space of oriented Lagrangians, whose convolution structure realizes the quantization and proves the strong Stone–von Neumann property.