Notes on quantization of symplectic vector spaces over finite fields
aa r X i v : . [ m a t h - ph ] A p r Notes on Canonical Quantization of SymplecticVector Spaces over Finite Fields
Shamgar Gurevich ∗ and Ronny Hadani † ∗ Department of Mathematics, University of California, Berkeley, CA 94720, USA [email protected] † Department of Mathematics, University of Chicago, IL 60637, USA [email protected]
Summary.
In these notes we construct a quantization functor, associating a Hilbertspace 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 symplecticgroup 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 thecase of the Heisenberg group, a question of Kazhdan about the possible existence ofa canonical Hilbert space attached to a coadjoint orbit of a general unipotent groupover F q . Key words:
Quantization functor, Weil representation, Quantization of La-grangian correspondences, Geometric intertwining operators.
AMS codes:
Quantization is a fundamental procedure in mathematics and in physics. Al-though it is widely used in both contexts, its precise nature remains to someextent unclear. From the physical side, quantization is the procedure by whichone associates to a classical mechanical system its quantum counterpart. Fromthe mathematical side, it seems that quantization is a way to construct in-teresting Hilbert spaces out of symplectic manifolds, suggesting a method forconstructing representations of the corresponding groups of symplectomor-phisms [14, 16].Probably, one of the principal manifestation of quantization in mathemat-ics appears in the form of the Weil representation [19, 20, 22] of the metaplecticgroup ρ : M p (2 n, R ) → U (cid:0) L ( R n ) (cid:1) , S. Gurevich and R. Hadani where
M p (2 n, R ) is a double cover of the linear symplectic group Sp (2 n, R ).The general ideology [23, 24] suggests that the Weil representation appearsthrough a quantization of the standard symplectic vector space (cid:0) R n , ω (cid:1) . Thismeans that there should exist a quantization functor H , associating to a sym-plectic manifold ( M, ω ) an Hilbert space H ( M ), such that when applied to (cid:0) R n , ω (cid:1) it yields the Weil representation in the form of H : Sp (2 n, R ) → U (cid:0) H (cid:0) R n , ω (cid:1)(cid:1) . As stated, this ideology is too naive since it does not account for the meta-plectic cover.
In these notes, we show that the quantization ideology can be made pre-cise when applied in the setting of symplectic vector spaces over the finitefield F q , where q is odd. Specifically, we construct a quantization functor H : Symp → Hilb , where
Symp denotes the (groupoid) category whose objects arefinite dimensional symplectic vector spaces over F q and morphisms are linearisomorphisms of symplectic vector spaces and Hilb denotes the category offinite dimensional Hilbert spaces.As a consequence, for a fixed symplectic vector space V ∈ Symp , we obtain,by functoriality, a homomorphism H : Sp ( V ) → U ( H ( V )), which we referto as the canonical model of the Weil representation of the symplectic group Sp ( V ). Properties of the quantization functor
In addition, we show that the functor H satisfies the following basic properties(cf. [24]): • Compatibility with Cartesian products.
The functor H is a monoidalfunctor: Given V , V ∈ Symp , we have a natural isomorphism H ( V × V ) ≃ H ( V ) ⊗ H ( V ) . • Compatibility with duality.
Given V = ( V, ω ) ∈ Symp , its symplecticdual is V = ( V, − ω ). There exists a natural non-degenerate pairing h· , ·i V : H (cid:0) V (cid:1) × H ( V ) → C . • Compatibility with linear symplectic reduction . Given V ∈ Symp , I ⊂ V an isotropic subspace in V and o I ∈ ∧ top I a non-zero vector, thereexists a natural isomorphism H ( V ) I ≃ H (cid:0) I ⊥ /I (cid:1) , (1) otes on Quantization of Symplectic Vector Spaces 3 where H ( V ) I stands for the subspace of I -invariant vectors in H ( V ) (anoperation which will be made precise in the sequel) and I ⊥ /I ∈ Symp isthe symplectic reduction of V with respect to I [3]. (A pair ( I, o I ), where o I ∈ ∧ top I is non-zero vector, is called an oriented isotropic subspace ). Quantization of oriented Lagrangian subspaces . A particular situa-tion is when I = L is a Lagrangian subspace. In this situation, L ⊥ /L = 0 and(1) yields an isomorphism H ( V ) L ≃ H (0) = C , which associates to 1 ∈ C a vector v L ◦ ∈ H ( V ). This means that we establish a mechanism which as-sociates to every oriented Lagrangian subspace in V a well defined vector in H ( V ) L ◦ v L ◦ ∈ H ( V ) . Interestingly, to the best of our knowledge (cf. [9]), this kind of structure,which exists in the setting of the Weil representation of the group Sp ( V ) , was not observed before. Quantization of oriented Lagrangian correspondences.
It is alsointeresting to consider simultaneously the compatibility of H with Cartesianproduct, duality, and linear symplectic reduction. The first and second prop-erties imply that H (cid:0) V × V (cid:1) is naturally isomorphic to the vector spaceHom ( H ( V ) , H ( V )). The third property implies that every oriented La-grangian L ◦ in V × V (i.e., oriented canonical relation from V to V (cf.[23, 24])) can be quantized into a well defined operator L ◦ A L ◦ ∈ Hom ( H ( V ) , H ( V )) . In this regard, a particular kind of oriented Lagrangian in V × V is thegraph Γ g of a symplectic linear map g : V → V , g ∈ Sp ( V ). The orientationis automatic in this case - it is induced from ω ∧− n , dim( V ) = 2 n , throughthe isomorphism p V : Γ g → V , where p V : V × V → V is the projection on the V -coordinate.A further and more detailed study of these properties will appear in asubsequent work. The strong Stone-von Neumann theorem
The main technical result of these notes is a proof ([10, 11] unpublished)of a stronger form of the Stone-von Neumann theorem for the Heisenberggroup over F q . In this regard we describe an algebro-geometric object (an ℓ -adic perverse Weil sheaf K ), which, in particular, implies the strong Stone-von Neumann theorem. The construction of the sheaf K is one of the maincontributions of this work.Finally, we note that our result answers, for the case of the Heisenberggroup, a question of Kazhdan [13] about the possible existence of a canonical S. Gurevich and R. Hadani
Hilbert space attached to a coadjoint orbit of a general unipotent group over F q . We devote the rest of the introduction to an intuitive explanation of themain ideas and results of these notes. Let (
V, ω ) be a 2 n -dimensional symplectic vector space over the finite field F q ,assuming q is odd. The vector space V considered as an abelian group admitsa non-trivial central extension H , called the Heisenberg group , which can bepresented as H = V × F q with center Z = Z ( H ) = { (0 , z ) : z ∈ F q } . Thegroup Sp = Sp ( V ) acts on H by group automorphisms via its tautologicalaction on the V -coordinate.The celebrated Stone-von Neumann theorem [18, 21] asserts that givena non-trivial central character ψ : Z → C × , there exists a unique (up toisomorphism) irreducible representation π : H → GL ( H ) such that the centeracts by ψ , i.e., π | Z = ψ · Id H . The representation π is called the Heisenbergrepresentation.
Choosing a Lagrangian subvector space L ∈ Lag ( V ) (the set Lag ( V ) iscalled the Lagrangian Grassmanian ) we can define a model ( π L , H, H L ) ofthe Heisenberg representation, where H L consists of functions f : H → C satisfying f ( z · l · h ) = ψ ( z ) f ( h ) for every l ∈ L , z ∈ Z and the action π L isgiven by right translation. The problematic issue in this construction is thatthere is no preferred choice of a Lagrangian subspace L ∈ V and consequentlynone of the spaces H L admit an action of the group Sp . In fact, an element g ∈ Sp induces an isomorphism g : H L →H gL , (2)for every L ∈ Lag ( V ). The strong Stone-von Neumann theorem
The strategy that we will employ is: “If you can not choose a preferred La-grangian subspace then work with all of them simultaneously”.We can think of the system of models {H L } as a vector bundle H onLag with fibers H | L = H L , the condition (2) means that H is equipped withan Sp -equivariant structure and what we seek is a canonical trivialization of H . More formally, we seek for a canonical system of intertwining morphisms F M,L ∈ Hom H ( H L , H M ), for every L, M ∈ Lag ( V ). The existence of such asystem is the content of the strong Stone-von Neumann theorem. Theorem 1 (Strong Stone-von Neumann theorem).
There exists acanonical system of intertwining morphisms { F M,L ∈ Hom H ( H L , H M ) } sat-isfying the multiplicativity property F N,M ◦ F M,L = F N,L , for every
N, M, L ∈ Lag ( V ) . otes on Quantization of Symplectic Vector Spaces 5 Remark 1 (Important remark).
It is important to note here that the precisestatement involves the finer notion of an oriented Lagrangian subspace [1, 17],but for the sake of the introduction we will ignore this technical nuance.The Hilbert space H ( V ) consists of systems of vectors ( v L ∈ H L ) L ∈ Lagsuch that F M,L ( v L ) = v M , for every L, M ∈ Lag ( V ). The vector space H ( V )can be thought of as the space of horizontal sections of H .As it turns out, the symplectic group Sp naturally acts on H ( V ). Wedenote this representation by ( ρ V , Sp, H ( V )), and refer to it as the canonicalmodel of the Weil representation. We proceed to explain the main underlyingidea behind the construction of the system { F M,L } . The construction will be close in spirit to the procedure of “analytic contin-uation”. We consider the subset U ⊂ Lag ( V ) , consisting of pairs ( L, M ) ∈ Lag ( V ) which are in general position, that is L ∩ M = 0. The basic idea isthat for a pair ( L, M ) ∈ U , F M,L can be given by an explicit formula - ansatz .The main statement is that this formula admits a unique multiplicative ex-tension to the set of all pairs. The extension is constructed using algebraicgeometry.
Extension to singular pairs
It will be convenient to work in the setting of kernels.
In more detail, everyintertwining morphism F ∈ Hom H ( H L , H M ) can be presented by a kernelfunction K ∈ C ( H, ψ ) satisfying K ( m · h · l ) = K ( h ), for every m ∈ M and l ∈ L (we denote by C ( H, ψ ) the subspace of functions f ∈ C ( H ) which are ψ -equivariant with respect to the center, that is f ( z · h ) = ψ ( z ) f ( h ), for every z ∈ Z ). Moreover, this presentation is unique when ( M, L ) ∈ U ; hence, in thiscase, we have a unique kernel K M,L representing our given F M,L . If we denoteby O the set U × H , we see that the collection { K M,L : (
M, L ) ∈ U } formsa function K O ∈ C ( O ) given by K O ( M, L ) = K M,L for every (
M, L ) ∈ U .The problem is how to (correctly) extend the function K O to the set X =Lag ( V ) × H . In order to do that, we invoke the procedure of geometrization,which we briefly explain below. Geometrization
A general ideology due to Grothendieck is that any meaningful set-theoreticobject is governed by a more fundamental algebro-geometric one. The pro-cedure by which one translates from the set theoretic setting to algebraicgeometry is called geometrization , which is a formal procedure by which setsare replaced by algebraic varieties and functions are replaced by certain sheaf-theoretic objects.
S. Gurevich and R. Hadani
The precise setting consists of a set X = X ( F q ) of rational points of analgebraic variety X , defined over F q and a complex valued function f ∈ C ( X )governed by an ℓ -adic Weil sheaf F .The variety X is a space equipped with an automorphism F r : X → X (called Frobenius), such that the set X is naturally identified with the set offixed points X = X F r .The sheaf F can be considered as a vector bundle on the variety X ,equipped with an endomorphism θ : F → F which lifts
F r .The procedure by which f is obtained from F is called Grothendieck’s sheaf-to-function correspondence and it can be described, roughly, as follows.Given a point x ∈ X , the endomorphism θ restricts to an endomorphism θ x : F | x → F | x of the fiber F | x . The value of f on the point x is defined to be f ( x ) = Tr( θ x : F | x → F | x ) . The function defined by this procedure is denoted by f = f F . Solution to the extension problem
Our extension problem fits nicely into the geometrization setting: The sets
O, X are sets of rational points of corresponding algebraic varieties O , X , theimbedding j : O ֒ → X is induced from an open imbedding j : O ֒ → X and,finally, the function K O comes from a Weil sheaf K O on the variety O .The extension problem is solved as follows: First extend the sheaf K O to asheaf K on the variety X and then take the corresponding function K = f K ,which establishes the desired extension. The reasoning behind this strategyis that in the realm of sheaves there exist several functorial operations ofextension, probably the most interesting one is called perverse extension [2].The sheaf K is defined as the perverse extension of K O . Apart from the introduction, the notes consists of three sections.In Section 2, all basic constructions are introduced and main statementsare formulated. We begin with the definition of the Heisenberg group and theHeisenberg representation. Next, we introduce the canonical system of inter-twining morphisms between different models of the Heisenberg representationand formulate the strong Stone von-Neumann theorem (Theorem 3). We pro-ceed to explain how to present an intertwining morphism by a kernel function,and we reformulate the strong Stone von-Neumann theorem in the setting ofkernels (Theorem 4). Using Theorem 3, we construct a quantization functor H . We finish this section by showing that H is a monoidal functor and that itis compatible with duality and the operation of linear symplectic reduction.In section 3, we construct a sheaf theoretic counterpart for the canonical sys-tem of intertwining morphisms (Theorem 5). This sheaf is then used to proveTheorem 4. Finally, in Section 4 we sketch the proof of Theorem 5. Completeproofs for the statements appearing in these notes will appear elsewhere. otes on Quantization of Symplectic Vector Spaces 7 We would like to thank our scientific advisor J. Bernstein for his interest andguidance, and for his idea about the notion of oriented Lagrangian subspace.It is a pleasure to thank D. Kazhdan for sharing with us his thoughts aboutthe possible existence of canonical Hilbert spaces. We thank A. Weinstein forteaching us some of his ideas concerning quantization, and for the opportunityto present this work in the symplectic geometry seminar, Berkeley, February2007. We acknowledge O. Gabber for his remark concerning the characteri-zation of the Weil representation. We would like to acknowledge M. Vergnefor her encouragement. Finally, we would like to thank O. Ceyhan and theorganizers of the conference AGAQ, held in Istanbul during June 2006, forthe invitation to present this work.
Let (
V, ω ) be a 2 n -dimensional symplectic vector space over the finite field F q . Considering V as an abelian group, it admits a non-trivial central exten-sion called the Heisenberg group. Concretely, the group H = H ( V ) can bepresented as the set H = V × F q with the multiplication given by( v, z ) · ( v ′ , z ′ ) = ( v + v ′ , z + z ′ + 12 ω ( v, v ′ )) . The center of H is Z = Z ( H ) = { (0 , z ) : z ∈ F q } . The symplectic group Sp = Sp ( V ) acts by automorphism of H through its tautological action onthe V -coordinate. One of the most important attributes of the group H is that it admits, prin-cipally, a unique irreducible representation. We will call this property TheStone-von Neumann property (S-vN for short). The precise statement goes asfollows. Let ψ : Z → C × be a non-trivial character of the center. For examplewe can take ψ ( z ) = e πip tr ( z ) . It is not hard to show Theorem 2 (Stone-von Neumann property).
There exists a unique (upto isomorphism) irreducible unitary representation ( π, H, H ) with the centeracting by ψ, i.e., π | Z = ψ · Id H . The representation π which appears in the above theorem will be calledthe Heisenberg representation . S. Gurevich and R. Hadani
Although the representation π is unique, it admits a multitude of differentmodels (realizations); in fact this is one of its most interesting and powerfulattributes. These models appear in families. In this work we will be interestedin a particular family of such models which are associated with Lagrangiansubspaces in V .Let us denote by Lag = Lag ( V ) the set of Lagrangian subspaces in V . Let C ( H, ψ ) denote the subspace of functions f ∈ C ( H ), satisfying the equivari-ance property f ( z · h ) = ψ ( z ) f ( h ), for every z ∈ Z .Given a Lagrangian subspace L ∈ Lag, we can construct a model ( π L , H, H L )of the Heisenberg representation: The vector space H L consists of functions f ∈ C ( H, ψ ) satisfying f ( l · h ) = f ( h ), for every l ∈ L and the Heisenbergaction is given by right translation ( π L ( h ) ⊲ f ) ( h ′ ) = f ( h ′ · h ), for f ∈ H L . Definition 1.
An oriented Lagrangian L ◦ is a pair L ◦ = ( L, o L ) , where L isa Lagrangian subspace in V and o L is a non-zero vector in V top L . Let Lag ◦ = Lag ◦ ( V ) denote the set of oriented Lagrangian subspaces in V .We associate to each oriented Lagrangian subspace L ◦ , a model ( π L ◦ , H, H L ◦ )of the Heisenberg representation simply by forgetting the orientation, taking H L ◦ = H L and π L ◦ = π L . Sometimes, we will use a more informative notation H L ◦ = H L ◦ ( V ) or H L ◦ = H L ◦ ( V, ψ ). Canonical system of intertwining morphisms
Given a pair ( M ◦ , L ◦ ) ∈ Lag ◦ , the models H L ◦ and H M ◦ are isomorphic asrepresentations of H by Theorem 2, moreover, since the Heisenberg represen-tation is irreducible, the vector space Hom H ( H L ◦ , H ◦ M ) of intertwining mor-phisms is one-dimensional. Roughly, the strong Stone-von Neumann propertyasserts the existence of a distinguished element F M ◦ ,L ◦ ∈ Hom H ( H L ◦ , H ◦ M ),for every pair ( M ◦ , L ◦ ) ∈ Lag ◦ . The precise statement involves the followingdefinition: Definition 2.
A system { F M ◦ ,L ◦ ∈ Hom H ( H L ◦ , H ◦ M ) : ( M ◦ , L ◦ ) ∈ Lag ◦ } ofintertwining morphisms is called multiplicative if for every triple ( N ◦ , M ◦ , L ◦ ) ∈ Lag ◦ the following equation holds F N ◦ ,L ◦ = F N ◦ ,M ◦ ◦ F M ◦ ,L ◦ . We proceed as follows. Let U ⊂ Lag ◦ denote the set of pairs ( M ◦ , L ◦ ) ∈ Lag ◦ which are in general position, i.e., L ∩ M = 0. For ( M ◦ , L ◦ ) ∈ U , wedefine F M ◦ ,L ◦ by the following explicit formula: F M ◦ ,L ◦ = C M ◦ ,L ◦ · e F M,L , (3) otes on Quantization of Symplectic Vector Spaces 9 where e F M,L : H L ◦ → H M ◦ is the averaging morphism e F M,L [ f ] ( h ) = X m ∈ M f ( m · h ) , for every f ∈ H L ◦ and C M ◦ ,L ◦ is a normalization constant given by C M ◦ ,L ◦ = ( G /q ) n · σ (cid:16) ( − n ) ω ∧ ( o L , o M ) (cid:17) , where n = dim ( V )2 , σ is the unique quadratic character (also called the Legen-dre character) of the multiplicative group G m = F × q , G is the one-dimensionalGauss sum G = X z ∈ F q ψ (cid:18) z (cid:19) , and ω ∧ is the pairing ω ∧ : V top L N V top M → F q induced by the symplecticform. Theorem 3 (The strong Stone - von Neumann property).
There existsa unique system { F M ◦ ,L ◦ } of intertwining morphisms satisfying1. Restriction. For every pair ( M ◦ , L ◦ ) ∈ U , F M ◦ ,L ◦ is given by (3).2. Multiplicativity. For every triple ( N ◦ , M ◦ , L ◦ ) ∈ Lag ◦ , F N ◦ ,L ◦ = F N ◦ ,M ◦ ◦ F M ◦ ,L ◦ . Theorem 3 will follow from Theorem 4 below.Granting the existence and uniqueness of the system { F M ◦ ,L ◦ } , we canwrite F M ◦ ,L ◦ in a closed form, for a general pair ( M ◦ , L ◦ ) ∈ Lag ◦ . In orderto do that we need to fix some additional terminology.Let I = M ∩ L . We have canonical tensor product decompositions ^ top M = ^ top I O ^ top
M/I, ^ top L = ^ top I O ^ top
L/I.
In terms of the above decompositions, the orientation can be written inthe form o M = ι M ⊗ o M/I , o L = ι L ⊗ o L/I . Using the same notations as before,we denote by e F M,L : H L ◦ → H M ◦ the averaging morphism e F M,L [ f ] ( h ) = X m ∈ M/I f ( m · h ) , for f ∈ H L ◦ and by C M ◦ ,L ◦ the normalization constant C M ◦ ,L ◦ = ( G ) k · σ (cid:18) ( − k ) ι M ι L · ω ∧ (cid:0) o L/I , o
M/I (cid:1)(cid:19) , where k = dim ( I ⊥ /I )2 . Proposition 1.
For every ( M ◦ , L ◦ ) ∈ Lag ◦ F M ◦ ,L ◦ = C M ◦ ,L ◦ · e F M,L . An explicit way to present an intertwining morphism is via a kernel function.Fix a pair ( M ◦ , L ◦ ) ∈ Lag ◦ and let C ( M \ H/L, ψ ) denote the subspace offunctions f ∈ C ( H, ψ ) satisfying the equivariance property f ( m · h · l ) = f ( h )for every m ∈ M and l ∈ L . Given a function K ∈ C ( M \ H/L, ψ ), we canassociate to it an intertwining morphism I [ K ] ∈ Hom H ( H L ◦ , H ◦ M ) defined by I [ K ] ( f ) = K ∗ f = m ! ( K ⊠ Z · M f ) , for every f ∈ H L ◦ . Here, K ⊠ Z · L f denotes the function K ⊠ f ∈ C ( H × H ),factored to the quotient H × Z · L H and m ! denotes the operation of summationalong the fibers of the multiplication mapping m : H × H → H . The function K is called an intertwining kernel . The procedure just described defines alinear transform I : C ( M \ H/L, ψ ) −→ Hom H ( H L ◦ , H M ◦ ) . An easy verification reveals that I is surjective, but it is injective onlywhen M, L are in general position.Fix a triple ( N ◦ , M ◦ , L ◦ ) ∈ Lag ◦ . Given kernels K ∈ C ( N \ H/M, ψ ) and K ∈ C ( M \ H/L, ψ ), their convolution K ∗ K = m ! ( K ⊠ Z · M K ) lies in C ( N \ H/L, ψ ). The transform I sends convolution of kernels to compositionof operators I [ K ∗ K ] = I [ K ] ◦ I [ K ] . Canonical system of intertwining kernels
Below, we formulate a slightly stronger version of Theorem 3, in the settingof kernels.
Definition 3.
A system { K M ◦ ,L ◦ ∈ C ( M \ H/L, ψ ) : ( M ◦ , L ◦ ) ∈ Lag ◦ } ofkernels is called multiplicative if for every triple ( N ◦ , M ◦ , L ◦ ) ∈ Lag ◦ thefollowing equation holds K N ◦ ,L ◦ = K N ◦ ,M ◦ ∗ K M ◦ ,L ◦ A multiplicative system of kernels { K M ◦ ,L ◦ } can be equivalently thoughtof as a single function K ∈ C (cid:0) Lag ◦ × H (cid:1) , K ( M ◦ , L ◦ ) = K M ◦ ,L ◦ , satisfyingthe following multiplicativity relation on Lag ◦ × Hp ∗ K ∗ p ∗ K = p ∗ K, (4) otes on Quantization of Symplectic Vector Spaces 11 where p ij (( L ◦ , L ◦ , L ◦ ) , h ) = (cid:0)(cid:0) L ◦ i , L ◦ j (cid:1) , h (cid:1) are the projections on the i, j copies of Lag ◦ and the left-hand side of (4) means fiberwise convolution,namely p ∗ K ∗ p ∗ K ( L ◦ , L ◦ , L ◦ ) = K ( L ◦ , L ◦ ) ∗ K ( L ◦ , L ◦ ). To simplify no-tations, we will sometimes suppress the projections p ij from (4) obtaining amuch cleaner formula K ∗ K = K. We proceed along lines similar to Section 2.3. For every ( M ◦ , L ◦ ) ∈ U ,there exists a unique kernel K M ◦ ,L ◦ ∈ C ( M \ H/L, ψ ) such that F M ◦ ,L ◦ = I [ K M ◦ ,L ◦ ], which is given by the following explicit formula K M ◦ ,L ◦ = C M ◦ ,L ◦ · e K M ◦ ,L ◦ , (5)where e K M ◦ ,L ◦ = (cid:0) ι − (cid:1) ∗ ψ , ι = ι M ◦ ,L ◦ is the isomorphism given by the com-position Z ֒ → H ։ M \ H/L . The system { K M ◦ ,L ◦ : ( M ◦ , L ◦ ) ∈ U } yields awell defined function K U ∈ C ( U × H ). Theorem 4 (Canonical system of kernels).
There exists a unique func-tion K ∈ C (cid:0) Lag ◦ × H (cid:1) satisfying1. Restriction. K | U = K U .2. Multiplicativity. K ∗ K = K . We note that the proof of the uniqueness part in Theorem 4 is easy,it follows from the fact that for every pair N ◦ , L ◦ ∈ Lag ◦ one can find athird M ◦ ∈ Lag ◦ such that the pairs N ◦ , M ◦ and M ◦ , L ◦ are in general po-sition. Therefore, by the multiplicativity property (Property 2), K N ◦ ,L ◦ = K N ◦ ,M ◦ ∗ K M ◦ ,L ◦ . The proof of the existence part will be algebro-geometric(see Section 3). Finally, we note that Theorem 3 follows from Theorem 4 bytaking F M ◦ ,L ◦ = I [ K M ◦ ,L ◦ ]. Let us denote by
Symp the category whose objects are symplectic vector spacesover F q and morphisms are linear isomorphisms of symplectic vector spaces.Using the canonical system of intertwining morphisms { F M ◦ ,L ◦ } we can as-sociate, in a functorial manner, a vector space H ( V ) to a symplectic vectorspace V ∈ Symp . The construction proceeds as follows.Let Γ ( V ) denote the total vector space Γ ( V ) = M L ◦ ∈ Lag ◦ ( V ) H L ◦ , Define H ( V ) to be the subvector space of Γ ( V ) consisting of se-quences ( v L ◦ ∈ H L ◦ : L ◦ ∈ Lag ◦ ) satisfying F M ◦ ,L ◦ ( v L ◦ ) = v M ◦ for every( M ◦ , L ◦ ) ∈ Lag ◦ ( V ). We will call the vector space H ( V ) the canonical vec-tor space associated with V . Sometimes we will use the more informativenotation H ( V ) = H ( V, ψ ). Proposition 2 (Functoriality).
The rule V
7→ H ( V ) establishes a con-travariant (quantization) functor H : Symp −→ Vect , where Vect denote the category of finite dimensional complex vector spaces.
Considering a fixed symplectic vector space V , we obtain as a consequencea representation ( ρ V , Sp ( V ) , H ( V )), with ρ V ( g ) = H (cid:0) g − (cid:1) , for every g ∈ Sp ( V ). The representation ρ V is isomorphic to the Weil representation andwe call it the canonical model of the Weil representation. Remark 2.
The canonical model ρ V can be viewed from another perspective:We begin with the total vector space Γ and make the following two obser-vations. First observation is that the symplectic group Sp acts naturally on Γ , the action is of a geometric nature, i.e., induced from the diagonal ac-tion on Lag ◦ × H . Second observation is that the system { F M ◦ ,L ◦ } defines an Sp -invariant idempotent (total Fourier transform) F : Γ → Γ given by F ( v L ◦ ) = 1 ◦ ) M M ◦ ∈ Lag ◦ F M ◦ ,L ◦ ( v L ◦ ) , for every L ◦ ∈ Lag ◦ and v L ◦ ∈ H L ◦ . The situation is summarized in thefollowing diagram: Sp (cid:8) Γ (cid:9) F. The canonical model is given by the image of F , that is, H ( V ) = F Γ . The nicething about this point of view is that it shows a clear distinction between oper-ators associated with action of the symplectic group and operators associatedwith intertwining morphisms. Finally, we remark that one can also considerthe Sp -invariant idempotent F ⊥ = Id − F and the associated representation (cid:16) ρ ⊥ V , Sp, H ( V ) ⊥ (cid:17) , with H ( V ) ⊥ = F ⊥ Γ . The meaning of this representationis unclear. Compatibility with Cartesian products
The category
Symp admits a monoidal structure given by Cartesian productof symplectic vector spaces. The category
Vect admits the standard monoidalstructure given by tensor product. With respect to these monoidal structures,the functor H is a monoidal functor. Proposition 3.
There exists a natural isomorphism α V × V : H ( V × V ) →H ( V ) ⊗ H ( V ) , where V , V ∈ Symp . otes on Quantization of Symplectic Vector Spaces 13 As a result, we obtain the following compatibility condition between thecanonical models of the Weil representation α V × V : ( ρ V × V ) | Sp ( V ) × Sp ( V ) −→ ρ V ⊗ ρ V . (6) Remark 3 ([5]).
Condition (6) has an interesting consequence in case theground field is F . In this case, the group Sp ( V ) is not perfect whendim( V ) = 2, therefore, a priori, the Weil representation is not uniquely de-fined in this particular situation. However, since the group Sp ( V ) becomesperfect when dim( V ) >
2, the canonical model gives a natural choice for theWeil representation in the singular dimension, dim( V ) = 2. Compatibility with symplectic duality
Let V = ( V, ω ) ∈ Symp and let us denote by V = ( V, − ω ) the symplectic dualof V . Proposition 4.
There exists a natural non-degenerate pairing h· , ·i V : H (cid:0) V , ψ (cid:1) × H ( V, ψ ) → C , where V ∈ Symp . Compatibility with symplectic reduction
Let V ∈ Symp and let I be an isotropic subspace in V considered as an abeliansubgroup in H ( V ). On the one hand, we can associate to I the subspace H ( V ) I of I -invariant vectors. On the other hand, we can form the symplecticreduction I ⊥ /I and consider the vector space H (cid:0) I ⊥ /I (cid:1) (note that since I is isotropic then I ⊂ I ⊥ and I ⊥ /I is equipped with a natural symplecticstructure). Roughly, we claim that the vector spaces H (cid:0) I ⊥ /I (cid:1) and H ( V ) I arenaturally isomorphic. The precise statement involves the following definition Definition 4.
An oriented isotropic subspace in V is a pair I ◦ = ( I, o I ) ,where I ⊂ V is an isotropic subspace and o I is a non-trivial vector in V top I . Proposition 5.
There exists a natural isomorphism α ( I ◦ ,V ) : H ( V ) I →H (cid:0) I ⊥ /I (cid:1) , where, V ∈ Symp and I ◦ an oriented isotropic subspace in V . The naturalitycondition is H ( f I ) ◦ α ( J ◦ ,U ) = α ( I ◦ ,V ) ◦ H ( f ) , for every f ∈ Mor
Symp ( V, U ) such that f ( I ◦ ) = J ◦ and f I ∈ Mor
Symp (cid:0) I ⊥ /I, J ⊥ /J (cid:1) is the induced mor-phism. As a result we obtain another compatibility condition between the canon-ical models of the Weil representation. In order to see this, fix V ∈ Symp andlet I ◦ be an oriented isotropic subspace in V . Let P ⊂ Sp ( V ) be the sub-group of elements g ∈ Sp ( V ) such that g ( I ◦ ) = I ◦ . The isomorphism α ( I ◦ ,V ) establishes the following isomorphism: α ( I ◦ ,V ) : ( ρ V ) | P −→ ρ I ⊥ /I ◦ π, (7)where π : P → Sp (cid:0) I ⊥ /I (cid:1) is the canonical homomorphism. In this section we are going to prove Theorem 4, by constructing a geometriccounterpart to the set-theoretic system of intertwining kernels. This will beachieved using geometrization.
We denote by k an algebraic closure of F q . Next we have to take some space torecall notions and notations from algebraic geometry and the theory of ℓ -adicsheaves. Varieties
In the sequel, we are going to translate back and forth between algebraicvarieties defined over the finite field F q and their corresponding sets of rationalpoints. In order to prevent confusion between the two, we use bold-face lettersto denote a variety X and normal letters X to denote its corresponding setof rational points X = X ( F q ). For us, a variety X over the finite field is aquasi-projective algebraic variety, such that the defining equations are givenby homogeneous polynomials with coefficients in the finite field F q . In thissituation, there exists a (geometric) Frobenius endomorphism
F r : X → X ,which is a morphism of algebraic varieties. We denote by X the set of pointsfixed by F r , i.e., X = X ( F q ) = X F r = { x ∈ X : F r ( x ) = x } . The category of algebraic varieties over F q will be denoted by Var F q . Sheaves
Let D b ( X ) denote the bounded derived category of constructible ℓ -adic sheaveson X [2, 4]. We denote by Perv ( X ) the Abelian category of perverse sheaves onthe variety X , i.e., the heart with respect to the autodual perverse t-structure otes on Quantization of Symplectic Vector Spaces 15 in D b ( X ). An object F ∈ D b ( X ) is called n -perverse if F [ n ] ∈ Perv ( X ). Finally,we recall the notion of a Weil structure (Frobenius structure) [4]. A Weilstructure associated to an object F ∈ D b ( X ) is an isomorphism θ : F r ∗ F−→F . A pair ( F , θ ) is called a Weil object. By an abuse of notation we oftendenote θ also by F r . We choose once an identification Q ℓ ≃ C , hence allsheaves are considered over the complex numbers. Remark 4.
All the results in this section make perfect sense over the field Q ℓ ,in this respect the identification of Q ℓ with C is redundant. The reason it isspecified is in order to relate our results with the standard constructions ofthe Weil representation [7, 12].Given a Weil object ( F , F r ∗ F ≃ F ) one can associate to it a function f F : X → C to F as follows f F ( x ) = X i ( − i Tr(
F r | H i ( F x ) ) . This procedure is called
Grothendieck’s sheaf-to-function correspondence .Another common notation for the function f F is χ F r ( F ), which is called the Euler characteristic of the sheaf F . We shall now start the geometrization procedure.
Replacing sets by varieties
The first step we take is to replace all sets involved by their geometric coun-terparts, i.e., algebraic varieties. The symplectic space (
V, ω ) is naturally iden-tified as the set V = V ( F q ), where V is a 2 n -dimensional symplectic vectorspace in Var F q . The Heisenberg group H is naturally identified as the set H = H ( F q ), where H = V × A is the corresponding group variety. Finally,Lag ◦ = Lag ◦ ( F q ), where Lag ◦ is the variety of oriented Lagrangians in V . Replacing functions by sheaves
The second step is to replace functions by their sheaf-theoretic counterparts[6]. The additive character ψ : F q −→ C × is associated via the sheaf-to-function correspondence to the Artin-Schreier sheaf L ψ living on A , i.e., wehave f L ψ = ψ. The Legendre character σ on F × q ≃ G m ( F q ) is associated to theKummer sheaf L σ on G m . The one-dimensional Gauss sum G is associatedwith the Weil object G = Z A L ψ ( z ) ∈ D b ( pt ) , where, for the rest of these notes, R = R ! denotes integration with compactsupport [2]. Grothendieck’s Lefschetz trace formula [8] implies that, indeed, f G = G . In fact, there exists a quasi-isomorphism G −→ H ( G )[ −
1] anddim H ( G ) =1, hence, G can be thought of as a one-dimensional vector space,equipped with a Frobenius operator, sitting at cohomological degree 1 . Our main objective, in this section, is to construct a multiplicative systemof kernels K : Lag ◦ × H −→ C extending the subsystem K U (see 2.4). Theextension appears as a direct consequence of the following geometrizationtheorem: Theorem 5 (Geometric kernel sheaf ).
There exists a geometrically ir-reducible [dim(
Lag ◦ ) + n + ] -perverse Weil sheaf K on Lag ◦ × H of pureweight w ( K ) = 0 , satisfying the following properties:1. Multiplicativity property. There exists an isomorphism K ≃ K ∗ K .
2. Function property. We have f K| U = K U . For a proof, see Section 4.
Proof of Theorem 4
Let K = f K . Invoking Theorem 5, we obtain that K is multiplicative (Prop-erty 1) and extends K U (Property 2). Hence, we see that K satisfies theconditions of Theorem 4. The nice thing about this construction is that ituses geometry and, in particular, the notion of perverse extension which hasno counterpart in the set-function theoretic setting. Section 4 is devoted to sketching the proof of Theorem 5.
The construction of the sheaf K is based on formula (5). Let U ⊂ Lag ◦ bethe open subvariety consisting of pairs ( M ◦ , L ◦ ) ∈ Lag ◦ in general position.The construction proceeds as follows: • Non-normalized kernel.
On the variety U × H define the sheaf e K U ( M ◦ , L ◦ ) = (cid:0) ι − (cid:1) ∗ L ψ , where ι = ι M ◦ ,L ◦ is the composition Z ֒ → H ։ M \ H / L . otes on Quantization of Symplectic Vector Spaces 17 • Normalization coefficient.
On the open subvariety U × H define the sheaf C ( M ◦ , L ◦ ) = G ⊗ n ⊗ L σ (cid:16) ( − n ) ω ∧ ( o L , o M ) (cid:17) [2 n ] ( n ) . (8) • Normalized kernels.
On the open subvariety U × H define the sheaf K U = C⊗ e K U . Finally, take K = j ! ∗ K U , (9)where j : U × H ֒ → Lag ◦ × H is the open imbedding, and j ! ∗ is the functorof perverse extension [2] (in our setting, j ! ∗ might better be called irreducibleextension, since the sheaves we consider are not perverse but perverse up to acohomological shift). It follows directly from the construction that the sheaf K is irreducible [dim( Lag ◦ ) + n + 1]-perverse of pure weight 0.The function property (Property 2) is clear from the construction. We areleft to prove the multiplicativity property (Property 2). We need to show that p ∗ K ≃ p ∗ K∗ p ∗ K , (10)where p ij : Lag ◦ × H → Lag ◦ × H are the projectors on the i, j copiesof Lag ◦ . We will need the following notations. Let U ⊂ Lag ◦ denote theopen subvariety consisting of triples ( L ◦ , L ◦ , L ◦ ) which are in general positionpairwisely. Let n k = dim( Lag ◦ k ) + n + 1. Lemma 1.
There exists, on U × H , an isomorphism p ∗ K ≃ p ∗ K∗ p ∗ K . Let V ⊂ Lag ◦ be the open subvariety consisting of triples ( L ◦ , L ◦ , L ◦ ) ∈ Lag ◦ such that L ◦ , L ◦ and L ◦ , L ◦ are in general position. Lemma 1 admitsa slightly stronger form. Lemma 2.
There exists, on V × H , an isomorphism p ∗ K ≃ p ∗ K∗ p ∗ K . We can now finish the proof of (10). Lemma 1 implies that the sheaves p ∗ K and p ∗ K∗ p ∗ K are isomorphic on the open subvariety U × H . The sheaf p ∗ K is irreducible [ n ]-perverse as a pullback by a smooth, surjective withconnected fibers morphism, of an irreducible [ n ]-perverse sheaf on Lag ◦ × H . Hence, it is enough to show that the sheaf p ∗ K∗ p ∗ K irreducible [ n ]-perverse. Let V ⊂ Lag ◦ be the open subvariety consisting of quadruples( L ◦ , L ◦ , L ◦ , L ◦ ) ∈ Lag ◦ such that the pairs L ◦ , L ◦ and L ◦ , L ◦ are in general position. Consider the projection p : V × H → Lag ◦ × H , it is clearlysmooth and surjective, with connected fibers. It is enough to show that thepull-back p ∗ ( p ∗ K∗ p ∗ K ) is irreducible [ n ]-perverse. Using Lemma 2 andalso invoking some direct diagram chasing one obtains p ∗ ( p ∗ K∗ p ∗ K ) ≃ p ∗ K ∗ p ∗ K ∗ p ∗ K . (11)The right-hand side of (11) is principally a subsequent application of aproperly normalized, Fourier transforms on p ∗ K , hence by the Katz-Laumontheorem [15] it is irreducible [ n ]-perverse.Let us summarize. We showed that both sheaves p ∗ K and p ∗ K∗ p ∗ K are irreducible [ n ]-perverse and are isomorphic on an open subvariety. Thisimplies that they must be isomorphic. This concludes the proof of the multi-plicativity property. References
1. Bernstein J.,
Private communication, Max-Planck Institute, Bonn, Germany (August, 2004).2. Beilinson A., Bernstein J. and Deligne P. Faisceaux pervers.
Analysis and topol-ogy on singular spaces, I, Asterisque, 100, Soc. Math. France, Paris (1982),5–171.3. Bates S, and Weinstein A., Lectures on the geometry of quantization.
BerkeleyMathematics Lecture Notes, 8. American Mathematical Society, Providence, RI (1997).4. Deligne P., La conjecture de Weil II.
Publ. Math. I.H.E.S 52 (1981), 313-428.5. Gabber O., Private communication (2006).6. Gaitsgory D., Informal introduction to geometric Langlands.
An introductionto the Langlands program, Jerusalem 2001 , Birkh¨auser, Boston, MA (2003)269-281.7. G´erardin P., Weil representations associated to finite fields.
J. Algebra 46 , no.1, (1977), 54-101.8. Grothendieck A., Formule de Lefschetz et rationalit´e des fonctions L . SeminaireBourbaki, Vol. 9, Exp. No. 279 (1964).9. Guillemin V. and Sternberg S., Some problems in integral geometry and somerelated problems in microlocal analysis.
Amer. J. Math. 101 (1979) 915-955.10. Gurevich S. and Hadani R., Heisenberg Realizations, Eigenfunctions andProof of the Kurlberg-Rudnick Supremum Conjecture. arXiv:math-ph/0511036 (2005).11. Hadani R., The Geometric Weil Representation and some applications,
Ph.D.Thesis, Tel-Aviv University (June, 2006).12. Howe R., On the character of Weil’s representation.
Trans. Amer. Math. Soc.177 (1973), 287–298.13. Kazhdan D.,
Private communication, Hebrew University, Jerusalem, Israel (March, 2004).14. Kirillov, A. A., Unitary representations of nilpotent Lie groups.
Uspehi Mat.Nauk 17 (1962) 67-110.otes on Quantization of Symplectic Vector Spaces 1915. Katz N. M. and Laumon G., Transformation de Fouri´er et majoration desommes exponenti`elles.
Inst. Hautes Etudes Sci. Publ. Math. No. 62 (1985),361–418.16. Kostant B., Quantization and Unitary Representations.
Lecture Notes in Math.170, Springer-Verlag (1970), 87-207.17. Lion G. and Vergne M., The Weil representation, Maslov index and theta series.
Progress in Mathematics, 6. Birkh¨auser, Boston, Mass (1980).18. Neumann J. von., Die Eindeutigkeit der Schr¨odingerschen Operationen.
Math.Ann. 104. (
Math. Scand., vol. 13 (1963) 31-43.20. Shale D., Linear symmetries of free boson fields.
Trans. Amer. Math. Soc. 103 (1962), 149–167.21. Stone M. H., Linear transformations in Hilbert space, III: operational methodsand group theory. Proc. Nat. Acad. Sci. U.S.A. 16 (1930) 172-175.22. Weil A., Sur certains groupes d’operateurs unitaires.
Acta Math. 111 (1964)143-211.23. Weinstein A., Symplectic geometry.
Bull. Amer. Math. Soc. (N.S.) 5, no. 1(