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 non...

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Datum:	2011
1. Verfasser:	Kordyukov, Y.A.
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spelling	irk-123456789-1476602019-02-16T01:25:38Z Classical and Quantum Dynamics on Orbifolds Kordyukov, Y.A. 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. 2011 Article Classical and Quantum Dynamics on Orbifolds / Y.A. Kordyukov // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 17 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58J40; 58J42; 58B34 http://dspace.nbuv.gov.ua/handle/123456789/147660 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Kordyukov, Y.A.
spellingShingle	Kordyukov, Y.A. Classical and Quantum Dynamics on Orbifolds Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kordyukov, Y.A.
author_sort	Kordyukov, Y.A.
title	Classical and Quantum Dynamics on Orbifolds
title_short	Classical and Quantum Dynamics on Orbifolds
title_full	Classical and Quantum Dynamics on Orbifolds
title_fullStr	Classical and Quantum Dynamics on Orbifolds
title_full_unstemmed	Classical and Quantum Dynamics on Orbifolds
title_sort	classical and quantum dynamics on orbifolds
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/147660
citation_txt	Classical and Quantum Dynamics on Orbifolds / Y.A. Kordyukov // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 17 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kordyukovya classicalandquantumdynamicsonorbifolds
first_indexed	2025-07-11T02:35:15Z
last_indexed	2025-07-11T02:35:15Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 106, 12 pages Classical and Quantum Dynamics on Orbifolds Yuri A. KORDYUKOV Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky Str., Ufa 450008, Russia E-mail: yurikor@matem.anrb.ru Received September 04, 2011, in final form November 20, 2011; Published online November 23, 2011 http://dx.doi.org/10.3842/SIGMA.2011.106 Abstract. 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. Key words: microlocal analysis; noncommutative geometry; symplectic reduction; quantiza- tion; foliation; orbifold; Hamiltonian dynamics; elliptic operators 2010 Mathematics Subject Classification: 58J40; 58J42; 58B34 1 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 obser- vables) 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 ∈ S1(T ∗X). Let ft 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 ∈ S0(T ∗X), the operator A(t) = eitPAe−itP is a pseudodifferential operator of order 0. The principal symbol at ∈ S0(T ∗X) of this operator is given by the formula at(x, ξ) = a(ft(x, ξ)), (x, ξ) ∈ T ∗X \ {0}. Here Sm(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. mailto:yurikor@matem.anrb.ru http://dx.doi.org/10.3842/SIGMA.2011.106 2 Yu.A. Kordyukov 2 Classical theory 2.1 Orbifolds In this section we briefly review the basic definitions concerning orbifolds. For more details on orbifold theory we refer the reader to the book [1]. Let X be a Hausdorff topological space. An n-dimensional orbifold chart on X is given by a triple (Ũ , GU , φU ), where Ũ ⊂ Rn is a connected open subset, GU is a finite group acting on Ũ smoothly and φU : Ũ → X is a continuous map, which is GU -invariant (φU ◦ g = φU for all g ∈ GU ) and induces a homeomorphism of Ũ/GU onto an open subset U = φU (Ũ) ⊂ X. An embedding λ : (Ũ , GU , φU )→ (Ṽ , GV , φV ) between two such charts is a smooth embedding λ : Ũ → Ṽ with φV ◦ λ = φU . An orbifold atlas on X is a family U = {(Ũ , GU , φU )} of orbifold charts, which cover X and are locally compatible: given any two charts (Ũ , GU , φU ) for φU (Ũ) = U ⊂ X and (Ũ , GV , φV ) for φV (Ṽ ) = V ⊂ X, and a point x ∈ U ∩ V , there exists an open neighborhood W of x and a chart (W̃ ,GW , φW ) for W such that there are embeddings λU : (W̃ ,GW , φW ) ↪→ (Ũ , GU , φU ) and λV : (W̃ ,GW , φW ) ↪→ (Ṽ , GV , φV ). An (effective) orbifold X of dimension n is a paracompact Hausdorff topological space equipped with an equivalence class of n-dimensional orbifold atlases. Throughout in the paper, X will denote a compact orbifold, dimX = n. Let x ∈ X and (Ũ , GU , φU ) be an orbifold chart such that x ∈ U = φU (Ũ). Take any x̃ ∈ Ũ such that φU (x̃) = x. Let Gx̃ ⊂ GU be the isotropy group for x̃. Up to conjugation, this group doesn’t depend on the choice of chart and will be called the local group at x. A point x ∈ X is called regular if the local group at x is trivial and it is called singular otherwise. Denote by Xreg the set of regular points of X and by Xsing the set of singular points of X. Xreg is an open and dense subset of X whose induced orbifold structure is a manifold structure. A function f : X → C is smooth iff for any orbifold chart (Ũ , GU , φU ) the composition f \|U ◦ φU is a smooth function on Ũ . The cotangent bundle T ∗X of X is an orbifold whose atlas is constructed as follows. Let (Ũ , GU , φU ) is an orbifold chart over U ⊂ X. Consider the local cotangent bundle T ∗Ũ = Ũ × Rn. It is equipped with a natural action of the group GU . The projection map T ∗Ũ → Ũ is GU -equivariant, so we obtain a natural projection pU : T ∗Ũ/GU → U , whose fiber p−1U (x) is homeomorphic to Rn/Gx̃. T ∗X is obtained by gluing together the bundles T ∗Ũ/GU over each chart U in the atlas of X. Namely, let (Ũ , GU , φU ) and (Ṽ , GV , φV ) be two orbifold charts for φU (Ũ) = U ⊂ X and φV (Ṽ ) = V ⊂ X respectively, and let x belong to U ∩ V . There exists an open neighborhood W of x and a chart (W̃ ,GW , φW ) for W such that there are embeddings λU : (W̃ ,GW , φW ) ↪→ (Ũ , GU , φU ) and λV : (W̃ ,GW , φW ) ↪→ (Ṽ , GV , φV ). These embeddings give rise to diffeomorphisms λU : W̃ → λU (W̃ ) ⊂ Ũ and λV : W̃ → λV (W̃ ) ⊂ Ṽ , which provide an equivariant diffeomorphism λUV = λV λ −1 U : λU (W̃ )→ λV (W̃ ), the transition function. Then the bundles pU : T ∗Ũ/GU → U and pV : T ∗Ṽ /GV → V are glued by the cotangent map T ∗λV U : T ∗λU (T ∗W̃ )→ T ∗λV (T ∗W̃ ). Like in the manifold case, the cotangent bundle T ∗X carries a canonical symplectic struc- ture. Here by a symplectic form on an orbifold Y we will understand an orbifold atlas U = {(Ũ , GU , φU )} together with a GU -invariant symplectic form ωU on Ũ for each (Ũ , GU , φU ) ∈ U such that, for any transition function λUV : λU (W̃ ) ⊂ Ũ → λV (W̃ ) ⊂ Ṽ as above, we have λ∗UV ωV = ωU . An orbifold Y equipped with a symplectic form ω is called a symplectic orbi- fold. The symplectic structure on T ∗X can be constructed as follows. Consider the orbifold chart (T ∗Ũ , GU , T ∗φU ) induced by an orbifold chart (Ũ , GU , φU ). Then T ∗Ũ carries a canonical symplectic form ωT ∗Ũ , which is invariant with respect to the lifted GU -action. These sym- plectic forms are compatible for two different orbifold charts and define a symplectic form on T ∗X. Classical and Quantum Dynamics on Orbifolds 3 The flow Ft on a symplectic orbifold (Y, ω) is Hamiltonian with a Hamiltonian H ∈ C∞(Y ) if, in any orbifold chart (Ũ , GU , φU ), the infinitesimal generator XH ∈ X (Ũ) of the flow satisfies a standard relation i(XH)ωU = d(H \|U ◦ φU ). 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 pre- sentations, see below). The flow Ft leaves the singular set of the orbifold Y invariant, and its restriction to the regular part Yreg 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. 2.2 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 x0 ∈ X, there exists an orbifold chart (Ũ , GU , φU ) over φU (Ũ) = U ⊂ X with x0 ∈ U and an orbifold chart (Ũ × Rk, GU , φ̃U ) over φ̃U (Ũ × Rk) = p−1(U) ⊂ E (called a local trivialization of E over (Ũ , GU , φU )) such that: 1) the action of GU on Ũ × Rk is an extension of the action of GU on Ũ given by g(x, v) = (gx, ρ(x, g)v), x ∈ Ũ , v ∈ Rk, where ρ is a smooth map from Ũ×GU to the algebra L(Rk) of linear maps in Rk satisfying ρ(gx, h) ◦ ρ(x, g) = ρ(x, hg), g, h ∈ GU , x ∈ Ũ ; (in other words, GU acts by vector bundle isomorphisms of the trivial vector bundle pr1 : Ũ × Rk → Ũ); 2) the map pr1 : Ũ × Rk → Ũ satisfies φU ◦ pr1 = p ◦ φ̃U . 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 : X → E is called C∞, if for each local trivialization (Ũ × Rk, GU , φ̃U ) over an orbifold chart (Ũ , GU , φU ) the restriction s \|U is covered by a GU -invariant smooth section s̃U : Ũ → Ũ × Rk: s \|U ◦ φU = φ̃U ◦ s̃U . We denote by C∞(X,E) the space of smooth section of E on X. Now we turn to pseudodifferential operators. Let X be a compact orbifold, and E a com- plex 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 x0 ∈ X and for any local trivialization (Ũ × Ck, GU , φ̃U ) of E over an orbifold chart (Ũ , GU , φU ) with x0 ∈ U , the operator C∞c (U,E \|U ) 3 f 7→ P (f) \|U ∈ C∞(U,E \|U ) is given by the restriction to GU -invariant functions of a (pseudo)differential operator P̃ of order m on C∞(Ũ ,Ck) that commutes with the induced GU action on C∞(Ũ ,Ck). 4 Yu.A. Kordyukov 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 Ψm(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 operator P̃ is unique up to a smoothing operator, so it is unique if P is a differential operator. A pseudodifferential operator P̃ on Ũ that commutes with the action of GU has a principal symbol σ(P̃ ) ∈ C∞(Ũ × (Rn \{0}),L(Ck)) that is invariant with respect to the natural GU -action. One can check that these locally defined func- tions 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 if P̃ is elliptic for all choices of orbifold charts. 2.3 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 ∈ S1(T ∗X). Let ft 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 gX on X. Its bicharacteristic flow is the geodesic flow of the metric gX 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 ∈ S0(T ∗X), the operator A(t) = eitPAe−itP is a pseudodifferential operator of order 0. Moreover, its principal symbol at ∈ S0(T ∗X) is given by at(x, ξ) = a(ft(x, ξ)), (x, ξ) ∈ T ∗X \ 0. 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. 2.4 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, \|TX\|1/2 ⊗E) with real scalar principal symbol p1 ∈ S1(T ∗X,End(π∗E)), p1(x, ξ) = h(x, ξ)idEx with h ∈ C∞(T ∗X\{0}). (Here \|TX\|1/2 denotes the half-density line orbibundle on X.) Let Hh be the associated Hamiltonian vector field and ft the associated Hamiltonian flow on T ∗X. Consider a local trivialization (Ũ ×Ck, GU , φ̃U ) over an orbifold chart (Ũ , GU , φU ). Let P̃ be the corresponding GU -invariant, matrix-valued first-order pseudodifferential operator on Ũ . The Classical and Quantum Dynamics on Orbifolds 5 subprincipal symbol of P̃ is a smooth matrix-valued function sub(P̃ ) ∈ C∞(Ũ×(Rn\{0}),L(Ck)) defined by sub(P̃ )(x, ξ) = p0(x, ξ)− 1 2i n∑ j=1 ∂2p1 ∂xj∂ξj (x, ξ), x ∈ Ũ , ξ ∈ Rn \ {0}, where pk is the homogeneous of degree k component in the asymptotic expansion of the complete symbol of P̃ . 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 ∇Hh := Hh + i sub(P̃ ) acting on C∞(Ũ × (Rn \ {0}),Ck). By [7], the operator ∇Hh is invariantly defined as a co- variant derivative (a partial connection) on the vector orbibundle π∗E on T ∗X \ {0} along the Hamiltonian vector field Hh. This determines a flow αt on π∗E by αt(x, ξ, v) = (x(t), ξ(t), v(t)), (x, ξ) ∈ T ∗X \ {0}, v ∈ (π∗E)(x,ξ) ∼= Ex, where (x(t), ξ(t)) = ft(x, ξ) and, in local coordinates, v(t) satisfies dv(t) dt = i sub(P̃ )(x(t), ξ(t))v(t). The induced flow α∗t on C∞(T ∗X \ {0}, π∗E) satisfies d dt α∗t f = ∇Hh f. There is also a flow Ad(αt) on End(π∗E), which, in its turn, induces a flow Ad(αt) ∗ on the space C∞(T ∗X \ {0},End(π∗E)). If f ∈ C∞(T ∗X \ {0},End(π∗E)), d dt Ad(αt) ∗f = [∇Hh , f ]. Theorem 2. For any A ∈ Ψ0(X,E) with the principal symbol a ∈ S0(T ∗X,End(π∗E)), the operator A(t) = eitPAe−itP is a pseudodifferential operator of order 0. Moreover, its principal symbol at∈S0(T ∗X,End(π∗E)) is given by at = Ad(αt) ∗a. 2.5 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. 6 Yu.A. Kordyukov 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). 2.6 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 vM the corresponding infinitesimal generator of the K-action on M . For any x ∈M , vectors of the form vM (x) with v ∈ k span the tangent space Tx(Kx) to the K-orbit, passing through x. Denote (T ∗KM)x = { ξ ∈ T ∗xM : 〈ξ, vM (x)〉 = 0 for any v ∈ k } . Since the action is locally free, the disjoint union T ∗KM = ⊔ x∈M (T ∗KM)x is a subbundle of the cotangent bundle T ∗M , called the conormal bundle. The conormal bun- dle T ∗KM is a K-invariant submanifold of T ∗M such that T ∗KM/K ∼= T ∗X. 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 〈J(x, ξ), v〉 = 〈ξ, vM (x)〉, (x, ξ) ∈ T ∗M, v ∈ k. Classical and Quantum Dynamics on Orbifolds 7 Thus, we see that T ∗KM = J−1(0), and the quotient T ∗KM/K is the Marsden–Weinstein reduced space M0 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 ∗KM . Let H̃ ∈ C∞(T ∗M)K be an arbitrary extension of H to a smooth K-invariant function on T ∗M . Let f̃t be the Hamiltonian flow of H̃ on T ∗M . Since H̃ is K-invariant, the flow f̃t preserves the conormal bundle T ∗KM , and its restriction to T ∗KM (denoted also by f̃t) commutes with the K-action on T ∗KM . So the flow f̃t on T ∗KM induces a flow ft on the quotient T ∗KM/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. 2.7 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. L2(X,E)) as a subspace C∞(M, E)K (resp. L2(M, E)K) of C∞(M, E) (resp. L2(M, E)), which consists of K-invariant functions on M . Let Π : L2(M, E) → L2(M, E)K ∼= L2(X,E) be the orthogonal projection onto L2(M, E)K in L2(M, E). It is clear that Π(C∞(M, E)) = C∞(M, E)K ∼= C∞(X,E). For any pseudodifferential operator B ∈ Ψm(M, E), define its transversal principal symbol σ(B) ∈ Sm(T ∗KM,End(π̃∗E)), where π̃ : T ∗M \ {0} → M is the bundle map, as the restriction of the principal symbol of B to T ∗KM . If B is K-invariant, then σ(B) is K-invariant, so it can be identified with an element of the space Sm(T ∗X,End(π∗E)). We have the following fact, relating pseudodifferential operators on M and X (see [2, Propo- sition 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 operator Ã : C∞(M, E) → C∞(M, E) (of the same order as A) which commutes with K, such that ΠÃΠ = A. The transverse principal symbol σ(Ã) ∈ S0(T ∗KM,End(π̃∗E))K of Ã coincides with the princi- pal symbol σ(A) ∈ S0(T ∗X,End(π∗E)) of A under the identification C∞(T ∗KM,End(π̃∗E))K ∼= C∞(T ∗X,End(π∗E)). Now we are ready to complete the proofs of Theorems 1 and 2. 2.8 Proofs of Theorems 1 and 2 Let P ∈ Ψ1(X) be an elliptic operator on X with real principal symbol p ∈ S1(T ∗X) and A ∈ Ψ0(X). Let X ∼= M/K be a quotient presentation for X. Take P̃ ∈ Ψ1(M) and Ã ∈ Ψ0(M) as in Proposition 3. Without loss of generality, we can also assume that P̃ is elliptic and its principal symbol is real. So we have ΠP̃Π = P, ΠÃΠ = A. Since P̃ is K-invariant, we have P̃Π = ΠP̃ . Using these facts, we easily derive that eitPAe−itP = ΠeitP̃ Ãe−itP̃Π. By the classical Egorov theorem, the operator eitP̃ Ãe−itP̃ is in Ψ0(M). Since it commutes with the K-action, we conclude that eitPAe−itP ∈ Ψ0(X). Moreover, for the transversal principal 8 Yu.A. Kordyukov symbol of eitP̃ Ãe−itP̃ , we have σ ( eitP̃ Ãe−itP̃ ) (ν) = σ(Ã)(f̃t(ν)), ν ∈ T ∗KM \ {0}, where f̃t is the Hamiltonian flow of p̃, the principal symbol of P̃ . By Proposition 3, we have σ ( eitP̃ Ãe−itP̃ ) (ν) = σ ( eitPAe−itP ) (ν), ν ∈ T ∗KM/K \ {0} ∼= T ∗X \ {0}, 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 forX and E is the lift of E to aK-equivariant smooth vector bundle onM given by Proposition 2. Take P̃ ∈ Ψ1(M, E) and Ã ∈ Ψ0(M, E) as in Proposition 3. Without loss of generality, we can assume that P̃ is elliptic and its principal symbol is scalar and real. Let f̃t be the Hamiltonian flow of h̃ ∈ C∞(T ∗M \ {0}), the principal symbol of P̃ . The subprincipal symbol of P̃ is invariantly defined as a partial connection∇Hh̃ on the vector orbibundle π̃∗E along the Hamiltonian vector field Hh̃. Therefore, we have the flow α̃∗t on C∞(T ∗M \ {0},End(π̃∗E)), which satisfies d dt α̃∗t f = ∇Hh̃ f, f ∈ C∞(T ∗M \ {0},End(π̃∗E)), and the flow Ad(α̃t) ∗ on C∞(T ∗M \ {0},End(π̃∗E)), which satisfies d dt Ad(α̃t) ∗f = [∇Hh̃ , f ], f ∈ C∞(T ∗M \ {0},End(π̃∗E)). By K-invariance, the restriction of α̃∗t to C∞(T ∗KM \ {0}, π̃∗E) takes C∞(T ∗KM \ {0}, π̃∗E)K to itself, and there is the following commutative diagram: C∞(T ∗KM \ {0}, π̃∗E)K α̃∗t−−−−→ C∞(T ∗KM \ {0}, π̃∗E)Ky∼= y∼= C∞(T ∗X \ {0}, π∗E) α∗t−−−−→ C∞(T ∗X \ {0}, π∗E) A similar statement holds for Ad(α̃t) ∗. 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) oK. As a linear space, C∞(M) o K = C∞(M × K). The product in C∞(M × K) is given, for any functions f, g ∈ C∞(M ×K), by (f ∗ g)(x, k) = ∫ K f(x, h)g ( h−1x, h−1k ) dh, (x, k) ∈M ×K, (1) the involution is given, for a function f ∈ C∞(M ×K), by f∗(x, k) = f(k−1x, k−1), (x, k) ∈M ×K. (2) Here dh denotes a fixed bi-invariant Haar measure on K. Classical and Quantum Dynamics on Orbifolds 9 It is useful to know that the crossed product algebra C∞(M)oK is associated with a certain groupoid, the transformation groupoid, G = M oK. As a set, G = M ×K. It is equipped with the source map s : M ×K → M given by s(x, k) = k−1x 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 L2(K, dk) given, for f ∈ C∞(M ×K) and ζ ∈ L2(K, dk), by Rx(f)ζ(k) = ∫ K f ( k−1x, k−1k1 ) ζ(k1) dk1. The completion of the involutive algebra C∞(M ×K) in the norm ‖f‖ = sup x∈M ‖Rx(f)‖ is called the reduced crossed product C∗-algebra and denoted by C(M) or K. Since K is compact, this algebra coincides with the full crossed product C∗-algebra C(M)oK, which is defined as the completion of C∞(M ×K) in the norm ‖k‖max = sup ‖π(k)‖, 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 C∞(M ×K) in L2(M) defined for f ∈ C∞(M ×K) and u ∈ L2(M) by R(f)u(x) = ∫ K f(x, k)u ( k−1x ) dk, x ∈M. (3) This representation extends to a ∗-representation of C(M) or K. The C∗-algebra C(M)orK 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). 3.2 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 , GU , φU ) defined in a neigh- borhood U ⊂ X of x such that there exists a K-equivariant diffeomorphism p−1(U) ∼= K ×GU Ũ . Recall that, by definition, K ×GU Ũ = (K × Ũ)/GU , where GU acts on K × Ũ by γ · (k, y) = ( kγ−1, γy ) , k ∈ K, y ∈ Ũ , γ ∈ GU , and the K-action on K ×GU Ũ is given by the left translations on K. 10 Yu.A. Kordyukov Now consider an orbifold chart (Ũ , GU , φU ) as in Proposition 4. For any a ∈ Sm(K ×K × Ũ × Rn), define an operator Ā : C∞c (K × Ũ)→ C∞(K × Ũ) by the formula Āu(k, y) = (2π)−n ∫ ei(y−y ′)ηa(k, k′, y, η)u(k′, y′) dk′ dy′ dη, (4) where u ∈ C∞c (K × Ũ), k ∈ K, y ∈ Ũ . Assume that the operator Ā commutes with the action of GU on K × Ũ . Then Ā defines an operator A : C∞c (K ×GU Ũ) ∼= C∞c ( p−1(U) ) → C∞(K ×GU Ũ) ∼= C∞ ( p−1(U) ) . If, in addition, the Schwartz kernel of the operator Ā is compactly supported in (K×Ũ)×(K×Ũ), then the operator A acts from C∞c (p−1(U)) to C∞c (p−1(U)) and can be extended in a trivial way to an operator in C∞(M). Such an operator will be called an elementary operator of class Ψm(M/K) associated to the orbifold chart (Ũ , GU , φU ). By definition, the class Ψm(M/K) consists of all operators A in C∞(M), which can be represented in the form A = d∑ i=1 Ai +K, where Ai is an elementary operator of class Ψ∗(M/K) associated to an orbifold chart (Ũi,GUi ,φUi) as in Proposition 4, i = 1, . . . , d, and K ∈ Ψ−∞(M). Remark 3. When the group K is discrete (and, therefore, finite), the algebra Ψ∗(M/K) is the crossed product algebra Ψ∗(X) oK. The principal symbol of the operator Ā given by (4) is a smooth function on K ×K × Ũ × (Rn \ {0}) given by σ(Ā)(k, k′, y, η) = am(k, k′, y, η), k, k′ ∈ K, y ∈ Ũ , η ∈ Rn \ {0}, where am is the degree m homogeneous component of the complete symbol a. This function is GU -invariant and defines a smooth function on (K ×K × Ũ × (Rn \ {0}))/GU . We put σ(A)((k, y, η), k′) = σ(Ā) ( k, (k′)−1k, y, η ) , (k, y, η) ∈ (K × Ũ × (Rn \ {0}))/GU , k′ ∈ K. (5) Let Sm(T ∗KM oK) be the space of all smooth functions on (T ∗KM \ {0}) oK homogeneous of degree m with respect to the R-action given by the multiplication in the fibers of the vector bundle π : T ∗KM →M . The principal symbol σ(A) of an operator A ∈ Ψm(M/K) given in local coordinates by the formula (5) is globally defined as an element of the space Sm(T ∗KM oK). One can introduce the structure of involutive algebra on S∗(T ∗KM o K) (see (1) and (2)), and show that the principal symbol mapping σ : Ψm(M/K)→ Sm(T ∗KM oK) satisfies, for any A ∈ Ψm1(M/K) and B ∈ Ψm2(M/K), σ(AB) = σ(A)σ(B), σ(A∗) = σ(A)∗. Example 1. For any f ∈ C∞(M ×K), the operator R(f) given by (3) belongs to Ψ0(M/K) and its principal symbol σ(R(f)) ∈ S0(T ∗KM oK) is given by σ(R(f)) = π∗Gf, where πG : T ∗KM ×K →M ×K is the natural projection. Classical and Quantum Dynamics on Orbifolds 11 3.3 Noncommutative Egorov theorem As above, let X be a compact orbifold and X ∼= M/K a quotient presentation for X. Assume that P is an elliptic, first-order pseudodifferential operator on X with real principal symbol p ∈ S1(T ∗X) ∼= S1(T ∗KM)K . By Proposition 3, there exists an elliptic, first-order pseudo- differential operator P̃ with real principal symbol p̃ ∈ S1(T ∗M)K , which commutes with K and whose restriction to C∞(M)K agrees with P . Denote by f̃t the Hamiltonian flow of p̃ on T ∗M \ {0}. Recall that the flow f̃t preserves the conormal bundle T ∗KM , and its restriction to T ∗KM commutes with the K-action on T ∗KM . Define a flow Ft on T ∗KM oK by the formula Ft(ν, k) = (f̃t(ν), k), (ν, k) ∈ T ∗KM oK. Observe that, due to K-invariance of the flow f̃t, the induced map F ∗t on C∞(T ∗KM oK) is an involutive algebra automorphism. The one-parameter automorphism group F ∗t of the algebra C∞(T ∗KMoK) is called the noncommutative bicharacteristic flow of the operator P (associated to the quotient presentation X ∼= M/K). Remark 4. One can show that the flow F ∗t is Hamiltonian with respect to the natural noncom- mutative Poisson structure on the algebra C∞(T ∗KMoK) and p is the corresponding Hamiltonian (see [13] for more details). The noncommutative version of the Egorov theorem for orbifolds reads as follows. Theorem 3. For any A ∈ Ψm(M/K) with the principal symbol a ∈ Sm(T ∗KMoK), the operator Φt(A) = eitP̃Ae−itP̃ is an operator of class Ψm(M/K). Moreover, the principal symbol a(t) ∈ Sm(T ∗KMoK) of Φt(A) is given by a(t) = F ∗t (a). Proof. The K orbits on M define a foliation F on M with compact leaves. The holonomy groupoid G of this foliation coincides with the transformation groupoid M oK. The conormal bundle of F coincides with T ∗KM . As it is well-known in foliation theory, the conormal bundle of the foliation carries a natural foliation, FN , on T ∗KM , called the linearized or the lifted foliation. In our case, this foliation is given by the K orbits on T ∗KM . The holonomy groupoid GFN of FN is the transformation groupoid T ∗KM oK. The algebra Ψ∗(M/K) of noncommutative pseudodifferential operators associated with the quotient presentation X ∼= M/K is a particular case of the algebra Ψ∗,−∞(M,F) of transversal pseudodifferential operators on the compact foliated manifold (M,F) introduced in [10]. In this setting, Theorem 3 is a straightforward consequence of the Egorov theorem for transversally elliptic operators proved in [11] (see also [12]). � Example 2. Suppose that X is a compact manifold (considered as an orbifold). Then a quotient presentation for X is just a K-principal bundle φ : M → X. Let gX be a Riemannian metric on X. Choose a bi-invariant metric on K and a connection on the principal bundle φ : M → X. There exists a unique K-invariant metric on X, which makes the map φ : (M, gM ) → (X, gX) into a Riemannian submersion, with the fibers isometric to K. Such a metric is sometimes called the Kaluza–Klein metric of the connection. Then, for a Hamiltonian H ∈ C∞(T ∗X \ {0}) given by H(x, ξ) = \|ξ\|gX , (x, ξ) ∈ T ∗X, 12 Yu.A. Kordyukov the Hamiltonian flow f̃t on T ∗M is the geodesic flow of the metric gM , and the reduced Hamil- tonian flow ft on T ∗X is the geodesic flow of the metric gX . The corresponding quantum dynamics on L2(X) and L2(M) are described by the operators P = √ ∆X and P̃ = √ ∆M respectively, where ∆X and ∆M are the Laplacians of the metrics gX and gM 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 whenK = O(n) andM is the orthonormal frame bundle F (X) ofX, the restriction of the geodesic flow f̃t to T ∗KM 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 ∗KM oK) and compare this notion with an appropriate notion of quantum ergodicity for the operator P itself. Acknowledgements The author was partially supported by the Russian Foundation of Basic Research (grant no. 10-01-00088). References [1] Adem A., Leida J., Ruan Y., Orbifolds and stringy topology, Cambridge Tracts in Mathematics, Vol. 171, Cambridge University Press, Cambridge, 2007. [2] Bucicovschi B., Seeley’s theory of pseudodifferential operators on orbifolds, math.DG/9912228. [3] Dryden E.B., Gordon C.S., Greenwald S.J., Webb D.L., Asymptotic expansion of the heat kernel for orbi- folds, Michigan Math. J. 56 (2008), 205–238. [4] Girbau J., Nicolau M., Pseudodifferential operators on V -manifolds and foliations. I, Collect. Math. 30 (1979), 247–265. 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Geom. 40 (2011), 47–65. http://dx.doi.org/10.1017/CBO9780511543081 http://arxiv.org/abs/math.DG/9912228 http://dx.doi.org/10.1307/mmj/1213972406 http://dx.doi.org/10.1007/s10455-009-9159-7 http://arxiv.org/abs/0805.1003 http://dx.doi.org/10.1007/s00220-006-0176-0 http://arxiv.org/abs/math.SP/0607616 http://dx.doi.org/10.1016/0040-9383(78)90013-7 http://dx.doi.org/10.1007/BF02677838 http://dx.doi.org/10.1007/s11040-004-6495-5 http://arxiv.org/abs/math.DG/0407435 http://dx.doi.org/10.1016/j.geomphys.2007.08.002 http://dx.doi.org/10.1016/j.geomphys.2007.08.002 http://arxiv.org/abs/0708.1660 http://dx.doi.org/10.1007/s11784-010-0016-x http://dx.doi.org/10.1007/s11784-010-0016-x http://arxiv.org/abs/0912.1923 http://dx.doi.org/10.1016/0034-4877(74)90021-4 http://dx.doi.org/10.1017/CBO9780511615450 http://dx.doi.org/10.1017/CBO9780511615450 http://dx.doi.org/10.2307/2944350 http://dx.doi.org/10.1007/s10455-010-9244-y 1 Introduction 2 Classical theory 2.1 Orbifolds 2.2 Pseudodifferential operators on orbifolds 2.3 Classical version of the Egorov theorem 2.4 The Egorov theorem for matrix-valued operators 2.5 Quotient presentations 2.6 Quotient presentations of the cotangent bundle 2.7 Pseudodifferential operators and quotient presentations 2.8 Proofs of Theorems 1 and 2 3 Noncommutative geometry 3.1 The operator algebras associated with a quotient orbifold 3.2 Noncommutative pseudodifferential operators on orbifolds 3.3 Noncommutative Egorov theorem References

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