Notes on the Self-Reducibility of the Weil Representation and Higher-Dimensional Quantum Chaos
aa r X i v : . [ m a t h - ph ] A p r Notes on the Self-Reducibility of the WeilRepresentation and Higher-DimensionalQuantum Chaos
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 discuss the self-reducibility property of the Weil repre-sentation. We explain how to use this property to obtain sharp estimates of certainhigher-dimensional exponential sums which originate from the theory of quantumchaos. As a result, we obtain the Hecke quantum unique ergodicity theorem for ageneric linear symplectomorphism A of the torus T = R N / Z N . Key words:
Hannay–Berry Model, Quantum Unique Ergodicity, Bounds onExponential Sums, Weil Representation, Self-Reducibility.
AMS codes:
In his celebrated 1964 Acta paper [34] Weil constructed a certain (projective)unitary representation of a symplectic group over a local field k (for example k could be R , C , or a p -adic field). This representation has many fascinat-ing properties which have gradually been brought to light over the last fewdecades. It now appears that this representation is a central object, bridg-ing various topics in mathematics and physics, including number theory, thetheory of theta functions and automorphic forms, invariant theory, harmonicanalysis, and quantum mechanics. Although it holds such a fundamental sta-tus, it is satisfying to observe that the Weil representation already appears inthe study of functions on linear spaces. Given a k -linear space L , there existsan associated (polarized) symplectic vector space V = L × L ∗ . The Weil rep-resentation of the group Sp = Sp ( V, ω ) can be realized on the Hilbert space
S. Gurevich and R. Hadani H = L ( L, C ) . Interestingly, some elements of the group Sp act by certainkinds of generalized Fourier transforms. In particular, there exists a specificelement w ∈ Sp (called the Weyl element) whose action is given, up to anormalization, by the standard Fourier transform. From this perspective, theclassical theory of harmonic analysis seems to be devoted to the study of aparticular operator in the Weil representation.In these notes we will be concerned only with the case of the Weil repre-sentations of symplectic groups over finite fields. The main technical part isdevoted to the study of a specific property of the Weil representation—the self-reducibility property . Briefly, this is a property concerning a relationshipbetween the Weil representations of symplectic groups of different dimensions.In parts of these notes we devoted some effort to developing a general theory.In particular, the results concerning the self-reducibility property apply alsoto the Weil representation over local fields.We use the self-reducibility property to bound certain higher-dimensionalexponential sums which originate from the theory of quantum chaos, therebyobtaining a proof of one of the main statements in the field—the Hecke quan-tum unique ergodicity theorem for a generic linear symplectomorphism of the2 N -dimensional torus. One of the main motivational problems in quantum chaos is [2, 3, 26, 30]describing eigenstates e HΨ = λΨ, Ψ ∈ H , of a chaotic Hamiltonian e H = Op ( H ) : H → H , where H is a Hilbert space. We deliberately use the notation Op ( H ) to empha-size the fact that the quantum Hamiltonian e H is a quantization of a classicalHamiltonian H : M → C , where M is a classical symplectic phase space (usu-ally the cotangent bundle of a configuration space M = T ∗ X , in which case H = L ( X )). In general, describing Ψ is considered to be an extremely com-plicated problem. Nevertheless, for a few mathematical models of quantummechanics rigorous results have been obtained. We shall proceed to describeone of these models. Hannay–Berry model
In [18] Hannay and Berry explored a model for quantum mechanics on the two-dimensional symplectic torus ( T , ω ). Hannay and Berry suggested to quantizesimultaneously the functions on the torus and the linear symplectic group Γ ≃ SL ( Z ). One of their main motivations was to study the phenomenon elf-Reducibility and Quantum Chaos 3 of quantum chaos in this model [26, 28]. More precisely, they considered anergodic discrete dynamical system on the torus which is generated by a hyper-bolic automorphism A ∈ Γ . Quantizing the system, the classical phase space( T , ω ) is replaced by a finite dimensional Hilbert space H , classical observ-ables, i.e., functions f ∈ C ∞ ( T ), by operators π ( f ) ∈ End( H ), and classicalsymmetries by a unitary representation ρ : Γ → U ( H ). Shnirelman’s theorem
Analogous with the case of the Schr¨odinger equation, consider the followingeigenstates problem ρ ( A ) Ψ = λΨ. A fundamental result, valid for a wide class of quantum systems which areassociated to ergodic classical dynamics, is Shnirelman’s theorem [31], assert-ing that in the semi-classical limit almost all (in a suitable sense) eigenstatesbecome equidistributed in an appropriate sense.A variant of Shnirelman’s theorem also holds in our situation [4]. Moreprecisely, we have that in the semi-classical limit ℏ → Ψ of the operator ρ ( A ) the corresponding Wignerdistribution h Ψ | π ( · ) Ψ i : C ∞ ( T ) → C approaches the phase space average R T ·| ω | . In this respect, it seems natural to ask whether there exist excep-tional sequences of eigenstates? Namely, eigenstates that do not obey theShnirelman’s rule ( scarred eigenstates). It was predicted by Berry [2, 3] that scarring phenomenon is not expected to be seen for quantum systems asso-ciated with generic chaotic classical dynamics. However, in our situation theoperator ρ ( A ) is not generic, and exceptional eigenstates were constructed.Indeed, it was confirmed mathematically in [8] that certain ρ ( A )-eigenstatesmight localize. For example, in that paper a sequence of eigenstates Ψ wasconstructed, for which the corresponding Wigner distribution approaches themeasure δ + | ω | on T . Hecke quantum unique ergodicity
A quantum system that obeys Shnirelman’s rule is also called quantum er-godic. Can one impose some natural conditions on the eigenstates so that noexceptional eigenstates will appear? Namely, quantum unique ergodicity willhold. This question was addressed in a paper by Kurlberg and Rudnick [25].In that paper, they formulated a rigorous notion of Hecke quantum unique er-godicity for the case ℏ = 1 /p . The following is a brief description of that work.The basic observation is that the degeneracies of the operator ρ ( A ) are cou-pled with the existence of symmetries. There exists a commutative group ofoperators that commutes with ρ ( A ), which can in fact be computed. In moredetail, the representation ρ factors through the quotient group Sp = SL ( F p ).We denote by T A ⊂ Sp the centralizer of the element A , now considered as S. Gurevich and R. Hadani an element of the quotient group. The group T A is called (cf. [25]) the Hecketorus corresponding to the element A . The Hecke torus acts semisimply on H . Therefore, we have a decomposition H = M χ : T A → C × H χ , where H χ is the Hecke eigenspace corresponding to the character χ . Considera unit eigenstate Ψ ∈ H χ and the corresponding Wigner distribution W χ : C ∞ ( T ) → C , defined by the formula W χ ( f ) = h Ψ | π ( f ) Ψ i . The main statementin [25] proves an explicit bound on the semi-classical asymptotic of W χ ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W χ ( f ) − Z T f | ω | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C f p / , where C f is a constant that depends only on the function f . In Rudnick’slectures at MSRI, Berkeley 1999 [27], and ECM, Barcelona 2000 [28], he con-jectured that a stronger bound should hold true, i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W χ ( f ) − Z T f | ω | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C f p / . (1)A particular case (which implies (1)) of the above inequality is when f = ξ ,where ξ is a non-trivial character. In this case, the integral R T ξ | ω | vanishesand in addition it turns out that C ξ = 2 + o (1). Hence, we obtain the followingsimplified form of (1) |W χ ( ξ ) | ≤ o (1) √ p , (2)for sufficiently large p. These stronger bounds were proved in the paper [13].It will be instructive to briefly recall the main ideas and techniques used in[13].
Geometric approach
The basic observation to be made is that the theory of quantum mechanics onthe torus, in the case ℏ = 1 /p , can be equivalently recast in the language ofthe representation theory of finite groups in characteristic p . We will endeavorto give a more precise explanation of this matter. Consider the quotient F p -vector space V = T ∨ /p T ∨ , where T ∨ ≃ Z is the lattice of characters on T .We denote by H = H ( V ) the Heisenberg group associated to V . The group Sp is naturally identified with the group of linear symplectomorphisms of V . Wehave an action of Sp on H . The Stone–von Neumann theorem (see Theorem 5)states that there exists a unique irreducible representation π : H → GL ( H ),with a non-trivial character ψ of the center of H . As a consequence of its elf-Reducibility and Quantum Chaos 5 uniqueness, its isomorphism class is fixed by Sp . This is equivalent to sayingthat H is equipped with a compatible projective representation ρ : Sp → P GL ( H ), which in fact can be linearized to an honest representation. Thisrepresentation is the celebrated Weil representation. Noting that Sp is thegroup of rational points of the algebraic group Sp (we use boldface lettersto denote algebraic varieties), it is natural to ask whether there exists analgebro-geometric object that underlies the representation ρ . The answer tothis question is positive. The construction is proposed in an unpublished letterof Deligne to Kazhdan [7], which appears now in [13, 16]. Briefly, the contentof this letter is a construction of representation sheaf K ρ on the algebraicvariety Sp . We obtain, as a consequence, the following general principle: Motivic principle.
All quantum mechanical quantities in the Hannay–Berrymodel are motivic in nature.
By this we mean that every quantum-mechanical quantity Q is associatedwith a vector space V Q (certain cohomology of a suitable ℓ -adic sheaf) endowedwith a Frobenius action Fr : V Q → V Q so that Q =Tr(Fr | V Q ) . In particular,it was shown in [13] that there exists a two-dimensional vector space V χ ,endowed with an action Fr : V χ → V χ , so that W χ ( ξ ) = Tr(Fr | V χ ) . (3)This, combined with the purity condition that the eigenvalues of Fr are ofabsolute value 1 / √ p, implies the estimate (2). The higher-dimensional Hannay–Berry model
The higher-dimensional Hannay–Berry model is obtained as a quantizationof a 2 N -dimensional symplectic torus ( T , ω ) acted upon by the group Γ ≃ Sp (2 N, Z ) of linear symplectic automorphisms. It was first constructed in[12], where, in particular, a quantization of the whole group of symmetries Γ was obtained. Consider a regular ergodic element A ∈ Γ , i.e., A generatesan ergodic discrete dynamical system and it is regular in the sense that ithas distinct eigenvalues over C . It is natural to ask whether quantum uniqueergodicity will hold true in this setting as well, as long as one takes intoaccount the whole group of hidden (Hecke) symmetries? Interestingly, theanswer to this question is NO! Several new results in this direction have beenannounced recently. In the case where the automorphism A is non-generic ,meaning that it has an invariant Lagrangian (and more generally co-isotropic)sub-torus T L ⊂ T , an interesting new phenomenon was revealed. There existsa sequence { Ψ ℏ } of Hecke eigenstates which are closely related to the physicalphenomena of localization , known in the physics literature (cf. [20, 24]) as scars . We will call them Hecke scars . These states are localized in the sensethat the associated Wigner distribution converges to the Haar measure µ onthe invariant Lagrangian sub-torus S. Gurevich and R. Hadani W Ψ ℏ ( f ) → Z T L f dµ, as ℏ → , (4)for every smooth observable f . These special kinds of Hecke eigenstates werefirst established in [10]. The semi-classical interpretation of the localizationphenomena (4) was announced in [23].The above phenomenon motivates the following definition: Definition 1.
An element A ∈ Γ is called generic if it is regular and admitsno non-trivial invariant co-isotropic sub-tori.Remark 1. The collection of generic elements constitutes an open subschemeof Γ . In particular, a generic element need not be ergodic automorphism of T . However, in the case where Γ ≃ SL ( Z ) every ergodic (i.e., hyperbolic)element is generic. An example of a generic element which is not ergodic isgiven by the Weyl element w = (cid:18) − (cid:19) .In these notes we will require the automorphism A ∈ Γ to be generic. Thiscase was first considered in [14], where using similar geometric techniques asin [13] the analogue of inequality (2) was obtained. For the sake of simplicity,let us assume that the automorphism A is strongly generic , i.e., it has nonon-trivial invariant sub-tori. Theorem 1 ([14]).
Let ξ be a non-trivial character of T . The following boundholds |W χ ( ξ ) | ≤ [2 + o (1)] N √ p N , (5) where p is a sufficiently large prime number. In particular, using the bound (5), we obtain the following statement forgeneral observable:
Corollary 1 (Hecke quantum unique ergodicity).
Consider an observ-able f ∈ C ∞ ( T ) and a sufficiently large prime number p. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W χ ( f ) − Z T f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C f √ p N , where µ = | ω | N is the corresponding volume form and C f is an explicit com-putable constant which depends only on the function f. In these notes, using the self-reducibility property of the Weil represen-tation, we improve the above estimates and obtain the following theorem: elf-Reducibility and Quantum Chaos 7
Theorem 2 (Sharp bound).
Let ξ be a non-trivial character of T . For suf-ficiently large prime number p we have |W χ ( ξ ) | ≤ [2 + o (1)] r p √ p N , (6) where the number r p is an integer between and N , that we will call thesymplectic rank of T A .Remark 2. It will be shown (see Subsection 6.2) that the distribution of thesymplectic rank r p (6) in the set { , ..., N } is governed by the Chebotarevdensity theorem applied to a suitable Galois group. For example, in the casewhere A ∈ Sp (4 , Z ) is strongly generic we havelim x →∞ { r p = r | p ≤ x } π ( x ) = , r = 1 , , where π ( x ) denotes the number of primes up to x . Remark 3.
For the more general version of Theorem 2, one that holds in thegeneral generic case (Definition 1), see Subsection 6.3.In order to witness the improvement of (6) over (5), it would be instructiveto consider the following extreme scenario. Assume that the Hecke torus T A acts on V ≃ F Np irreducibly. In this case it turns out that r p = 1. Hence, (6)becomes |W χ ( ξ ) | ≤ o (1) √ p N , which constitutes a significant improvement over the coarse topological es-timate (5). Let us elaborate on this. Recall the motivic interpretation (3)of the Wigner distribution. In [14] an analogous interpretation was givento the higher-dimensional Wigner distributions, realizing them as W χ ( ξ ) =Tr(Fr | V χ ) , where, by the purity condition, the eigenvalues of Fr are of absolutevalue 1 / √ p N . But, in this setting the dimension of V χ is not 2, but 2 N , i.e.,the Frobenius looks like Fr = λ ∗ ∗ ∗· ∗ ∗· ∗ λ N . Hence, if we use only this amount of information, then the best estimate whichcan be obtained is (5). Therefore, in this respect the problem that we con-front is showing cancellations between different eigenvalues, more preciselyangles, of the Frobenius operator acting on a high-dimensional vector space,i.e., cancellations in the sum P N j =1 e iθ j , where the angles 0 ≤ θ j < π are S. Gurevich and R. Hadani defined via λ j = e iθ j / √ p N . This problem is of a completely different nature,which is not accounted for by standard cohomological techniques (we thankR. Heath-Brown for pointing out to us [19] about the phenomenon of can-cellations between Frobenius eigenvalues in the presence of high-dimensionalcohomologies).
Remark 4.
Choosing a realization
H ≃ C ( F Np ), the matrix coefficient W χ ( ξ )is equivalent to an exponential sum of the form h Ψ | π ( ξ ) Ψ i = X x ∈ F Np Ψ ( x ) e πip ξ + x Ψ ( x + ξ − ) . (7)Here one encounters two problems. First, it is not so easy to describe theeigenstates Ψ . Second, the sum (7) is a high-dimensional exponential sum(over F p ), which is known to be hard to analyze using standard techniques.The crucial point that we explain in these notes is that it can be realized,essentially, as a one-dimensional exponential sum over F q , where q = p N . Let us explain the main idea underlying the proof of estimate (6). Let usassume for the sake of simplicity that the Hecke torus is completely inert, i.e.,acts irreducibly on the vector space V ≃ F Np . Representation theoretic interpretation of the Wigner distribution
The Hecke eigenstate Ψ is a vector in a representation space H . The space H supports the Weil representation of the symplectic group Sp ≃ Sp (2 N, k ), k = F p . The vector Ψ is completely characterized in representation theoreticterms, as being a character vector of the Hecke torus T A . As a consequence,all quantities associated to Ψ , and in particular the Wigner distribution W χ are characterized in terms of the Weil representation. The main observationto be made is that the Hecke state Ψ can be characterized in terms of anotherWeil representation, this time of a group of much smaller dimension. In fact, itcan be characterized, roughly, in terms of the Weil representation of SL ( K ) ,K = F p N . Self-reducibility property
A fundamental notion in our study is that of a symplectic module structure .A symplectic module structure is a triple (
K, V, ω ), where K is a finite di-mensional commutative algebra over k, equipped with an action on the vec-tor space V , and ω is a K -linear symplectic form satisfying the propertyTr K/k ( ω ) = ω. Let us assume for the sake of simplicity that K is a field. Let elf-Reducibility and Quantum Chaos 9 Sp = Sp ( V, ω ) be the group of K -linear symplectomorphisms with respect tothe form ω . There exists a canonical embedding ι : Sp ֒ → Sp. (8)We will be mainly concerned with symplectic module structures whichare associated to maximal tori in Sp . More precisely, it will be shown thatassociated to a maximal torus T ⊂ Sp there exists a canonical symplecticmodule structure ( K, V, ω ) so that T ⊂ Sp . The most extreme situation iswhen the torus T ⊂ Sp is completely inert, i.e., acts irreducibly on the vectorspace V . In this particular case, the algebra K is in fact a field with dim K V =2 which implies that Sp ≃ SL ( K ), i.e., using (8) we get T ⊂ SL ( K ) ⊂ Sp.
Let us denote by ( ρ, Sp, H ) the Weil representation of Sp . The main ob-servation now is (cf. [9]) the following: Theorem 3 (Self-reducibility property).
The restricted representation ( ρ = ι ∗ ρ, SL ( K ) , H ) is the Weil representation of SL ( K ) . Applying the self-reducibility property to the Hecke torus T A , it followsthat the Hecke eigenstates Ψ can be characterized in terms of the Weil repre-sentation of SL ( K ). Therefore, in this respect, Theorem 2 is reduced to theresult obtained in [13]. Let A ∈ Γ be a generic linear symplectomorphism. As in harmonic analysis,one would like to use Theorem 2 concerning the Hecke eigenstates in order toextract information on the spectral theory of the operator ρ ( A ) itself. For thesake of simplicity, let us assume that A is strongly generic, i.e., it acts on thetorus T with no non-trivial invariant sub-tori. Next, a possible reformulationof the quantum unique ergodicity statement, one which is formulated for theautomorphism A itself instead of the Hecke group of symmetries, is presented.The element A acts via the Weil representation ρ on the space H anddecomposes it into a direct sum of ρ ( A )-eigenspaces H = M λ ∈ Spec ( ρ ( A )) H λ . (9)Considering an ρ ( A )-eigenstate Ψ and the corresponding projector P Ψ one usually studies the Wigner distribution h Ψ | π ( ξ ) Ψ i = Tr( π ( ξ ) P Ψ ) which,due to the fact that rank ( P Ψ ) = 1 , is sometimes called pure state. In thesame way, we might think about an Hecke–Wigner distribution h Ψ | π ( ξ ) Ψ i =Tr( π ( ξ ) P χ ) , attached to a T A -eigenstate Ψ , as a pure Hecke state . Followingvon Neumann [33] we suggest the possibility of looking at the more general statistical state , defined by a non-negative, self-adjoint operator D , called thevon Neumann density operator, normalized to have Tr( D ) = 1 . For example, to the automorphism A we can attach the natural family of density operators D λ = m λ P λ, where P λ is the projector on the eigenspace H λ (9) , and m λ =dim( H λ ) . Consequently, we obtain a family of statistical states W λ ( · ) = Tr( π ( · ) D λ ) . Theorem 4.
Let ξ be a non-trivial character of T . For a sufficiently largeprime number p , and every statistical state W λ , we have |W λ ( ξ ) | ≤ (2 + o (1)) r p √ p N , (10) where r p is an explicit integer ≤ r p ≤ N which is determined by A. Theorem 4 follows from the fact that the Hecke torus T A acts on thespaces H λ , and hence, one can use the Hecke eigenstates, and the bound(6). In particular, using (10) we obtain for a general observable the followingbound: Corollary 2.
Consider an observable f ∈ C ∞ ( T ) and a sufficiently largeprime number p. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W λ ( f ) − Z T f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C f √ p N , where µ = | ω | N is the corresponding volume form and C f is an explicit com-putable constant which depends only on the function f. Bounds of higher-dimensional exponential sums.
The main results of thesenotes are a sharp estimates of certain higher-dimensional exponential sumsattached to tori in Sp (2 N, F q ). This is the content of Theorems 12 and 14and is obtained using the self-reducibility property of the Weil represen-tation as stated in Theorems 9 and 10.2. Hecke quantum unique ergodicity theorem.
The main application of thesenotes is the proof of the
Hecke quantum unique ergodicity theorem , i.e.,Theorems 17 and 18, for generic linear symplectomorphism of the torusin any dimension. The proof of the theorem is a direct application of thesharp bound on the higher-dimensional exponential sums.3.
Multiplicities formula.
Exact formula for the multiplicities, i.e., the di-mensions of the character spaces for the action of maximal tori in theWeil representation are derived. This is obtained first for the SL ( F q )case in Theorem 8 using the character formula presented in Theorem 7.Then, as a direct application of the self-reducibility property, the formulais extended in Theorem 11 to the higher-dimensional cases. elf-Reducibility and Quantum Chaos 11 In addition, a formulation of the quantum unique ergodicity statementfor quantum chaos problems, close in spirit to the von Neumann idea aboutdensity operator, is suggested in Theorem 4. The statement includes only thequantum operator A rather than the whole Hecke group of symmetries [25].The proof of the statement uses the Hecke operators as a harmonic analysistool. Apart from the introduction, the notes consist of five sections.In Section 2 we give some preliminaries on representation theory whichare used in the notes. In Subsection 2.3 we recall the invariant presentationof the Weil representation over finite fields [16], and we discuss applicationsto multiplicities. Section 3 constitutes the main technical part of this work.Here we develop the theory that underlies the self-reducibility property ofthe Weil representation. In particular, in Subsection 3.1 we introduce the no-tion of symplectic module structure. In Subsection 3.2 we prove the existenceof symplectic module structure associated with a maximal torus in Sp . Fi-nally, we establish the self-reducibility property of the Weil representation,i.e., Theorem 10, and apply this property to get information on multiplicitiesin Subsection 3.4. Section 4 is devoted to an application of the theory devel-oped in previous sections to estimating higher-dimensional exponential sumswhich originate from the mathematical theory of quantum chaos. In Section5 we describe the higher-dimensional Hannay–Berry model of quantum me-chanics on the torus. Finally, in Section 6 we present the main application ofthese notes—the proof of the Hecke quantum unique ergodicity theorem forgeneric linear symplectomorphisms of the 2 N -dimensional torus. Remark 5.
Complete proofs for the statements appearing in these notes willbe given elsewhere.
It is a pleasure to thank our teacher J. Bernstein for his interest and guidancein this project. We thank D. Kazhdan for several interesting discussions. Weappreciate the support of the Technion, and in particular of M. Baruch. Wethank D. Kelmer for sharing with us computer simulation data. Finally, wewould like to thank O. Ceyhan and the organizers of the conference AGAQ,held in Istanbul during June 2006, and J. Wolf and the organizers of theconference Lie Groups, Lie Algebras and Their Representations, Berkeley,November 2006, for the invitation to present this work.
In this section, we denote by k = F q the finite field of q elements and oddcharacteristic. Let (
V, ω ) be a 2 N -dimensional symplectic vector space over the finite field k . There exists a two-step nilpotent group H = H ( V, ω ) associated to thesymplectic vector space (
V, ω ). The group H is called the Heisenberg group.
It can be realized as the set H = V × k, equipped with the multiplication rule( v, z ) · ( v ′ , z ′ ) = ( v + v ′ , z + z ′ + ω ( v, v ′ )) . The center of H is Z ( H ) = { (0 , z ) : z ∈ k } . Fix a non-trivial centralcharacter ψ : Z ( H ) −→ C × . We have the following fundamental theorem: Theorem 5 (Stone–von Neumann).
There exists a unique ( up to isomorphism ) irreducible representation ( π, H, H ) with central character ψ , i.e., π ( z ) = ψ ( z ) Id H for every z ∈ Z ( H ) . We call the representation π appearing in Theorem 5, the Heisenberg rep-resentation associated with the central character ψ . Remark 6.
The representation π , although it is unique, admits a multitude ofdifferent models (realizations). In fact, this is one of its most interesting andpowerful attributes. In particular, to any Lagrangian splitting V = L ′ ⊕ L ,there exists a model ( π L ′ ,L , H, C ( L )), where C ( L ) denotes the space of complexvalued functions on L . In this model, we have the following actions: • π L ′ ,L ( l ′ )[ f ]( x ) = ψ ( ω ( l ′ , x )) f ( x ); • π L ′ ,L ( l )[ f ]( x ) = f ( x + l ); • π L ′ ,L ( z )[ f ]( x ) = ψ ( z ) f ( x ),where l ′ ∈ L ′ , x, l ∈ L , and z ∈ Z ( H ) . The above model is called the Schr¨odinger realization associated with thesplitting V = L ′ ⊕ L . Given a linear operator A : H → H we can associate to it a function on thegroup H defined as follows W ( A )( h ) = H Tr( Aπ ( h − )) . (11)The transform W : End( H ) → C ( H ) is called the Weyl transform [21, 35].
The Weyl transform admits a left inverse π : C ( H ) → End( H ) given by theextended action π ( K ) = P h ∈ H K ( h ) π ( h ). elf-Reducibility and Quantum Chaos 13 Let Sp = Sp ( V, ω ) denote the group of linear symplectic automorphisms of V . The group Sp acts by group automorphisms on the Heisenberg groupthrough its tautological action on the vector space V . A direct consequence ofTheorem 5 is the existence of a projective representation e ρ : Sp → P GL ( H ).The classical construction of e ρ out of the Heisenberg representation π is dueto Weil [34]. Considering the Heisenberg representation π and an element g ∈ Sp , one can define a new representation π g acting on the same Hilbertspace via π g ( h ) = π ( g ( h )). Clearly both π and π g have central character ψ ;hence, by Theorem 5, they are isomorphic. Since the space Hom H ( π, π g ) isone-dimensional, choosing for every g ∈ Sp a non-zero representative e ρ ( g ) ∈ Hom H ( π, π g ) gives the required projective representation. In more concreteterms, the projective representation e ρ is characterized by the formula e ρ ( g ) π ( h ) e ρ (cid:0) g − (cid:1) = π ( g ( h )) , (12)for every g ∈ Sp and h ∈ H . It is a peculiar phenomenon of the finite fieldsetting that the projective representation e ρ can be linearized into an honestrepresentation. This linearization is unique, except in the case the finite fieldis F and dim V = 2 (for the canonical choice in the latter case see [17]). Theorem 6.
There exists a canonical unitary representation ρ : Sp −→ GL ( H ) , satisfying the formula (12). Invariant presentation of the Weil representation
An elegant description of the Weil representation can be obtained [16] usingthe Weyl transform (see Subsection 2.2). Given an element g ∈ Sp , the oper-ator ρ ( g ) can be written as ρ ( g ) = π ( K g ), where K g is the Weyl transform K g = W ( ρ ( g )). The homomorphism property of ρ is manifested as K g ∗ K h = K gh for every g, h ∈ Sp, where ∗ denotes (properly normalized) group theoretic convolution on H .Finally, the function K can be explicitly described on an appropriate subsetof Sp [16]. Let U ⊂ Sp denote the subset consisting of all elements g ∈ Sp such that g − I is invertible. For every g ∈ U and v ∈ V we have K g ( v ) = ν ( g ) ψ ( ω ( κ ( g ) v, v )) , (13)where κ ( g ) = g + Ig − I is the Cayley transform [21, 36], and ν ( g ) = ( G/q ) N σ (det( g − I )) , with σ the unique quadratic character of the multiplicative group F × q , and G = P z ∈ Z ( H ) ψ ( z ) the quadratic Gauss sum. Let J denote the semi-direct product J = Sp ⋉ H. The group J is sometimesreferred to as the Jacobi group. The compatible pair ( π, ρ ) is equivalent toa single representation τ : J −→ GL ( H ) of the Jacobi group defined by theformula τ ( g, h ) = ρ ( g ) π ( h ). It is an easy exercise to verify that the Egorovidentity (12) implies the multiplicativity of the map τ .In these notes, we would like to adopt the name Heisenberg–Weil repre-sentation when referring to the representation τ . The invariant presentation (11) and formula (13) imply [16] a formula for thecharacter of the 2 N -dimensional Heisenberg–Weil representation over a finitefield (cf. [9, 22]). Theorem 7 (Character formulas [16]).
The character ch ρ of the Weilrepresentation, when restricted to the subset U , is given bych ρ ( g ) = σ (( − N det( g − I )) , (14) and the character ch τ of the Heisenberg–Weil representation, when restrictedto the subset U × H , is given by ch τ ( g, v, z ) = ch ρ ( g ) ψ ( ω ( κ ( g ) v, v ) + z ) . (15) We would like to apply the formula (15) to the study of the multiplicitiesarising from actions of tori via the Weil representation (cf. [1, 9, 32]). Let usstart with the two-dimensional case (see Theorem 11 for the general case).Let T ⊂ Sp ≃ SL ( F q ) be a maximal torus. The torus T acts semisimply on H , decomposing it into a direct sum of character spaces H = L χ : T → C × H χ . As a consequence of having the explicit formula (14), we obtain a simpledescription for the multiplicities m χ = dim H χ . Denote by σ : T → C × theunique quadratic character of T. Theorem 8 (Multiplicities formula).
We have m χ = 1 for any character χ = σ. Moreover, m σ = 2 or , depending on whether the torus T is split orinert, respectively. What about the multiplicities for action of tori in the Weil representationof higher-dimensional symplectic groups? This problem can be answered (seeTheorem 11) using the self-reducibility property of the Weil representation. elf-Reducibility and Quantum Chaos 15
In this section, unless stated otherwise, the field k is an arbitrary field ofcharacteristic different from two. Let K be a finite-dimensional commutative algebra over the field k . Let Tr : K → k be the trace map, associating to an element x ∈ K the trace of the k -linear operator m x : K → K obtained by left multiplication by the element x . Consider a symplectic vector space ( V, ω ) over k. Definition 2.
A symplectic K -module structure on ( V, ω ) is an action K ⊗ k V → V, and a K -linear symplectic form ω : V × V → K such thatTr ◦ ω = ω. (16)Given a symplectic module structure ( K, V, ω ) on a symplectic vector space(
V, ω ) , we denote by Sp = Sp ( V, ω ) the group of K -linear symplectomorphismswith respect to the form ω . The compatibility condition (16) gives a naturalembedding ι : Sp ֒ → Sp. (17)
Let T ⊂ Sp be a maximal torus. A particular case
In order to simplify the presentation, let us assume first that T acts irreduciblyon the vector space V , i.e., there exists no non-trivial T -invariant subspaces.Let A = Z ( T, End( V )) , be the centralizer of T in the algebra of all linear en-domorphisms. Clearly (due to the assumption of irreducibility) A is a divisionalgebra. Moreover, we have Claim.
The algebra A is commutative.In particular, this claim implies that A is a field extension of k . Let us nowdescribe a special quadratic element in the Galois group Gal( A/k ). Denoteby ( · ) t : End( V ) → End( V ) the symplectic transpose characterized by theproperty ω ( Rv, u ) = ω ( v, R t u ) , for all v, u ∈ V , and every R ∈ End( V ). It can be easily verified that ( · ) t preserves A , leaving the subfield k fixed, hence, it defines an element Θ ∈ Gal(
A/k ) , satisfying Θ = Id. Denote by K = A Θ the subfield of A consistingof elements fixed by Θ . We have the following proposition: Proposition 1 (Hilbert’s Theorem 90).
We have dim K V = 2 . Corollary 3.
We have dim K A = 2 . As a corollary, we have the following description of T . Denote by N A/K : A → K the standard norm map. Corollary 4.
We have T = S ( A ) = (cid:8) a ∈ A : N A/K ( a ) = 1 (cid:9) The symplectic form ω can be lifted to a K -linear symplectic form ω ,which is invariant under the action of the torus T . This is the content of thefollowing proposition: Proposition 2 (Existence of canonical symplectic module structure).
There exists a canonical T -invariant K -linear symplectic form ω : V × V → K satisfying the property Tr ◦ ω = ω. Concluding, we obtained a T -invariant symplectic K -module structure on V . Let Sp = Sp ( V, ω ) denote the group of K -linear symplectomorphisms withrespect to the symplectic form ω . We have (17) a natural embedding Sp ⊂ Sp .The elements of T commute with the action of K , and preserve the symplecticform ω (Proposition 2); hence, we can consider T as a subgroup of Sp . ByProposition 1 we can identify Sp ≃ SL ( K ) , and using (17) we obtain T ⊂ SL ( K ) ⊂ Sp. (18)To conclude we see that T consists of the K -rational points of a maximal torus T ⊂ SL (in this case T consists of the rational points of an inert torus). General case
Here, we drop the assumption that T acts irreducibly on V . By the sameargument as before, the algebra A = Z ( T, End( V )) is commutative, yet, itmay no longer be a field. The symplectic transpose ( · ) t preserves the algebra A , and induces an involution Θ : A → A. Let K = A Θ be the subalgebraconsisting of elements a ∈ A fixed by Θ . Following the same argument as inthe proof of Proposition 1, we can show that V is a free K -module of rank 2.Following the same arguments as in the proof of Proposition 2, we can showthat there exists a canonical symplectic form ω : V × V → K , which is K -linear and invariant under the action of the torus T . Concluding, associated toa maximal torus T there exists a T -invariant symplectic K -module structure( K, V, ω ) . (19)Denote by Sp = Sp ( V, ω ) the group of K -linear symplectomorphisms withrespect to the form ω. We have a natural embedding elf-Reducibility and Quantum Chaos 17 ι S : Sp ֒ → Sp (20)and we can consider T as a subgroup of Sp . Finally, we have Sp ≃ SL ( K ) , and T consists of the K -rational points of a maximal torus T ⊂ SL . In particular,the relation (18) holds also in this case: T ⊂ SL ( K ) ⊂ Sp.
We shall now proceed to give a finer description of all objects discussed sofar. The main technical result is summarized in the following lemma:
Lemma 1 (Symplectic decomposition).
We have a canonical decomposi-tion ( V, ω ) = M α ∈ Ξ ( V α , ω α ) , (21) into ( T, A ) -invariant symplectic subspaces. In addition, we have the followingassociated canonical decompositions T = Q T α , where T α consists of elements t ∈ T such that t | V β = Id forevery β = α .2. A = L A α , where A α consists of elements a ∈ A such that a | V β = Idfor every β = α . Moreover, each sub-algebra A α is preserved under theinvolution Θ. K = L K α , where K α = A Θα . Moreover, K α is a field and dim K α V α = 2.4. ω = L ω α , where ω α : V α × V α → K α is a K α -linear T α -invariant sym-plectic form satisfying Tr ◦ ω α = ω α . Definition 3.
We will call the set Ξ ( ) the symplectic type of T and thenumber | Ξ | the symplectic rank of T . Using the results of Lemma 1, we have an isomorphism Sp ≃ Y Sp α , (22)where Sp α = Sp ( V α , ω α ) denotes the group of K α -linear symplectomorphismswith respect to the form ω α . Moreover, for every α ∈ Ξ we have T α ⊂ Sp α . Inparticular, under the identifications Sp α ≃ SL ( K α ) , there exist the followingsequence of inclusions T = Y T α ⊂ Y SL ( K α ) = SL ( K ) ⊂ Sp, (23)and for every α ∈ Ξ the torus T α coincides with the K α -rational points of amaximal torus T α ⊂ SL . In this subsection we assume that the field k is a finite field of odd char-acteristic (although, the results continue to hold true also for local fields ofcharacteristic = 2, i.e., with the appropriate modification, replacing the group Sp with its double cover f Sp ). Let ( τ, J, H ) be the Heisenberg–Weil represen-tation associated with a central character ψ : Z ( J ) = Z ( H ) → C × . Recallthat J = Sp ⋉ H, and τ is obtained as a semi-direct product, τ = ρ ⋉ π , ofthe Weil representation ρ and the Heisenberg representation π . Let T ⊂ Sp be a maximal torus. A particular case
For clarity of presentation, assume first that T acts irreducibly on V . Usingthe results of the previous section, there exists a symplectic module structure( K, V, ω ) (in this case
K/k is a field extension of degree [ K : k ] = N ). Thegroup Sp = Sp ( V, ω ) is imbedded as a subgroup ι S : Sp ֒ → Sp.
Our goal is todescribe the restriction ( ρ = ι ∗ S ρ, Sp, H ) . (24)Define an auxiliary Heisenberg group H = V × K, (25)with the multiplication given by ( v, z ) · ( v ′ , z ′ ) = ( v + v ′ , z + z ′ + ω ( v, v ′ )).There exists homomorphism ι H : H → H, (26)given by ( v, z ) ( v, Tr( z )). Consider the pullback ( π = ι ∗ H π, H, H ) . We have
Proposition 3.
The representation ( π = ι ∗ H π, H, H ) is the Heisenberg repre-sentation associated with the central character ψ = ψ ◦ Tr.
The group Sp acts by automorphisms on the group H through its tauto-logical action on the V -coordinate. This action is compatible with the actionof Sp on H , i.e., we have ι H ( g · h ) = ι S ( g ) · ι H ( h ) for every g ∈ Sp , and h ∈ H .The description of the representation ρ (24) now follows easily (cf. [9]). Theorem 9 (Self-reducibility property (particular case)).
The repre-sentation ( ρ, Sp, H ) is the Weil representation associated with the Heisenbergrepresentation ( π, H, H ) .Remark 7. We can summarize the result in a slightly more elegant mannerusing the Jacobi groups. Let J = Sp ⋉ H and J = Sp ⋉ H be the Jacobigroups associated with the symplectic spaces ( V, ω ) and (
V, ω ) respectively.We have a homomorphism ι : J → J, given by ι ( g, h ) = ( ι S ( g ) , ι H ( h )).Let ( τ, J, H ) be the Heisenberg–Weil representation of J associated with acharacter ψ of the center Z ( J ) (note that Z ( J ) = Z ( H )), then the pullback( ι ∗ τ, J, H ) is the Heisenberg–Weil representation of J , associated with thecharacter ψ = ψ ◦ Tr of the center Z ( J ). elf-Reducibility and Quantum Chaos 19 The general case
Here, we drop the assumption that T acts irreducibly on V . Let ( K, V, ω ) bethe associated symplectic module structure (19). Using the results of Subsec-tion 3.2, we have decompositions(
V, ω ) = M α ∈ Ξ ( V α , ω α ) , ( V, ω ) = M α ∈ Ξ ( V α , ω α ) , (27)where ω α : V α × V α → K α . Let (cf. 25) H = V × K, be the Heisenberg groupassociated with ( V, ω ). There exists (cf. (26) a homomorphism ι H : H → H. Let us describe the pullback π = ι ∗ H π of the Heisenberg representation. First,we note that the decomposition (27) induces a corresponding decompositionof the Heisenberg group, H = Q H α , where H α is the Heisenberg groupassociated with ( V α , ω α ). We have the following proposition Proposition 4.
There exists an isomorphism ( π, H, H ) ≃ ( O π α , Y H α , O H α ) , where ( π α , H α , H α ) is the Heisenberg representation of H α associated withthe central character ψ α = ψ ◦ Tr K α /k . Let ι S : Sp ֒ → Sp, be the embedding (20). Our next goal is to describe therestriction ρ = ι ∗ S ρ. Recall that we have a decomposition Sp = Q Sp α (see(22)). In terms of this decomposition we have (cf. [9]) Theorem 10 (Self-reducibility property—general case).
There existsan isomorphism ( ρ, Sp, H ) ≃ ( O ρ α , Y Sp α , O H α ) , where ( ρ α , Sp α , H α ) is the Weil representation associated with the Heisenbergrepresentation π α .Remark 8. As before, we can state an equivalent result using the Jacobi groups J = Sp ⋉ H and J = Sp ⋉ H . We have a decomposition J = Q J α , where J α = Sp α ⋉ H α . Let τ be the Heisenberg–Weil representation of J associatedwith a character ψ of the center Z ( J ) (note that Z ( J ) = Z ( H )). Then thepullback τ = ι ∗ τ is isomorphic to N τ α , where τ α is the Heisenberg–Weilrepresentation of J α , associated with the character ψ α = ψ ◦ Tr K α /k of thecenter Z ( J α ). Let us specialize to the case where the filed k is a finite field of odd char-acteristic. Let T ⊂ Sp be a maximal torus. The torus T acts, via the Weilrepresentation ρ , on the space H , decomposing it into a direct sum of T -character spaces H = L χ : T → C × H χ . Consider the problem of determining themultiplicities m χ = dim( H χ ). Using Lemma 1, we have (see (23)) a canonicaldecomposition of T T = Y T α , (28)where each of the tori T α coincides with a maximal torus inside Sp ≃ SL ( K α ) , for some field extension K α ⊃ k. In particular, by (28) we havea decomposition H χ = O χ α : T α → C × H χ α , (29)where χ = Q χ α : Q T α → C × . Hence, by Theorem 10, and the result aboutthe multiplicities in the two-dimensional case (see Theorem (8)), we can com-pute the integer m χ as follows . Denote by σ α the quadratic character of T α (note that by Theorem (8) the quadratic character σ α cannot appear in thedecomposition (29) if the torus T α is inert). Theorem 11.
We have m χ = 2 l , where l = { α : χ α = σ α } . In this section we present an application of the self-reducibility technique tobound higher-dimensional exponential sums attached to tori in Sp = Sp ( V, ω ) , where ( V, ω ) is a 2 N -dimensional symplectic vector space over the finite field F p , p = 2. These exponential sums originated from the theory of quantumchaos (see Sections 5 and 6). Let ( τ, J, H ) be the Heisenberg–Weil represen-tation associated with a central character ψ : Z ( J ) = Z ( H ) → C × . Recallthat J = Sp ⋉ H , and τ is obtained as a semi-direct product τ = ρ ⋉ π ofthe Weil representation ρ and the Heisenberg representation π . Consider amaximal torus T ⊂ Sp.
The torus T acts semisimply on H , decomposing itinto a direct sum of character spaces H = L χ : T → C × H χ . We shall study commoneigenstates Ψ ∈ H χ . In particular, we will be interested in estimating matrixcoefficients of the form h Ψ | π ( ξ ) Ψ i where ξ ∈ V is not contained in any proper T -invariant subspace. It will be convenient to assume first that the torus T iscompletely inert (i.e., acts irreducibly on V ). In this case one can show (seeTheorem 11) that dim H χ = 1 for every χ . Below we sketch a proof of thefollowing estimate. elf-Reducibility and Quantum Chaos 21 Theorem 12.
For ξ ∈ V which is not contained in any proper T -invariantsubspace, we have |h Ψ | π ( ξ ) Ψ i| ≤ o (1) √ p N . Let us explain way it is not easy to get such a bound by a direct calculation.Choosing a Schr¨odinger realization (see Remark 6), we can identify H = C ( F Np ). Under this identification, the matrix coefficient is equivalent to a sum h Ψ | π ( ξ ) Ψ i = X x ∈ F Np Ψ ( x ) e πip ξ + x Ψ ( x + ξ − ) . (30)In this respect two problems are encountered. First, it is not easy to describethe eigenstates Ψ . Second, the sum (30) is a high-dimensional exponentialsum, which is known to be hard to analyze using standard techniques.Interestingly enough, representation theory suggests a remedy for bothproblems. Our strategy will be to interpret the matrix coefficient h Ψ | π ( ξ ) Ψ i in representation theoretic terms, and then to show, using the self-reducibilitytechnique, that (30) is equivalent to a 1-dimensional sum over F q , q = p N . Representation theory and dimensional reduction of (30)
The torus T acts irreducibly on the vector space V . Invoking the result ofSection 3.2, there exists a canonical symplectic module structure ( K, V, ω )associated to T . Recall that in this particular case the algebra K is in facta field, and dim K V = 2 (in our case K = F q , where q = p N ). Let J = Sp ⋉ H be the Jacobi group associated to the (two-dimensional) symplecticvector space ( V, ω ) over K . There exists a natural homomorphism ι : J → J .Invoking the results of Section 3.3, the pullback τ = ι ∗ τ is the Heisenberg–Weilrepresentation of J , i.e., τ = ρ ⋉ π .Let Ψ ∈ H χ . Denote by P χ the orthogonal projector on the vector space H χ . We can write P χ in terms of the Weil representation ρP χ = 1 | T | X B ∈ T χ − ( B ) ρ ( B ) . (31)Since dim H χ = 1 (Theorem 11) we realize that h Ψ | π ( ξ ) Ψ i = Tr( P χ π ( ξ )) . (32)Substituting (31) in (32), we can write h Ψ | π ( ξ ) Ψ i = 1 | T | X B ∈ T χ − ( B )Tr( ρ ( B ) π ( ξ )) . Noting that Tr( ρ ( B ) π ( ξ )) is nothing other than the character ch τ ( B · ξ )of the Heisenberg–Weil representation τ . and that | T | = p N + 1, we deducethat it is enough to prove that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X B ∈ T χ − ( B )ch τ ( B · ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ q, (33)where q = p N . Now, note that the left-hand side of (33) is a one-dimensionalexponential sum over F q , which is defined completely in terms of the two-dimensional Heisenberg–Weil representation τ . Estimate (33) is then a par-ticular case of the following theorem, proved in [13]. Theorem 13.
Let ( V, ω ) be a two-dimensional symplectic vector space over afinite field k = F q , and ( τ, J, H ) be the corresponding Heisenberg–Weil repre-sentation. Let T ⊂ Sp be a maximal torus. We have the following estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X B ∈ T χ ( B ) ch τ ( B · ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ q, (34) where χ is a character of T , and = ξ ∈ V is not an eigenvector of T . In this subsection we state and prove the analogue of Theorem 12, where wedrop the assumption of T being completely inert. In what follows, we use theresults of Subsections 3.2 and 3.3.Let ( K, V, ω ) be the symplectic module structure associated with the torus T . The algebra K is no longer a field, but decomposes into a direct sum offields K = L α ∈ Ξ K α . We have canonical decompositions( V, ω ) = M ( V α , ω α ) , ( V, ω ) = M ( V α , ω α ) . Recall that V α is a two-dimensional vector space over the field K α . TheJacobi group J decomposes into J = Q J α , where J α = Sp α ⋉ H α is theJacobi group associated to ( V α , ω α ) . The pullback ( τ = ι ∗ τ, J, H ) decomposesinto a tensor product ( N τ α , Q J α , N H α ) , where τ α is the Heisenberg–Weilrepresentation of J α . The torus T decomposes into T = Q T α , where T α is amaximal torus in Sp α . Consequently, the character χ : T → C × decomposesinto a product χ = Πχ α : ΠT α → C × , and the space H χ decomposes into atensor product H χ = O H χ α . (35)It follows from the above decomposition that it is enough to estimatematrix coefficients with respect to pure tensor eigenstates, i.e., eigenstates Ψ of the form Ψ = N Ψ α , where Ψ α ∈ H χ α . For a vector of the form ξ = L ξ α ,we have DO Ψ α | π ( ξ ) O Ψ α E = Y h Ψ α | π ( ξ α ) Ψ α i . (36) elf-Reducibility and Quantum Chaos 23 Hence, we need to estimate the matrix coefficients h Ψ α | π ( ξ α ) Ψ α i , but theseare defined in terms of the two-dimensional Heisenberg–Weil representation τ α . In addition, we recall the assumption that the vector ξ ∈ V is not con-tained in any proper T -invariant subspace. This condition in turn impliesthat no summand ξ α is an eigenvector of T α . Hence, we can use Lemma 13,obtaining |h Ψ α | π ( ξ α ) Ψ α i| ≤ / √ p [ K α : F p ] . (37)Consequently, using (36) and (37) we obtain (cid:12)(cid:12)(cid:12)DO Ψ α | π ( ξ ) O Ψ α E(cid:12)(cid:12)(cid:12) ≤ | Ξ | / √ p P [ K α : F p ] = 2 | Ξ | / √ p [ K : F p ] = 2 | Ξ | / √ p N . Recall that the number r p = | Ξ | is called the symplectic rank of the torus T . The main application of the self-reducibility property, presented in thesenotes, is summarized in the following theorem. Theorem 14.
Let ( V, ω ) be a N -dimensional vector space over the finitefield F p , and ( τ, J, H ) the corresponding Heisenberg–Weil representation. Let Ψ ∈ H χ be a unit χ -eigenstate with respect to a maximal torus T ⊂ Sp . Wehave the following estimate: |h Ψ | π ( ξ ) Ψ i| ≤ m χ · (2 + o (1)) r p √ p N , where ≤ r p ≤ N is the symplectic rank of T , m χ = dim H χ , and ξ ∈ V isnot contained in any T -invariant subspace. We shall proceed to describe the higher-dimensional Hannay–Berry model ofquantum mechanics on toral phase spaces. This model plays an important rolein the mathematical theory of quantum chaos as it serves as a model wheregeneral phenomena, which are otherwise treated only on a heuristic basis, canbe rigorously proven.
Our classical phase space is the 2 N -dimensional symplectic torus ( T , ω ) . Wedenote by Γ the group of linear symplectic automorphisms of T . Note that Γ ≃ Sp (2 N, Z ) . On the torus T we consider an algebra of complex functions(observables) A = F ( T ). We denote by Λ ≃ Z N the lattice of characters(exponents) of T . The form ω induces a skew-symmetric form on Λ , whichwe denote also by ω , and we assume it takes integral values on Λ and isnormalized so that R T | ω | N = 1 . We take our classical mechanical system to be of a very simple nature. Let A ∈ Γ be a generic element (see Definition 1), i.e., A is regular and admitsno invariant co-isotropic sub-tori. The last condition can be equivalently re-stated in dual terms, namely, requiring that A admits no invariant isotropicsubvectorspaces in Λ Q = Λ ⊗ Z Q . The element A generates, via its action asan automorphism A : T −→ T , a discrete time dynamical system. Before we employ the formal model, it is worthwhile to discuss the generalphenomenological principles of quantization which are common to all models.Principally, quantization is a protocol by which one associates a Hilbert space H to the classical phase space, which in our case is the torus T ; In addition,the protocol gives a rule f Op ( f ) : H → H , by which one associates an operator on the Hilbert space to every classicalobservable, i.e., a function f ∈ F ( T ) . This rule should send a real functioninto a self-adjoint operator. In addition, in the presence of classical symmetrieswhich in our case are given by the group Γ , the Hilbert space H should supporta (projective unitary) representation Γ → P GL ( H ), γ U ( γ ) : H → H , which is compatible with the quantization rule Op ( · ).More precisely, quantization is not a single protocol, but a one-parameterfamily of protocols, parameterized by a parameter ℏ called the Planck con-stant. Accepting these general principles, one searches for a formal model bywhich to quantize. In this work we employ a model called the non-commutativetorus model. Denote by A the algebra of trigonometric polynomials on T , i.e., A consists offunctions f which are a finite linear combinations of characters. We shall con-struct a one-parametric deformation of A called the non-commutative torus[6, 29].Let ~ = 1 /p, where p is an odd prime number, and consider the additivecharacter ψ : F p −→ C × , ψ ( t ) = e πitp . We give here the following presentationof the algebra A ~ . Let A ℏ be the free non-commutative C -algebra generatedby the symbols s ( ξ ), ξ ∈ Λ, and the relations s ( ξ ) s ( η ) = ψ ( ω ( ξ, η )) s ( ξ + η ) . (38)Here we consider ω as a map ω : Λ × Λ −→ F p .Note that A ℏ satisfies the following properties: elf-Reducibility and Quantum Chaos 25 • As a vector space A ℏ is equipped with a natural basis s ( ξ ), ξ ∈ Λ . Hencewe can identify the vector space A ℏ with the vector space A for each valueof ℏ , A ℏ ≃ A . (39) • Substituting ℏ = 0 we have A = A . Hence, we see that indeed A ℏ is adeformation of the algebra of (algebraic) functions on T . • The group Γ acts by automorphisms on the algebra A ℏ , via γ · s ( ξ ) = s ( γξ ),where γ ∈ Γ and ξ ∈ Λ . This action induces an action of Γ on the categoryof representations of A ℏ , taking a representation π and sending it to therepresentation π γ , where π γ ( f ) = π ( γf ), f ∈ A ℏ .Using the identification (39) , we can describe a choice for the quantizationof the functions. We just need to pick a representation of the quantum algebra A ℏ . But what representation to pick? It turns out that, we have a canonicalchoice. All the irreducible algebraic representations of A ℏ are classified [12]and each of them is of dimension p N . We have
Theorem 15 (Invariant representation [12]).
Let ℏ = 1 /p where p is aprime number. There exists a unique (up to isomorphism) irreducible repre-sentation π : A ℏ → End ( H ℏ ) which is fixed by the action of Γ . Namely, π γ isisomorphic to π for every γ ∈ Γ . Let ( π, A ℏ , H ) be a representative of the special representation definedin Theorem 15. For every element γ ∈ Γ we have an isomorphism e ρ ( γ ) : H → H intertwining the representations π and π γ , namely, it satisfies e ρ ( γ ) π ( f ) e ρ ( γ ) − = π ( γf ) , for every f ∈ A ℏ and γ ∈ Γ . The isomorphism e ρ ( γ ) is not unique but unique up to a scalar (this is a consequence of Schur’slemma and the fact that π and π γ are irreducible representations). It is easyto realize that the collection { e ρ ( γ ) } constitutes a projective representation e ρ : Γ → P GL ( H ). Let ℏ = 1 /p where p is an odd prime = 3 . We have thefollowing linearization theorem (cf. [11, 13])
Theorem 16 (Linearization).
The projective representation e ρ can be lin-earized uniquely to an honest representation ρ : Γ → GL ( H ) that factorsthrough the finite quotient group Sp ≃ Sp (2 N, F p ) .Remark 9. The representation ρ : Sp → GL ( H ) is the celebrated Weil repre-sentation, here obtained via quantization of the torus. Recall that we started with a classical dynamic on T , generated by a generic(i.e., regular with no non-trivial invariant co-isotropic sub-tori) element A ∈ Γ . Using the Weil representation, we can associate to A the unitary operator ρ ( A ) : H → H , which constitutes the generator of discrete time quantumdynamics. We would like to study the ρ ( A )-eigenstates ρ ( A ) Ψ = λΨ, which satisfy additional symmetries. This we do in the next section. It turns out that the operator ρ ( A ) has degeneracies namely, its eigenspacesmight be extremely large. This is manifested in the existence of a group ofhidden symmetries commuting with ρ ( A ) (note that classically the group oflinear symplectomorphisms of T that commute with A , i.e., T A ( Z ) , does notcontribute much to the harmonic analysis of ρ ( A )). These symmetries canbe computed. Indeed, let T A = Z ( A, Sp ) , be the centralizer of the element A in the group Sp . Clearly T A contains the cyclic group h A i generated bythe element A, but it often happens that T A contains additional elements.The assumption that A is regular (i.e., has distinct eigenvalues) implies thatfor sufficiently large p the group T A consists of the F p -rational points of amaximal torus T A ⊂ Sp , i.e., T A = T A ( F p ) (more precisely, p large enoughso that it does not divides the discriminant of A ). The group T A is called the Hecke torus. It acts semisimply on H , decomposing it into a direct sum ofcharacter spaces H = L χ : T A → C × H χ . We shall study common eigenstates Ψ ∈H χ , which we call Hecke eigenstates and will be assumed to be normalizedso that k Ψ k H = 1. In particular, we will be interested in estimating matrixcoefficients of the form h Ψ | π ( f ) Ψ i , where f ∈ A is a classical observable onthe torus T (see Subsection 5.4). We will call these matrix coefficients Hecke–Wigner distributions . It will be convenient for us to start with the followingcase.
Let us assume first that the automorphism A acts on T with no invariantsub-tori. In dual terms, this means that the element A acts irreducibly on the Q -vector space Λ Q = Λ ⊗ Z Q . We denote by r p the symplectic rank of T A , i.e., r p = | Ξ | where Ξ = Ξ ( T A ) is the symplectic type of T A (see Definition 3). By definition we have1 ≤ r p ≤ N (for example, we get the two extreme cases: d p = 1 when thetorus T A acts irreducibly on V ≃ F Np , and d p = N when T A splits). We have Theorem 17.
Consider a non-trivial exponent = ξ ∈ Λ and a sufficientlylarge prime number p. Then for every normalized Hecke eigenstate Ψ ∈ H χ the following bound holds: |h Ψ | π ( ξ ) Ψ i| ≤ m χ · r p √ p N , (40) where m χ = dim( H χ ) . elf-Reducibility and Quantum Chaos 27 The lattice Λ constitutes a basis for A , hence, using the bound (40) we obtain Corollary 5 (Hecke quantum unique ergodicity—strongly genericcase).
Consider an observable f ∈ A and a sufficiently large prime num-ber p . For every normalized Hecke eigenstate Ψ we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h Ψ | π ( f ) Ψ i − Z T f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C f √ p N , where µ = | ω | N is the corresponding volume form and C f is an explicit com-putable constant which depends only on the function f. Remark 10.
In Subsection 6.2 we will elaborate on the distribution of thesymplectic rank r p (40) and in Subsection 6.3 the more general statementswhere A ∈ Γ is any generic element (see Definition 1) will be stated andproved. Proof of Theorem 17
The proof is by reduction to the bound on the Hecke–Wigner distributionsobtained in Section 4, namely reduction to Theorem 14. Our first goal is to in-terpret the Hecke–Wigner distribution h Ψ | π ( ξ ) Ψ i in terms of the Heisenberg–Weil representation. Step 1.
Replacing the non-commutative torus by the finite Heisenberggroup.
Note that the Hilbert space H is a representation space of both thealgebra A ℏ and the group Sp . We will show next that the representation( π, A ℏ , H ) is equivalent to the Heisenberg representation of some finite Heisen-berg group. The representation π is determined by its restriction to the lattice Λ . However, the restriction π | Λ : Λ → GL ( H ) , is not multiplicative and in fact constitutes (see Formula (38)) a projectiverepresentation of the lattice given by π ( ξ ) π ( η ) = ψ ( ω ( ξ, η )) π ( ξ + η ) . (41)It is evident from (41) that the map π | Λ factors through the quotient F p -vectorspace V Λ → V = Λ/pΛ → GL ( H ) . (42)The vector space V is equipped with a symplectic structure ω obtained viaspecialization of the corresponding form on Λ. Let H = H ( V, ω ) be the Heisen-berg group associated with (
V, ω ). Recall that as a set H = V × F p and themultiplication is given by( v, z ) · ( v ′ , z ′ ) = ( v + v ′ , z + z ′ + ω ( v, v ′ )) . (43) From formula (41), the factorization (42), and the multiplication rule (43) welearn that the map π : V → GL ( H ) , given by (42), lifts to an honest repre-sentation of the Heisenberg group π : H → GL ( H ) . Finally, the pair ( ρ, π ),where ρ is the Weil representation obtained using quantization of the torus(see Theorem 16) glues into a single representation τ = ρ ⋉ π of the Jacobigroup J = Sp ⋉ H , which is of course nothing other than the Heisenberg–Weilrepresentation τ : J → GL ( H ) . (44)Having the Heisenberg–Weil representation at our disposal we proceed to Step 2.
Reformulation.
Let V and T A be the algebraic group scheme de-fined over Z so that Λ = V ( Z ) and for every prime p we have V = V ( F p ) and T A = T A ( F p ) . In this setting for every prime number p we can consider thelattice element ξ ∈ Λ as a vector in the F p -vector space V .Let ( τ, J, H ) be the Heisenberg–Weil representation (44) and consider anormalized Hecke eigenstate Ψ ∈ H χ . We need to verify that for a sufficientlylarge prime number p we have |h Ψ | π ( ξ ) Ψ i| ≤ m χ · r p √ p N , (45)where m χ denotes the multiplicity m χ = dim H χ and r p is the symplecticrank of T A . This verification is what we do next.
Step 3.
Verification.
We need to show that we meet the conditions ofTheorem 14. What is left to check is that for sufficiently large prime number p the vector ξ ∈ V is not contained in any T A -invariant subspace of V. Letus denote by O ξ the orbit O ξ = T A · ξ. We need to show that for sufficientlylarge p we have Span F p { O ξ } = V. (46)The condition (46) is satisfied since it holds globally. In more details, ourassumption on A guarantees that it holds for the corresponding objects overthe field of rational numbers Q , i.e., Span Q { T A ( Q ) · ξ } = V ( Q ). Hence (46)holds for a sufficiently large prime number p. We would like to compute the asymptotic distribution of the symplectic rank r p (45) in the set { , ..., N } , i.e., δ ( r ) = lim x →∞ { r p = r ; p ≤ x } π ( x ) , (47)where π ( x ) denotes the number of prime numbers up to x. We fix an algebraic closure Q of the field Q , and denote by G the Galoisgroup G = Gal( Q / Q ) . Consider the vector space V = V ( Q ). By extension of elf-Reducibility and Quantum Chaos 29 scalars the symplectic form ω on V ( Q ) induces a Q -linear symplectic formon V , which we will also denote by ω. Let T denote the algebraic torus T = T A ( Q ) . The action of T on V is completely reducible, decomposing it intoone-dimensional character spaces V = L χ ∈ X V χ . Let Θ be the restriction of the symplectic transpose ( · ) t : End( V ) → End( V ) to T . The involution Θ acts on the set of characters X by χ Θ ( χ ) = χ − and this action is compatible with the action of the Galois group G on X by conjugation χ gχg − , where χ ∈ X and g ∈ G . This means (recallthat A is strongly generic) that we have a transitive action of G on the set X /Θ. Consider the kernel K = ker( G → Aut( X /Θ )) , and the correspondingfinite Galois group Q = G/ K . Considering Q as a subgroup of Aut( X /Θ ) wedefine the cycle number c ( C ) of a conjugacy class C ⊂ Q to be the number ofirreducible cycles that compose a representative of C. By a direct applicationof the Chebotarev theorem [5] we get
Proposition 5 (Chebotarev’s theorem).
The distribution δ ( ) obeys δ ( r ) = | C r || Q | , where C r = ∪ C ⊂ Qc ( C )= r C. Let us now treat the more general case where the automorphism A acts on T ina generic way (Definition 1). In dual terms, this means that the torus T ( Q ) = T A ( Q ) acts on the symplectic vector space V ( Q ) = Λ ⊗ Z Q decomposing itinto an orthogonal symplectic direct sum( V ( Q ) , ω ) = M α ∈ Ξ ( V α ( Q ) , ω α ) , (48)with an irreducible action of T ( Q ) on each of the spaces V α ( Q ). For anexponent ξ ∈ Λ define its support with respect to the decomposition (48) by S ξ = Supp( ξ ) = { α ; P α ξ = 0 } , where P α : V ( Q ) → V ( Q ) is the projectoronto the space V α ( Q ) and denote by d ξ the dimension d ξ = P α ∈ S ξ dim V α ( Q ) . The decomposition (48) induces a decomposition of the torus T ( Q ) into aproduct of completely inert tori T ( Q ) = Y α ∈ Ξ T α ( Q ) . (49)Consider now a sufficiently large prime number p and specialize all theobjects involved to the finite filed F p . The Hecke torus T = T ( F p ) acts on the quantum Hilbert space H decomposing it into an orthogonal direct sum H = L χ : T → C × H χ . The decomposition (49) induces decompositions on the levelof groups of points T = Q α ∈ Ξ T α , where T α = T α ( F p ) , on the level of characters χ = Π α χ : Q α T α → C × , and on the level of character spaces H χ = N α H χ α . For each torus T α we denote by r p,α = r p ( T α ) its symplectic rank (seeDefinition 3) and we consider the integer | S ξ | ≤ r p,ξ ≤ d ξ given by r p,ξ = Π α ∈ S ξ r p,α . Let us denote by m χ ξ the dimension m χ ξ = P α ∈ S ξ dim H χ α . Finally, wecan state the theorem for the generic case. We have
Theorem 18 (Hecke quantum unique ergodicity—generic case).
Con-sider a non-trivial exponent = ξ ∈ Λ and a sufficiently large prime number p. Then for every normalized Hecke eigenstate Ψ ∈ H χ the following boundholds: |h Ψ | π ( ξ ) Ψ i| ≤ m χ ξ · r p,ξ √ p d ξ . (50)Considering the decomposition (48) we denote by d the dimension d =min α V α ( Q ) . Since the lattice Λ constitutes a basis for the algebra A of ob-servables on T , then using the bound (50) we obtain Corollary 6.
Consider an observable f ∈ A and a sufficiently large primenumber p. Then for every normalized Hecke eigenstate Ψ we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h Ψ | π ( f ) Ψ i − Z T f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C f √ p d , where µ = | ω | N is the corresponding volume form and C f is an explicit com-putable constant which depends only on the function f. The proof of Theorem 18 is a straightforward application of Theorem 17.Indeed, considering the decomposition (48) of the torus T ( Q ) to a productof completely inert tori T α ( Q ), we may apply the theory developed for thestrongly generic case in Subsection (6.1) to each of the tori T α ( Q ) to deduceTheorem 18. Remark 11.
As explained in Subsection (6.2) the distribution of the symplecticrank r p,ξ is determined by the Chebotarev theorem applied to (now a productof) suitable finite Galois groups Q α attached to the tori T α , α ∈ S ξ (49) . Remark 12.
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