Classical and Quantum Dynamics on Orbifolds

We present two versions of the Egorov theorem for orbifolds. The first one is a straightforward extension of the classical theorem for smooth manifolds. The second one considers an orbifold as a singular manifold, the orbit space of a Lie group action, and deals with the corresponding objects in noncommutative geometry.


Introduction
The Egorov theorem is a fundamental fact in microlocal analysis and mathematical physics. It relates the evolution of pseudodifferential operators on a compact manifold (quantum observables) determined by a first-order elliptic operator with the corresponding evolution of classical observables -the bicharacteristic flow on the space of symbols. This theorem is the rigorous version of the classical-quantum correspondence in quantum mechanics.
Let P be an elliptic, first-order pseudodifferential operator on a compact manifold X with real principal symbol p ∈ S 1 (T * X). Let f t be the bicharacteristic flow of the operator P , that is, the Hamiltonian flow of p on T * X. The Egorov theorem states that, for any pseudodifferential operator A of order 0 with the principal symbol a ∈ S 0 (T * X), the operator A(t) = e itP Ae −itP is a pseudodifferential operator of order 0. The principal symbol a t ∈ S 0 (T * X) of this operator is given by the formula Here S m (T * X) denote the space of smooth functions on T * X \ {0}, homogeneous of degree m with respect to a fiberwise R-action on T * X.
The main purpose of this paper is to extend the Egorov theorem to orbifolds. We present two versions of the Egorov theorem. The first one is a straightforward extension of the classical theorem. The second one concerns with noncommutative geometry. It considers an orbifold as a singular manifold, the orbit space of a Lie group action, and deals with the corresponding noncommutative objects.
Spectral theory of elliptic operators on orbifolds has received much attention recently (see, for instance, a brief survey in the introduction of [3]). In [17], the Duistermaat-Guillemin trace formula was extended to compact Riemannian orbifolds. This formula has been applied in [6] to an inverse spectral problem on some orbifolds. We believe that our results will play an important role for the study of spectral asymptotics for elliptic operators on orbifolds, in particular, for the study of problems related to quantum ergodicity.

Orbifolds
The flow F t on a symplectic orbifold (Y, ω) is Hamiltonian with a Hamiltonian H ∈ C ∞ (Y ) if, in any orbifold chart (Ũ , G U , φ U ), the infinitesimal generator X H ∈ X (Ũ ) of the flow satisfies a standard relation Since Y is not a manifold, this equation can not be reduced to a system of first-order ordinary differential equations on a manifold. Nevertheless, one can show the existence and uniqueness of the Hamiltonian flow with an arbitrary Hamiltonian H (for instance, using quotient presentations, see below). The flow F t leaves the singular set of the orbifold Y invariant, and its restriction to the regular part Y reg of Y is the Hamiltonian flow of the function H Yreg in the usual sense. We refer the reader to [16] for more information on Hamiltonian dynamics on singular symplectic spaces.

Pseudodifferential operators on orbifolds
Here we recall basic facts about pseudodifferential operators on orbifolds (see [2,4,5] for details). We start with some information about orbibundles.
A (real) vector orbibundle over an orbifold X is given by an orbifold E and a surjective continuous map p : E → X such that, for any x 0 ∈ X, there exists an orbifold chart (Ũ , 1) the action of G U onŨ × R k is an extension of the action of G U onŨ given by where ρ is a smooth map fromŨ × G U to the algebra L(R k ) of linear maps in R k satisfying (in other words, G U acts by vector bundle isomorphisms of the trivial vector bundle pr 1 : Moreover, any two local trivializations are compatible in a natural way. The tangent bundle and the cotangent bundle of an orbifold X are examples of real vector orbibundles over X.
A section s : Now we turn to pseudodifferential operators. Let X be a compact orbifold, and E a complex vector orbibundle over X. A linear mapping P : C ∞ (X, E) → C ∞ (X, E) is a (pseudo) differential operator on X of order m iff: (1) the Schwartz kernel of P is smooth outside of a neighborhood of the diagonal in X × X.
(2) for any x 0 ∈ X and for any local trivialization (Ũ × C k , G U ,φ U ) of E over an orbifold is given by the restriction to G U -invariant functions of a (pseudo)differential operatorP of order m on C ∞ (Ũ , C k ) that commutes with the induced G U action on C ∞ (Ũ , C k ).
All our pseudodifferential operators are assumed to be classical (or polyhomogeneous), that is, their complete symbols can be represented as an asymptotic sum of homogeneous components. Denote by Ψ m (X, E) the class of pseudodifferential operators of order m acting on C ∞ (X, E).
It is not hard to show [2,Proposition 3.3] that the operatorP is unique up to a smoothing operator, so it is unique if P is a differential operator. A pseudodifferential operatorP onŨ that commutes with the action of G U has a principal symbol σ(P ) ∈ C ∞ (Ũ × (R n \ {0}), L(C k )) that is invariant with respect to the natural G U -action. One can check that these locally defined functions determine a global smooth section σ(P ) of the vector orbibundle End(π * E) on T * X \ {0}, the principal symbol of P . (Here π : T * X → X is the bundle map and π * E is the pull-back of the orbibundle E under the map π.) The pseudodifferential operator P on X is elliptic ifP is elliptic for all choices of orbifold charts.

Classical version of the Egorov theorem
Let X be a compact orbifold, and P an elliptic, first-order pseudodifferential operator on X with real principal symbol p ∈ S 1 (T * X). Let f t be the bicharacteristic flow of the operator P , that is, the Hamiltonian flow of p on the cotangent bundle T * X.
As an example, one can consider P = √ ∆ X , where ∆ X is the Laplace-Beltrami operator associated to a Riemannian metric g X on X. Its bicharacteristic flow is the geodesic flow of the metric g X on T * X.
The classical version of the Egorov theorem for orbifolds reads as follows.
Theorem 1. For any pseudodifferential operator A of order 0 with the principal symbol a ∈ S 0 (T * X), the operator is a pseudodifferential operator of order 0. Moreover, its principal symbol a t ∈ S 0 (T * X) is given by The proof of Theorem 1 will be given in Section 2.8.
Remark 1. The classical Egorov theorem plays a crucial role in the proof of the well-known result due to Shnirelman, stating that the ergodicity of the bicharacteristic flow of a first-order elliptic pseudodifferential operator on a compact manifold implies quantum ergodicity for the operator itself. We will discuss these issues for orbifolds elsewhere.

The Egorov theorem for matrix-valued operators
Using the results of [7], one can easily extend Theorem 1 to pseudodifferential operators acting on sections of a vector orbibundle over a compact orbifold. Let X be a compact orbifold, E a complex vector orbibundle on X, and P an elliptic, first-order pseudodifferential operator acting on C ∞ (X, |T X| 1/2 ⊗ E) with real scalar principal symbol p 1 ∈ S 1 (T * X, End(π * E)), p 1 (x, ξ) = h(x, ξ)id Ex with h ∈ C ∞ (T * X \{0}). (Here |T X| 1/2 denotes the half-density line orbibundle on X.) Let H h be the associated Hamiltonian vector field and f t the associated Hamiltonian flow on T * X.
Consider a local trivialization (Ũ × C k , G U ,φ U ) over an orbifold chart (Ũ , G U , φ U ). LetP be the corresponding G U -invariant, matrix-valued first-order pseudodifferential operator onŨ . The subprincipal symbol ofP is a smooth matrix-valued function sub(P ) ∈ C ∞ (Ũ ×(R n \{0}), L(C k )) defined by where p k is the homogeneous of degree k component in the asymptotic expansion of the complete symbol ofP . If E is trivial, the subprincipal symbol turns out to be well defined as a function on T * X. In the general case, we consider the first-order differential operator [7], the operator ∇ H h is invariantly defined as a covariant derivative (a partial connection) on the vector orbibundle π * E on T * X \ {0} along the Hamiltonian vector field H h .
This determines a flow α t on π * E by There is also a flow Ad(α t ) on End(π * E), which, in its turn, induces a flow Ad(α t ) * on the space Theorem 2. For any A ∈ Ψ 0 (X, E) with the principal symbol a ∈ S 0 (T * X, End(π * E)), the operator is a pseudodifferential operator of order 0. Moreover, its principal symbol a t ∈S 0 (T * X, End(π * E)) is given by a t = Ad(α t ) * a.

Quotient presentations
We will need the following well-known fact from orbifold theory due to Kawasaki [8,9] (see, for instance, [2,15] for a detailed proof).
Proposition 1. Let M be a smooth manifold and K a compact Lie group acting on M with finite isotropy groups. Then the quotient X = M/K (with the quotient topology) has a natural orbifold structure. Conversely, any orbifold is a quotient of this type.
Any representation of an orbifold X as the quotient X ∼ = M/K of an action of a compact Lie group K on a smooth manifold M with finite isotropy groups will be called a quotient presentation for X. There is a classical example of a quotient presentation for an orbifold X due to Satake. Choose a Riemannian metric on X. It can be shown that the orthonormal frame bundle M = F (X) of the Riemannian orbifold X is a smooth manifold, the group K = O(n) acts smoothly, effectively and locally freely on M , and M/K ∼ = X.
Remark 2. More generally, one can consider realizations of an orbifold as the leaf space of a foliated manifold with all leaves compact and all holonomy groups finite (a generalized Seifert fibration). The holonomy groupoid of such a foliation is a proper effective groupoid, which provides a characterization of orbifolds in terms of groupoids.
Note that if X ∼ = M/K is a quotient presentation for X, then the pull-back by the natural projection M → X is an isomorphism C ∞ (X) ∼ = C ∞ (M ) K between the smooth functions on X and the K-invariant functions on M .
There is the following extension of Proposition 1 observed by Kawasaki (see, for instance, [2] for a detailed proof).

Proposition 2.
Let E be a smooth vector bundle over a smooth manifold M and K a compact Lie group acting on E by vector bundle isomorphisms such that isotropy groups on M are finite. Then the quotient map E = E/K → X = M/K has a canonical structure of a vector orbibundle. Conversely, any vector orbibundle is a quotient of this type.

Moreover (see, for instance, [2, Proposition 2.4])
, if X ∼ = M/K is a quotient presentation for X, E is a vector orbibundle on X, and E is the smooth vector bundle over M given by Proposition 2, then C ∞ (M, E) K ∼ = C ∞ (X, E).

Quotient presentations of the cotangent bundle
A quotient presentation X ∼ = M/K for the orbifold X gives rise to a quotient presentation for the cotangent bundle T * X of X in the following way. The action of K on M induces an action of K on the cotangent bundle T * M . Denote by k the Lie algebra of K. For any v ∈ k, denote by v M the corresponding infinitesimal generator of the K-action on M . For any x ∈ M , vectors of the form v M (x) with v ∈ k span the tangent space T x (Kx) to the K-orbit, passing through x. Denote Since the action is locally free, the disjoint union is a subbundle of the cotangent bundle T * M , called the conormal bundle. The conormal bundle T * K M is a K-invariant submanifold of T * M such that This gives a quotient presentation for T * X. This construction is a particular case of the symplectic reduction. Indeed, the K-action on T * M is a Hamiltonian action with the corresponding momentum map J : T * M → k * given by Thus, we see that and the quotient T * K M/K is the Marsden-Weinstein reduced space M 0 at 0 ∈ k * [14]. Using quotient presentations, one can show the existence of Hamiltonian flows on T * X. Let X ∼ = M/K be a quotient presentation for X. Consider a Hamiltonian H ∈ C ∞ (T * X) as a smooth K-invariant function on T * K M . LetH ∈ C ∞ (T * M ) K be an arbitrary extension of H to a smooth K-invariant function on T * M . Letf t be the Hamiltonian flow ofH on T * M . SinceH is K-invariant, the flowf t preserves the conormal bundle T * K M , and its restriction to T * K M (denoted also byf t ) commutes with the K-action on T * K M . So the flowf t on T * K M induces a flow f t on the quotient T * K M/K = T * X, which is called the reduced flow. One can show that this flow is a Hamiltonian flow on T * X with Hamiltonian H.

Pseudodifferential operators and quotient presentations
Let X be a compact orbifold and E a complex vector orbibundle on X. Let X ∼ = M/K be a quotient presentation for X and let E be the lift of E to a smooth vector bundle over M given by Proposition 2. Let us consider C ∞ (X, E) (resp. L 2 (X, E)) as a subspace For any pseudodifferential operator B ∈ Ψ m (M, E), define its transversal principal symbol σ(B) ∈ S m (T * K M, End(π * E)), whereπ : T * M \ {0} → M is the bundle map, as the restriction of the principal symbol of B to T * K M . If B is K-invariant, then σ(B) is K-invariant, so it can be identified with an element of the space S m (T * X, End(π * E)).
We have the following fact, relating pseudodifferential operators on M and X (see [2,Proposition 3.4

] and [17, Proposition 2.3]).
Proposition 3. Given a linear operator A : C ∞ (X, E) → C ∞ (X, E), A is a pseudodifferential operator on X iff there exists a pseudodifferential operatorÃ : C ∞ (M, E) → C ∞ (M, E) (of the same order as A) which commutes with K, such that ΠÃΠ = A.

Proofs of Theorems 1 and 2
Let P ∈ Ψ 1 (X) be an elliptic operator on X with real principal symbol p ∈ S 1 (T * X) and A ∈ Ψ 0 (X). Let X ∼ = M/K be a quotient presentation for X. TakeP ∈ Ψ 1 (M ) andÃ ∈ Ψ 0 (M ) as in Proposition 3. Without loss of generality, we can also assume thatP is elliptic and its principal symbol is real. So we have ΠP Π = P, ΠÃΠ = A.
By the classical Egorov theorem, the operator e itPÃ e −itP is in Ψ 0 (M ). Since it commutes with the K-action, we conclude that e itP Ae −itP ∈ Ψ 0 (X). Moreover, for the transversal principal symbol of e itPÃ e −itP , we have wheref t is the Hamiltonian flow ofp, the principal symbol ofP . By Proposition 3, we have , that immediately completes the proof of Theorem 1.
The proof of Theorem 2 is similar. Under the assumptions of Theorem 2, let X ∼ = M/K be a quotient presentation for X and E is the lift of E to a K-equivariant smooth vector bundle on M given by Proposition 2. TakeP ∈ Ψ 1 (M, E) andÃ ∈ Ψ 0 (M, E) as in Proposition 3. Without loss of generality, we can assume thatP is elliptic and its principal symbol is scalar and real. Letf t be the Hamiltonian flow ofh ∈ C ∞ (T * M \ {0}), the principal symbol ofP . The subprincipal symbol ofP is invariantly defined as a partial connection ∇ Hh on the vector orbibundleπ * E along the Hamiltonian vector field Hh. Therefore, we have the flowα * t on C ∞ (T * M \ {0}, End(π * E)), which satisfies d dtα A similar statement holds for Ad(α t ) * . Taking into account these facts, Theorem 2 is a direct consequence of Egorov's theorem in [7].
3 Noncommutative geometry 3.1 The operator algebras associated with a quotient orbifold Let X be a compact orbifold. The choice of quotient presentation X ∼ = M/K for X allows us to consider X as the orbit space of a Lie group action, which is a typical object of noncommutative geometry. So we can use some notions and ideas of noncommutative geometry.
First, one can consider the smooth crossed product algebra C ∞ (M ) ⋊ K. As a linear space, the involution is given, for a function f ∈ C ∞ (M × K), by Here dh denotes a fixed bi-invariant Haar measure on K.
It is useful to know that the crossed product algebra C ∞ (M ) ⋊ K is associated with a certain groupoid, the transformation groupoid, G = M ⋊ K. As a set, G = M × K. It is equipped with the source map s : M × K → M given by s(x, k) = k −1 x and the target map r : M × K → M given by r(x, k) = x.
For any x ∈ M , there is a natural * -representation of the algebra C ∞ (M × K) in the Hilbert space L 2 (K, dk) given, for f ∈ C ∞ (M × K) and ζ ∈ L 2 (K, dk), by The completion of the involutive algebra C ∞ (M × K) in the norm is called the reduced crossed product C * -algebra and denoted by C(M ) ⋊ r K.
Since K is compact, this algebra coincides with the full crossed product C * -algebra C(M )⋊K, which is defined as the completion of C ∞ (M × K) in the norm where supremum is taken over the set of all * -representations π of the algebra C ∞ (M × K) in Hilbert spaces.
There is also a natural representation of This representation extends to a * -representation of C(M ) ⋊ r K. The C * -algebra C(M ) ⋊ r K can be naturally called the orbifold C * -algebra associated to the quotient presentation X ∼ = M/K. From the point of view of noncommutative geometry, this algebra is a noncommutative analogue of the algebra of continuous functions on the quotient space M/K. It is Morita equivalent to the commutative algebra C(X).

Noncommutative pseudodifferential operators on orbifolds
Let X be a compact orbifold. One can introduce the algebra Ψ * (M/K) of noncommutative pseudodifferential operators on X associated with a quotient presentation X ∼ = M/K.
First, let us start with a local definition. Constructing an appropriate slice for the K-action on M , one can give the following local description of the quotient map p : M → X (see, for instance, [2, Proposition 2.1] for details).

Proposition 4.
For any x ∈ X, there exists an orbifold chart (Ũ , G U , φ U ) defined in a neighborhood U ⊂ X of x such that there exists a K-equivariant diffeomorphism Recall that, by definition, K × G UŨ = (K ×Ũ )/G U , where G U acts on K ×Ũ by γ · (k, y) = kγ −1 , γy , k ∈ K, y ∈Ũ , γ ∈ G U , and the K-action on K × G UŨ is given by the left translations on K.
Now consider an orbifold chart (Ũ , G U , φ U ) as in Proposition 4. For any a ∈ S m (K × K × U × R n ), define an operator the Hamiltonian flowf t on T * M is the geodesic flow of the metric g M , and the reduced Hamiltonian flow f t on T * X is the geodesic flow of the metric g X .
The corresponding quantum dynamics on L 2 (X) and L 2 (M ) are described by the operators P = √ ∆ X andP = √ ∆ M respectively, where ∆ X and ∆ M are the Laplacians of the metrics g X and g M respectively. It is well known that the operator ∆ M can be expressed in terms of Bochner Laplacians acting on sections of vector bundles over X associated with the principal bundle φ : M → X.
In the case when K = O(n) and M is the orthonormal frame bundle F (X) of X, the restriction of the geodesic flowf t to T * K M is closely related with the frame flow on the frame bundle F (X) on X (see [12] for more details).
Remark 5. In this case, both classical and quantum dynamical systems are noncommutative. It would be interesting to extend some basic results on quantum ergodicity to this setting. For instance, one can introduce the notion of ergodicity for the bicharacteristic flow F * t on the noncommutative algebra C ∞ (T * K M ⋊ K) and compare this notion with an appropriate notion of quantum ergodicity for the operator P itself.