› ~ramacher › 1410.1096v1.pdfquantum ergodicity and reduction benjamin kuster and pablo ramacher...

QUANTUM ERGODICITY AND REDUCTION

BENJAMIN KUSTER AND PABLO RAMACHER

Abstract. We prove an equivariant quantum ergodicity theorem for Schrodinger operators on a

closed connected Riemannian manifold M with an isometric effective action of a compact connected

Lie group G, relying on recent results on singular equivariant asymptotics. The theorem requiresonly that the reduced Hamiltonian flow on the principal stratum of the symplectic reduction of M is

ergodic, and it implies an equivariant version of the Shnirelman-Zelditch-Colin-de-Verdiere theorem,

as well as an equidistribution theorem for the eigenfunctions of the Laplace-Beltrami operator in eachisotypic component of L2(M). In case that G is trivial, one recovers the classical results.

Contents

1. Introduction 12. Setup and background 73. An equivariant trace formula 224. Generalized equivariant semiclassical Weyl law 295. Equivariant classical ergodicity 326. Equivariant quantum ergodicity 337. Equivariant quantum limits for the Laplace-Beltrami operator 408. An example 45References 50

1. Introduction

1.1. General background. Let M be a closed connected Riemannian manifold with Riemannianmeasure dM of dimension n, and denote by ∆ the Laplace-Beltrami operator on M with domainC∞(M). One of the central problems of ergodic theory is to study the properties of eigenfunctionsof −∆ in the limit of large eigenvalues. Concretely, let uj be an orthonormal basis of L2(M) ofeigenfunctions of −∆ with respective eigenvalues Ej, repeated according to their multiplicity. AsEj → ∞, one is interested among other things in the pointwise convergence of the uj , bounds of theLp-norms of the uj for 1 ≤ p ≤ ∞, and the weak convergence of the measures |uj |2dM .

One of the main conjectures in ergodic theory is the Rudnick-Sarnak-Conjecture [21] on quantumunique ergodicity (QUE). It says that if M has negative sectional curvature, the whole sequence|uj |2dM converges weakly to the normalized Riemannian measure (volM)−1dM as Ej → ∞. It hasbeen verified in certain arithmetic situations by Lindenstrauss [17], but in general, the conjectureis still very open. The guiding idea behind these questions is the correspondence principle of semi-classical physics. To explain this in more detail, consider the unit cotangent bundle S∗M of M , whichcorresponds to the phase space of a classical free particle moving with constant velocity. Each point inS∗M represents a state of the classical system, its motion being given by the geodesic flow in S∗M , andclassical observables correspond to functions a ∈ C∞(S∗M). On the other hand, by the Kopenhageninterpretation of quantum mechanics, quantum observables correspond to self-adjoint operators A inthe Hilbert space L2(M). The elements ψ ∈ L2(M) are interpreted as the states of the quantummechanical system, and the expectation value for measuring the property A while the system is in the

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state ψ is given by 〈Aψ,ψ〉L2(M). The transition between the classical and the quantum-mechanical

picture is given by a map

Sl(M) 3 a 7−→ Op~(a),

where Op~(a) is a pseudodifferential operator depending on Planck’s constant ~, and C∞c (T ∗M) ⊂Sl(M) ⊂ C∞(T ∗M), l ∈ R, denotes a suitable space of symbol functions. Op~(a) is regarded as the“quantization” of the classical observable a. The correspondence principle then says that in the limit ofhigh energies, the quantum mechanical system should behave like the corresponding classical system.Next, consider the distributions1

µj : C∞(S∗M) −→ C, a 7−→ 〈Op~(a)uj , uj〉L2(M) ,

which define a family of measures on S∗M . If it exists, the distribution limit µ = limj→∞ µj constitutesa so-called microlocal lift or quantum limit for the eigenfunction sequence uj, and the measure onS∗M defined by a quantum limit is a probability measure invariant under the geodesic flow. Since themeasure on S∗M defined by a quantum limit µ projects to a weak limit of the measures µj = |uj |2dM ,one can reduce the equidistribution problem to the classification of quantum limits. So far, the mostgeneral result, going back to work of Shnirelman [22], Zelditch [26], and Colin de Verdiere [6], is the

following. If d(S∗M) is the Liouville measure on S∗M , then 1N

∑Nj=1 µj converges weakly to d(S∗M)

as N → ∞. Furthermore, if the geodesic flow on S∗M is ergodic, then there exists a subsequenceujkk∈N of density 1 such that the measures defined by µjk converge weakly to d(S∗M) and µjk todM . Intuitively, the geodesic flow being ergodic means that the geodesics are distributed on S∗Min a sufficiently chaotic way with respect to Liouville measure. In general, a measure-preservingtransformation T : X → X on a measure space (X, µ) is called ergodic if T−1(A) = A impliesµ(A) ∈ 0, µ(X) for every measurable set A ⊂ X.

1.2. Problem. In this article, we will address the problem of finding quantum limits for sequences ofeigenfunctions of Schrodinger operators in case that M carries additional symmetries. Thus, let G bea compact connected Lie group, and let us assume that G acts on M in an isometric and effective way.Note that the action may be singular, that is, there might be various different orbit types. Considerfurther the Schrodinger operator

P (h) = −h2∆ + V, P (h) : C∞(M)→ C∞(M), h ∈ R>0,

with a G-invariant potential V ∈ C∞(M,R), where V is identified with the operator given by pointwise

multiplication with V . The Schrodinger operator P (h) has a unique self-adjoint extension as anunbounded operator in L2(M)

P (h) : H2(M)→ L2(M),

where H2(M) ⊂ L2(M) denotes the second Sobolev space. We will call P (h) a Schrodinger operator,too. For each h > 0, the spectrum of P (h) is discrete and bounded from below, and its eigenvaluesform a sequence unbounded towards +∞. When studying the spectral asymptotics of Schrodingeroperators, one often uses the semiclassical method. Instead of studying the spectral properties of P (h)for fixed h = ~ and large energies, one considers fixed energy intervals, allowing h > 0 to become small.The two methods are essentially equivalent. In the special case V ≡ 0, the Schrodinger operator isjust a rescaled version of −∆ so that the semiclassical method can be used to study the spectralasymptotics of the Laplace-Beltrami operator. Now, since P (h) commutes with the G-action, onecan use representation theory to study the spectrum of P (h) in a more refined way. Indeed, by thePeter-Weyl theorem, the left-regular representation of G on L2(M) has an orthogonal decompositioninto isotypic components given by

L2(M) =⊕χ∈G

L2χ(M), L2

χ(M) = Tχ L2(M),

1Here we are regarding s ∈ C∞(S∗M) as an element in C∞(T ∗M) by extending it using Ehresmann’s theorem and

a cutoff function. By this theorem, there is an open tubular neighbourhood N of S∗M in T ∗M with radius δ, such that

N is diffeomorphic to (−δ, δ)× S∗M [10].

QUANTUM ERGODICITY AND REDUCTION 3

with associated projections Tχ : L2(M)→ L2χ(M). Since P (h) commutes with each Tχ, we can consider

the restricted operators P (h)χ := P (h)|L2χ(M)

: L2χ(M)→ L2

χ(M), and study their spectral asymptotics.

Conversely, each eigenspace of the Schrodinger operator P (h) constitutes a unitary G-module, and itsdecomposition into a direct sum of irreducible G-representations represents the so-called fine structureof the spectrum of P (h). Note that, so far, it is a priori irrelevant whether the group action has variousdifferent orbit types or not.On the other hand, the principal symbol of the Schrodinger operator is given by the G-invariant symbolfunction

p : T ∗M → R, (x, ξ) 7→ ‖ξ‖2x + V (x),

which can be considered as a Hamiltonian on the cotangent bundle T ∗M with its canonical symplecticform, defining a Hamiltonian flow ϕt : T ∗M → T ∗M . For any regular value c ∈ R of p, we can study therestriction of ϕt to the hypersurface Σc := p−1(c) ⊂ T ∗M , the special case V ≡ 0, c = 1 giving thegeodesic flow on S∗M . There is a canonical hypersurface measure dµc on Σc, induced by the measureon T ∗M defined by the canonical symplectic form. In the special case Σc = S∗M , dµc is commonlycalled the Liouville measure. Again, the underlying symmetries can be used to simplify the study of thesystem by performing a procedure called symplectic reduction [18]. Namely, let J : T ∗M → g denotethe momentum map of the Hamiltonian G-action on T ∗M , and consider the space Ω = J−1(0), as

well as the symplectic quotient Ω = Ω/G. In contrast to the situation encountered in the Peter-Weyltheorem, the orbit structure of the underlying G-action on M is not at all irrelevant for the symplecticreduction. Indeed, if G acts with more than one orbit type, the space Ω need not be a manifold. Ingeneral, Ω is a stratified space, where each stratum is a connected component of the union of all orbits

of one particular type. Similarly, Ω has a stratification into orbit types as well. In particular, Ω has a

principal stratum Ωreg, which is the connected smooth manifold consisting of all orbits with minimal

isotropy type. Moreover, Ωreg carries a canonical symplectic structure. The Hamiltonian flow on T ∗M

induces a flow ϕt : Ωreg → Ωreg, which is the Hamiltonian flow associated to the reduced Hamiltonian

p : Ωreg → R induced by p. Therefore, we call ϕt the reduced Hamiltonian flow.Now, let c be a regular value of p and assume, in spite that ϕt itself might not be ergodic on

(Σc, dµc), that the reduced Hamiltonian flow ϕt is ergodic on Σc := p−1(c) ⊂ Ωreg with respect to

the measure dµc on Σc induced by the symplectic measure on Ωreg. Consider further an orthonormalbasis

uχj (h)

of a specific isotypic component L2

χ(M) consisting of eigenfunctions of P (h). It is then

a natural question whether there is a non-trivial family of sets Λ(h)h>0, Λ(h) ⊂ N, such that thedistributions

µχj (h) : C∞c (Σc) −→ C, a 7−→⟨Oph(a)uχj (h), uχj (h)

⟩L2(M)

converge, as j ∈ Λ(h) and h→ 0, to a particular distribution limit with density 1, which would answerthe corresponding question for the measures µχj (h) = |uχj (h)|2dM . In particular, in the special caseV ≡ 0, c = 1, the problem is equivalent to finding quantum limits for sequences of eigenfunctions ofthe Laplace-Beltrami operator in each isotypic component of L2(M).

The general idea behind the approach sketched above can be summarized as follows. One the

one hand, the motivation behind symplectic reduction is that the reduced symplectic manifold Ωreg

will in general be much simpler than the cotangent bundle T ∗M . Hence, ergodicity of the reducedHamiltonian flow may be a much weaker requirement than ergodicity of the non-reduced one. On theother hand, although we require only ergodicity of the reduced Hamiltonian flow, we are still consideringeigenfunctions of Schrodinger operators on the non-reduced manifold M . In this way, we can rely on

the simpler space Ωreg to get information about the possibly very complicated eigenfunctions on theoriginal manifold M .

1.3. Main results. To study the problem described above, we shall use semiclassical methods. Incase of the Laplace-Beltrami operator, it would also be possible to study this problem via the originalhigh-energy approach of Shnirelman, Zelditch and Colin de Verdiere.


To describe ours, we need to fix some additional notation. Let H be a principal isotropy group of theG-action and κ the dimension of the principal orbits. In the whole paper we assume κ < n = dim M .

Denote by Ωreg the principal stratum of Ω and by Ωreg the principal stratum of Ω. Choose χ ∈ G,and denote by dχ the dimension of the irreducible G-representations of isomorphism class χ, and by[πχ|H : 1] the multiplicity of the trivial representation in the H-representation πχ|H . For m ∈ R, thenotation Ψm

h (M) will denote the set of semiclassical pseudodifferential operators on M of order m.Our first main result is

Result 1 (Generalized equivariant semiclassical Weyl law, Theorem 4.1). Let B ∈ Ψ0h(M) be a

G-equivariant semiclassical pseudodifferential operator with principal symbol σ(B) = [b], and assumethat b ∈ S0(M) is G-invariant and independent of h. Let c ∈ R be a regular value of the reduced symbol

function p and Σc = p−1(c). Then, for each β ∈(

0, 12(κ+2)

)and χ ∈ G we have

(2π)n−κhn−κ−β∑

j ∈ N, uj(h) ∈ L2χ(M),

Ej(h) ∈ [c, c+ hβ ]

〈Buj(h), uj(h)〉L2(M)

= dχ [πχ|H : 1]

ˆΣc

b dµc +O(h

12(κ+2)

−β(log h−1)Λ−1),

where b(G · (x, ξ)) := b(x, ξ), and Λ is a natural number bounded by the number of orbit types.

Result 1 relies on an equivariant trace formula for Schrodinger operators with remainder estimatewhich is the content of Theorem 3.1. Its proof reduces to the asymptotic description of certain oscil-latory integrals that have recently been studied in [20] using resolution of singularities. The involvedphase functions are given in terms of the underlying G-action on M , and if singular orbits occur,the corresponding critical sets are no longer smooth, so that a partial desingularization process hasto be implemented in order to obtain asymptotics with remainder estimates via the stationary phaseprinciple. Relying on Result 1, and after studying classical ergodicity in the equivariant situation, weobtain

Result 2 (Equivariant quantum ergodicity for Schrodinger operators, Theorem 6.6). Choosea Hilbert basis uj(h)j∈N of L2(M) such that each uj(h) is an eigenfunction of P (h), and denote theeigenvalue associated to uj(h) by Ej(h). Let c ∈ R be a regular value of the reduced symbol function

p, and suppose that the reduced flow ϕt is ergodic on Σc. Let χ ∈ G, β ∈(0, 1

2(κ+2)

), and set

Jχ(h) := j ∈ N : Ej(h) ∈ [c, c+ hβ ], uj(h) ∈ L2χ(M).

Then, there is a h0 > 0 such that for each h ∈ (0, h0] we have a subset Λχ(h) ⊂ Jχ(h) such that

limh→0

#Λχ(h)

#Jχ(h)= 1,

and the following holds. For each G-equivariant A ∈ Ψ0h(M) and G-invariant a ∈ S0(M) with σ(A) =

[a], and for arbitrary ε > 0, there is a hε ∈ (0, h0] such that∣∣∣ 〈Auj(h), uj(h)〉L2(M) −ˆ

Σc

a dµc

∣∣∣ < ε ∀ j ∈ Λχ(h), ∀h ∈ (0, hε].

Let us remark that the remainder estimates in Result 1, and consequently the desingularizationprocess implemented in [20], are crucial in obtaining the sharp energy localization in Result 2. In thespecial case of the Laplace-Beltrami operator, Result 2 becomes an equivariant version of the classicalquantum ergodicity theorem of Shnirelman [22], Zelditch [26], and Colin de Verdiere [6], yielding

Result 3 (Equivariant quantum limits for the Laplacian, Theorem 7.1). Assume that the re-

duced geodesic flow is ergodic, and choose χ ∈ G. Let uχj j∈N be an orthonormal basis of L2χ(M) of


eigenfunctions of −∆. Then, there is a subsequence uχjkk∈N of density 1 in uχj j∈N such that for all

G-invariant functions a ∈ C∞(S∗M) one has2

⟨Op(a)uχjk , u

χjk

⟩L2(M)

−→

S∗Mreg

a d(S∗Mreg) as k →∞,

where a : S∗Mreg → C is induced by a, and d(S∗Mreg) is the Liouville measure on the unit spherebundle, and we wrote Op for Op1.

Projecting from S∗Mreg onto Mreg, we immediately get for any G-invariant f ∈ C(M) thatˆ

M

f(x)|uχjk(x)|2 dM(x) −→

Mreg

f(x)dM(x)

volOxas k →∞,

where volOx denotes the volume of the G-orbit Ox = G · x, see Corollary 7.3. This actually amountsto a statement on convergence of measures on the topological Hausdorff space M/G, and applyingsome elementary tools from representation theory, we finally deduce from Theorem 7.1 the following

Result 4 (Representation-theoretic equidistribution theorem, Theorem 7.6). Assume that the

reduced geodesic flow is ergodic, and let χ ∈ G. By the Peter-Weyl theorem and spectral theorem,choose an orthogonal decomposition L2

χ(M) =⊕

i∈N Vχi into irreducible unitary G-modules of class χ

such that each V χi is contained in some eigenspace of the Laplace-Beltrami operator. For each i ∈ N,select some vi ∈ V χi with ‖vi‖L2(M) = 1. The function

Θχi : M → R, x 7→

ˆ

G

|vi(g−1x)|2 dg

is independent of the choice of vi and G-invariant, and we regard it as a function on M/G. Then,there is a subsequence V χikk∈N of density 1 in V χi i∈N such that for all f ∈ C(M/G) one has

(1.1)

ˆ

M/G

f Θχikd(M/G)

k→∞−→

M/G

f(O)d(M/G)(O)

volO.

Here, d(M/G) := π∗dM denotes the pushforward measure defined by the canonical projection π : M →M/G, and volO is the non-zero function assigning to a G-orbit O its volume.

Note that Result 4 is a statement about limits of representations, or multiplicities, and not eigen-functions, since it assigns to each χ-isotypic G-module in L2(M) a measure on M/G, and then considersthe limiting measure. In essence, it can therefore be regarded as a representation-theoretic result inwhich the spectral theory for the Laplacian only enters in choosing a concrete decomposition of eachisotypic component. In the case of the trivial group G = e, there is only one isotypic componentin L2(M) associated to the trivial representation, and choosing the family V χi i∈N is equivalent tochoosing a Hilbert basis of L2(M) of eigenfunctions of the Laplace-Beltrami operator. Result 4 thenreduces to the classical equidistribution theorem for the Laplacian.

In Section 8, we consider the example of the singular SO(2)-action on the 2-sphere given by rotationsaround a symmetry axis in order to illustrate our results. We show that in spite of its simplicity, thisexample provides already non-trivial insights. In fact, although the 2-sphere is a prototypical exampleof a manifold with non-ergodic geodesic flow, the reduced geodesic flow with respect to the consideredsingular SO(2)-action turns out to be ergodic. Moreover, the spherical harmonics are sufficiently wellunderstood so that we can independently verify Result 4 in this case.

2See Footnote 1.


1.4. Comments and outlook. We would like to close this introduction by making some comments,and indicating some possible research lines for the future.

Weaker versions of Results 2 and 3 can be proven by the same methods employed here with a lesssharp energy localization in an interval [c, d] of non-zero volume, instead of the interval [c, c + hβ ].The point is that for these weaker statements no remainder estimate in the semiclassical Weyl lawis necessary, as noted at the end of Section 6.2. Thus, in principle, these weaker results could havealso been obtained in the late 1970’s using heat kernel methods as in [8] or [4]. In contrast, for thestronger versions of equivariant quantum ergodicity proven in Results 2 and 3, remainder estimatesin Weyl’s law, and in particular the results obtained in [20] for general group actions via resolutionof singularities, are necessary. However, the weaker versions would still be strong enough to implyResult 4. Therefore, in principle, the representation-theoretic equidistribution theorem could havebeen proven already when Shnirelman formulated his theorem more than 40 years ago.

In the case of non-singular group actions, where G acts on M with only one orbit type, M :=M/G is a closed smooth manifold with Riemannian metric induced by the G-invariant Riemannian

metric on M . In that case, the geodesic flow on M and the reduced geodesic flow on M agree. One

can then apply the classical Shnirelman-Zelditch-Colin-de-Verdiere equidistribution theorem to M ,

yielding an equidistribution statement for the eigenfunctions of the Laplacian ∆M

on M in terms of

weak convergence of measures on M . On the other hand, one can apply Result 4 to M , yielding

also a statement about weak convergence of measures on M , but this time with measures related toeigenfunctions of the Laplacian ∆M on M in a single isotypic component of L2(M). It is then anobvious question how these two results are related. The answer to it seems to be rather difficult ingeneral, since – in spite of the presence of the group action – the geometry of M may be much more

complicated than that of M . Consequently, the eigenfunctions of ∆M , even those in the trivial isotypiccomponent L2

χ0(M), that is, those that are G-invariant, may be much harder to understand than the

eigenfunctions of ∆M

. Indeed, if π : M → M denotes the orbit projection, it is a result of Watson [24]that π∗∆

M= ∆M π∗ if, and only if, all G-orbits are minimal submanifolds of M . Hence, only in that

case we can identify Eig(∆M , λ) ∩ L2χ0

(M) with Eig(∆M, λ) for each eigenvalue λ, where Eig denotes

the eigenspace of an operator belonging to a certain eigenvalue. In the even more restrictive situationof totally geodesic orbits, Bergery-Bourguignon [1] proved that integration along orbits commuteswith the Laplacian. Now, if all orbits are minimal submanifolds or even totally geodesic, they allhave the same volume, so that the orbit-volume function is constant and cancels out in (1.1). In thisparticular situation, it is easy to see that the simultaneous application of the Shnirelman-Zelditch-Colin-de-Verdiere equidistribution theorem and the results of Watson and Bergery-Bourguignon implyour results, but only for the trivial isotypic component. In cases where the orbit volume is not constant,we do not know of any significant results about the relation between the eigenfunctions of ∆

Mand ∆M .

It would therefore be interesting to study more concrete, non-trivial examples in which the G-actionis non-singular.

As further lines of research, we would like to mention that, in particular in view of Result 4, itmight be possible to deepen our understanding of equivariant quantum ergodicity via representationtheory. Also, it would be interesting to see whether our results can be generalized to G-vector bundles.Finally, it seems natural to ask whether an equivariant version of the QUE conjecture could hold. Thisconjecture assumes that the manifold M under consideration must have negative sectional curvature,and the conjecure does not hold if one merely requires ergodicity of the geodesic flow on M [11]. Con-sequently, for the formulation of an analogous conjecture in our context, one would need an equivariantversion of this condition which should certainly be stronger than the condition of the reduced geodesicflow being ergodic, since in the case G = e it should reduce to the condition of M having negativesectional curvature. In the particular case of the SO(2)-action on the 2-sphere studied in Section 8,we actually show that the representation-theoretic equidistribution theorem for the Laplacian appliesto the full sequence of spherical harmonic functions in a fixed isotopic component, so that equivariantQUE holds in this case.


2. Setup and background

In this section we shall prepare some material needed in the sequel, and fix some global notation.

2.1. Semiclassical pseudodifferential operators and symmetries. In what follows, we shall givea brief overview of the theory of semiclassical pseudodifferential operators. For a detailed introduction,we refer the reader to [27, Chapters 9 and 14] and [7]. Semiclassical analysis developed out of the theoryof ordinary pseudodifferential operators, and a thorough exposition can be found in [23]. Let M be asmooth manifold of dimension n.

Definition 2.1. Let m ∈ R and (Uα, γα)α∈A, γα : M ⊃ Uα → Vα ⊂ Rn, be an atlas for M . Then

Sm(M) := a ∈ C∞(T ∗M), (γ−1α )∗a ∈ Sm(Vα × Rn) ∀ α ∈ A,

where one identifies T ∗Uα with Vα × Rn, and for an open subset E ⊂ Rn

Sm(E) :=a ∈ C∞(E × Rn) : for all compact K ⊂ E and multiindices s, t

there exists CKs,t > 0 such that |∂sx∂tξa(x, ξ)| ≤ CKs,t 〈ξ〉m−|t| ∀x ∈ K,(2.1)

where 〈ξ〉 :=√

1 + |ξ|2.

The definition is independent of the choice of atlas. We will call an element of Sm(M) a sym-bol function. A symbol function may depend on h. If this is the case, one requires that there is ah0 > 0 such that the constants Cs,t in (2.1) are independent of h for 0 < h ≤ h0. For simplicity ofnotation we usually do not make a possible h-dependence explicit when working with symbol functions.

Next, let Ψmh (M) denote the C-linear space of all semiclassical pseudodifferential operators P :

C∞c (M)→ C∞(M) of order m. Such operators are locally of the form

Au(x) =1

(2πh)n

ˆRn

ˆRneih 〈x−y,ξ〉a

(x+ y

2, ξ)u(y) dy dξ =

ˆRn

ˆRnei〈x−y,ξ〉a

(x+ y

2, hξ)u(y) dy dξ,

where a ∈ Sm(Rn), u ∈ C∞c (Rn), and dξ = (2π)−n dξ. This class emerges from the usual class ofpseudodifferential operators of order m by substituting in the amplitude ξ by hξ. This so-called Weylquantization is motivated by the fact that the classical Hamiltonian H(x, ξ) = ξ2 should correspond tothe quantum Laplacian −h2∆. We write Ψm(M) := Ψm

1 (M) for the linear space of classical pseudo-differential operators of order m. From the classical theorems about pseudodifferential operators oneinfers in particular the following relation between symbol functions and semiclassical pseudodifferentialoperators.

Theorem 2.1 ([14, page 86], [27, Theorem 14.1]). There is a C-linear map

(2.2) Ψmh (M)→ Sm(M)/

(hSm−1(M)

), P 7→ σ(P )

which assigns to a semiclassical pseudodifferential operator its principal symbol. Moreover, for eachchoice of a covering Uαα∈A of M and a partition of unity ϕαα∈A subordinate to Uαα∈A, thereis a C-linear map called quantization, and given by

(2.3) Sm(M)→ Ψmh (M), s 7→ Oph,Uα,ϕαα∈A(s).

Any choice of such a map induces the same C-linear bijection

Ψmh (M)/

(hΨm−1

h (M)) σ

Oph

Sm(M)/(hSm−1(M)

),(2.4)

which means in particular that the bijection exists and is independent from the choice of covering andpartition of unity.


We will write Op := Op1 for the usual non-semiclassical quantization map, and we do not make a

difference in our notation between the maps on the quotients (2.4) and the maps (2.2), (2.3) obtainedby precomposition with the quotient projections. However, we will call an element in a quotient spaceSm(M)/

(hSm−1(M)

)a principal symbol, whereas we call the elements of Sm(M) symbol functions, as

introduced above. Operations on principal symbols such as pointwise multiplication with other prin-cipal symbols or smooth functions and composition with smooth functions are defined by performingthe corresponding operations on the symbol functions. For a semiclassical pseudodifferential operatorA, we will use the notation

σ(A) = [a]

to express that the principal symbol σ(A) is the equivalence class in Sm(M)/(hSm−1(M)

)defined by

a symbol function a ∈ Sm(M).

The following theorem says that from the linear isomorphisms of the preceding theorem for variousm ∈ R, we actually get an isomorphism of R-graded algebras.

Theorem 2.2 ([27, Theorem 14.1]). If A ∈ Ψmh (M) and B ∈ Ψm′

h (M) then AB ∈ Ψm+m′

h (M) andσ(AB) = σ(A)σ(B).

Next, let us suppose that M carries certain symmetries in form of a Lie group G acting smoothly on

M . It induces a Hamiltonian G-action on T ∗M and a representation of G on the space L2(M) givenby

(2.5) G× L2(M)→ L2(M), (g, f) 7→(Lgf : x 7→ f(g−1 · x)

).

We will usually just write gf for Lgf . Clearly, we can define an action similar to (2.5) also on C∞(M).

Let G be the character ring of G, which we identify with the set of isomorphism classes of irreducible

representations of G. An irreducible character χ ∈ G defines an equivalence class of irreducible unitaryG-representations with dimension dχ = χ(e), where e ∈ G is the unit element. By the Peter-Weyltheorem, the left-regular representation (2.5) of G on L2(M) decomposes orthogonally into isotypiccomponents according to

(2.6) L2(M) =⊕χ∈G

L2χ(M).

The projection in L2(M) onto L2χ(M) is given by

Tχ : L2(M)→ L2χ(M), f 7→

x 7→ dχ

ˆ

G

χ(g)f(g−1x) dg

,

where dg is the normalized Haar measure on G. For an operator A in L2(M), we call

Aχ := TχATχ

the reduced operator associated to the irreducible representation χ. Further, an operator A is calledG-equivariant if it commutes with the G-representation on L2(M), that is, if P (gf) = gP (f) for allf ∈ L2(M). If A is G-equivariant, it commutes with each projection Tχ, so that Aχ = ATχ = TχA.Note that in general, the kernel of Aχ is much larger than that of A because of the projection ontoL2χ(M). However, we can consider the restricted operator

A|χ := A|L2χ(M) : L2

χ(M)→ L2χ(M)

which has the advantage over Aχ that the dimension of the eigenspace of A|χ associated to an eigenvalueλ is never larger than the dimension of the eigenspace of A associated to λ.


We consider now G-equivariant operators and G-invariant symbol functions, as well as the relationbetween them. A symbol function s in Sm(M) is called G-invariant if it fulfills s(g · η) = s(η) for allη ∈ T ∗M .

Proposition 2.1 (Quantization of G-invariant symbol functions). Let s ∈ Sm(M) be G-invariant modulo hSm−1(M). Then Oph(s) ∈ Ψm

h (M)/(hΨm−1

h (M))

is G-equivariant.

Proof. Let g ∈ G, and denote the action of G on M by lg(x) = g · x, and the endomorphism

T ∗M → T ∗M induced by lg as lg. By direct calculation, using the definition of the quantizationmaps Oph,Uα,ϕαα∈A from Theorem 2.1, one verifies that

Oph,(Uα,ϕα)α∈A(s) = Lg−1 Oph,(lg−1 (Uα),ϕαlg)

α∈A(s lg) Lg.

Let us first assume that s is G-invariant so that s lg = s. Then we get

(2.7) Oph,(lg(Uα),ϕαlg−1)α∈A

(s) = Lg−1 Oph,(Uα,ϕα)α∈A(s) Lg.

From (2.7) we read off that Lg−1 Oph,(Uα,ϕα)α∈A(s) Lg is just the quantization of the symbol

function s with respect to a different open cover and partition of unity. This implies that Lg−1 Oph,(Uα,ϕα)α∈A

(s) Lg is in fact an element of Ψmh (M), and Theorem 2.1 gives

(2.8) Oph(s) =[Oph,(Uα,ϕα)α∈A

(s)]

=

[Oph,(lg(Uα),ϕαlg−1)

α∈A(s)

]=[Lg−1 Oph,(Uα,ϕα)α∈A

(s) Lg]

= Lg−1 Oph(s) Lg in Ψmh (M)/

(hΨm−1

h (M)).

Finally, since Oph,(Uα,ϕα)α∈A

(hSm−1(M)

)⊂ hΨm−1

h (M), the observations made so far imply that

(2.8) also holds if s is only G-invariant modulo hSm−1(M).

2.2. Equivariant Schrodinger operators on closed Riemannian G-manifolds. Let now M bea connected closed3 Riemannian manifold of dimension n, carrying an isometric effective action of acompact connected Lie group G. Let V ∈ C∞(M,R) be a G-invariant potential, that is, V (g·x) = V (x)for all g ∈ G, x ∈M . We then consider the Schrodinger operator

P (h) : C∞(M)→ C∞(M), P (h) = −h2∆ + V,

parametrized by h ∈ R>0, where the symbol ∆ denotes the Laplace-Beltrami operator and V isidentified with the operator given by pointwise multiplication with V . Since the G-action is isometric,each Schrodinger operator is equivariant with respect to the left-regular G-representation on L2(M),

so that P (h) Lg f = Lg P (h) f for all f ∈ C∞(M). Note that P (h) is essentially self-adjoint, henceit has a unique self-adjoint extension

(2.9) P (h) : H2(M)→ L2(M)

in the Hilbert space L2(M), the domain being the second Sobolev space, see [27, Theorem 14.7 i].By continuity, P (h) is equivariant. We will call P (h) a Schrodinger operator, too, and use the same

notation for P (h) and P (h) whenever the precise meaning can be inferred from the context. Let uscollect a list of well known facts about the operator P (h).

Theorem 2.3 ([27, Theorem 14.7 ii]). The eigenfunctions of P (h) are in C∞(M). There existsa Hilbert basis ujj∈N of L2(M) such that each uj is an eigenfunction of P (h). The associatedeigenvalues Ejj∈N of P (h) fulfill

(2.10) limj→∞

Ej =∞.

Note that (2.10) implies that all eigenspaces of P (h) are finite-dimensional since the eigenvalues are

repeated in Ejj∈N according to their multiplicity.

3i.e. compact without boundary


Theorem 2.4 ([27, Theorems 14.9 and 14.10]). Let S(R) denote the Schwartz space of rapidly de-creasing functions, and f ∈ S(R). Then, the operator f(P (h)) defined by the spectral theorem forunbounded self-adjoint operators is an element in Ψ−∞h (M) =

⋂m∈Z Ψm

h (M). Furthermore, f(P (h))

extends to a bounded operator f(P (h)) : L2(M) → L2(M) of trace class, and the principal symbol off(P (h)) is

(2.11) σ (f(P (h))) = f σ (P (h)) ,

whereσ(P (h)) = [p], p = ‖·‖2T∗M + V τ,

is the principal symbol of P (h), with τ : T ∗M→M denoting the cotangent bundle.

2.3. Actions of compact Lie groups and symplectic reduction. In what follows, we recall someessential facts from the theory of compact Lie group actions on smooth manifolds needed in the sequel.For a detailed introduction, we refer the reader to [3]. Let X be a smooth manifold of dimension n andG a compact Lie group acting locally smoothly on X. For x ∈ X, denote by Gx the isotropy groupand by G · x = Ox the orbit of x so that

Gx = g ∈ G, g · x = x,Ox = G · x = g · x ∈ X, g ∈ G.

Note that G · x and G/Gx are homeomorphic. The equivalence class of an orbit Ox under equivarianthomeomorphisms, written [Ox], is called its orbit type. The conjugacy class of a stabiliser group Gx iscalled its isotropy type, and written (Gx). If K1 and K2 are closed subgroups of G, a partial orderingof orbit and isotropy types is given by

[G/K1] ≤ [G/K2]⇐⇒ (K2) ≤ (K1)⇐⇒ K2 is conjugate to a subgroup of K1.

The set of all orbits is denoted by X/G, and is a compact, topological Hausdorff space [3, Theorem3.1]. In the following we shall assume that it is connected. One of the central results in the theory ofcompact group actions is

Theorem 2.5 (Principal Orbit theorem, [3, Theorem IV.3.1]). There exists a maximum orbit type[Omax] with associated minimal isotropy type (H). The union X(H) of orbits of isotropy type (H) isopen and dense in X, and its image in X/G is connected.

We call [Omax] the principal orbit type of the G-action on X and a representing orbit a principal

orbit. Similarly, we call the isotropy type (H) and an isotropy group Gx ∼ H principal. Casually, wewill identify orbit types with isotropy types and say an orbit of type (H) or even an orbit of type H,making no distinction between equivalence classes and their representants. Note that if X is compact,the reduced space X(H)/G is a smooth manifold of dimension n − κ, where κ is the dimension ofOmax, since G acts with only one orbit type on X(H).

Let now (X, ω) be a connected symplectic paracompact manifold with an action of a Lie group Gthat leaves ω invariant. Denote by g the Lie algebra of G. Note that G acts on g with the adjointaction and on g∗ dually with the coadjoint action. A smooth equivariant map

J : X→ g∗

is called momentum map for the G-action on X if for each X ∈ g the function

JX : X→ R, x 7→ J(x)(X),

is related to the symplectic form ω and the fundamental vector field X on X associated to X accordingto

dJX = −X yω.It is clear from the definition that a momentum map is unique up to addition of a constant function.If a momentum map exists, the action of G on X is called Hamiltonian.


Remark 2.6. An important class of examples of Hamiltonian group actions is given by those actionsof Lie groups on cotangent bundles which are induced from Lie group actions on the base manifoldsby dualizing the derivatives. Indeed, let M be a smooth manifold with smooth action of G and

X = T ∗Mτ→ M the cotangent bundle, with induced G-action and with standard symplectic form

ω = −dθ, where θ is the tautological or Liouville one-form on T ∗M . Then there is a momentum mapJ given explicitly by the formula

T ∗M 3 η 7→ J(η)(X) := η(Xτ(η)

), X ∈ g.

Next, we recall some central results of the theory of symplectic reduction of Sjamaar, Lerman andBates. For a detailed exposition of these facts we refer the reader to [18]. Assume that (X, ω) carriesa global Hamiltonian action of G. Let J : X → g∗ be the corresponding momentum map, µ a valueof J, and Gµ the isotropy group of µ with respect to the co-adjoint action on g∗. Consider further anisotropy group K ⊂ G of the G-action on X, let η ∈ J−1(µ), and Xη

K be the connected component ofXK = ζ ∈ X : Gζ = K containing η. We then have the following

Theorem 2.7 ([18, Theorem 8.1.1]).

(1) The set J−1(µ) ∩Gµ ·XηK is a smooth submanifold of X.

(2) The quotient

Ω(K)µ := (J−1(µ) ∩Gµ ·Xη

K)/Gµ

possesses a differentiable structure such that the projection π(K)µ : J−1(µ)∩Gµ ·Xη

K → Ω(K)µ is

a surjective submersion.

(3) There exists a unique symplectic form ω(K)µ on Ω

(K)µ such that (ι

(K)µ )∗ω = (π

(K)µ )∗(ω

(K)µ ), where

ι(K)µ : J−1(µ) ∩Gµ ·Xη

K → X denotes the inclusion.(4) Let p ∈ C∞(X) be a G-invariant function, Hp = s-grad p its Hamiltonian vector field, and

ϕt the corresponding flow. Then ϕt leaves invariant the components of J−1(µ) ∩Gµ ·XηK and

commutes with the Gµ-action, yielding a reduced flow ϕµt on Ω(K)µ given by π

(K)µ ϕt ι(K)

µ =

ϕµt π(K)µ .

(5) The reduced flow ϕµt on Ω(K)µ is Hamiltonian, and its Hamiltonian p

(K)µ : Ω

(K)µ → R satisfies

p(K)µ π(K)

µ = p ι(K)µ .

Remark 2.8. With the notation above we have G · XK = X(K). Indeed, for x ∈ XK , the isotropygroup of x is K. If g′g · x = g · x for some g, g′ ∈ G, then g−1g′g · x = x, hence g−1g′g ∈ K, i.e.g′ ∈ (K). That shows G ·XK ⊂ X(K). On the other hand, if x ∈ X(K), then (Gx) = (K), hence forevery g′ ∈ Gx, there is a k ∈ K and a g ∈ G such that g′ = gkg−1. But then kg−1x = g−1x, so thatg−1x ∈ XK , and in particular x ∈ G ·XK .

Let now M be a connected closed Riemannian manifold of dimension n, carrying an isometriceffective action of a compact connected Lie group G. In all what follows, the principal isotropy type ofthe action will be denoted by (H) and the dimension of the principal orbits in M by κ. Furthermore,we shall always assume that κ < n, and write

(2.12) Ω := J−1(0) =⊔x∈M

AnnTx(G · x),

where J : T ∗M → g∗ is the momentum map associated to the G-action on T ∗M . Note that as soonas there are two orbits G · x, G · x′ in M of different dimensions, their annihilators AnnTx(G · x) andAnnTx(G · x′) have different dimensions, so that Ω is not a vector bundle. Further, let

Mreg := M(H), Ωreg := Ω ∩ T ∗M(H),

where M(H) and T ∗M(H) denote the union of orbits of principal type in M and T ∗M , respectively.By Theorem 2.5, Mreg is open in M , hence Mreg is a smooth submanifold. We then define

Mreg := Mreg/G.


Mreg is a smooth manifold, since G acts with only one orbit type on Mreg. Moreover, because the

Riemannian metric g on M is G-invariant, it induces a Riemannian metric g on Mreg. On the otherhand, by Theorem 2.7, Ωreg is a smooth submanifold of T ∗M , and the quotient

Ωreg := Ωreg/G

possesses a unique differentiable structure such that the projection π : Ωreg → Ωreg is a surjective

submersion. Furthermore, there exists a unique symplectic form ω on Ωreg such that ι∗ω = π∗ω, whereι : Ωreg → T ∗M denotes the inclusion and ω the canonical symplectic form on T ∗M . Consider now

the inclusion j : (T ∗Mreg∩Ωreg)/G → Ωreg. The symplectic form ω on Ωreg induces a symplectic formj∗ω on (T ∗Mreg ∩ Ωreg)/G. The following lemma allows us to understand that induced symplecticstructure on (T ∗Mreg ∩ Ωreg)/G in more concrete terms.

Lemma 2.9. Let ω denote the canonical symplectic form on the cotangent bundle T ∗Mreg. Then the

two 2(n− κ)-dimensional symplectic manifolds ((T ∗Mreg ∩ Ωreg)/G, j∗ω) and (T ∗Mreg, ω) are canoni-

cally symplectomorphic.

Remark 2.10. If M = Mreg, the previous lemma simply asserts that T ∗(M/G) is isomorphic to

Ω ⊂ (T ∗M)/G as symplectic manifolds.

Proof. First, we apply Theorem 2.7 once to the manifold T ∗M and once to the manifold T ∗Mreg.Noting that the momentum map of the G-action on T ∗Mreg agrees with the restriction of the mo-mentum map of the G-action on T ∗M to T ∗Mreg, we get that j∗ω is the unique symplectic form on(T ∗Mreg ∩ Ωreg)/G which fulfills

(2.13) i∗ω = Π∗j∗ω,

where Π : T ∗Mreg → T ∗Mreg/G is the orbit projection, i : T ∗Mreg ∩ Ωreg → T ∗Mreg is the inclusion,and ω is the canonical symplectic form on T ∗Mreg.

We are now going to construct in a canonical way a diffeomorphism Φ : (T ∗Mreg∩Ωreg)/G→ T ∗Mreg

such that Φ∗ω also fulfills (2.13), that is, such that i∗ω = Π∗Φ∗ω holds. We will then be able to concludefrom the uniqueness statement associated to (2.13) that j∗ω = Φ∗ω, so that Φ is a symplectomorphism.

Denote by π : Mreg → Mreg ≡ Mreg/G the orbit projection. Together with the pointwise derivatives,π induces a morphism of smooth vector bundles

∂π : TMreg → TMreg,

which has the fiberwise kernel ker ∂π|x = Tx(G · x). The Riemannian metrics on Mreg and Mreg

provide us with the usual isomorphisms of smooth vector bundles α : TMreg ' T ∗Mreg and β :

TMreg ' T ∗Mreg. For each x ∈ Mreg, we have α−1(Ann Tx(G · x)) = (Tx(G · x))⊥, so that ∂π isfiberwise injective when restricted to α−1(T ∗Mreg ∩ Ωreg). However, the dimensions of (Tx(G · x))⊥

and Tπ(x)Mreg are both n−κ. So denoting the restriction of ∂π to α−1(T ∗Mreg∩Ωreg) by ∂π it followsthat

∂π : α−1(T ∗Mreg ∩ Ωreg)→ TMreg

together with the map π : Mreg → Mreg is a morphism of smooth vector bundles which is fiberwise

an isomorphism. For x ∈ Mreg, denote the restriction of ∂π to Tx(G · x)⊥ by γx. That defines an

isomorphism γx : (Tx(G · x))⊥'→ Tπ(x)Mreg. Looking at the definition of ∂π, we see that ∂π is

G-invariant and that for each ξ ∈ Tπ(x)Mreg, ∂π−1(ξ) = G · γ−1x (ξ), the orbit of γ−1

x (ξ) under theG-action on TMreg induced from the G-action on Mreg. We conclude that ∂π induces a diffeomorphism

∂π : (α−1(T ∗Mreg ∩ Ωreg))/G→ TMreg.

Because the G-action on M is isometric, α is G-invariant. Therefore, ∂π α−1 is G-invariant andinduces a diffeomorphism

˜∂π α−1 : (T ∗Mreg ∩ Ωreg)/G→ TMreg.


Composition with β yields the map

(2.14) Φ := β ˜∂π α−1 : (T ∗Mreg ∩ Ωreg)/G→ T ∗Mreg.

As a composition of diffeomorphisms, Φ is a diffeomorphism. Moreover, the Riemannian metric on

Mreg that is used to define β is defined as the induced Riemannian metric obtained from the G-invariantRiemannian metric on Mreg. Therefore, one can check that the map Φ is actually independent fromthe choice of G-invariant metric on Mreg. That makes the definition of Φ canonical. Now, in view of(2.13), we want to show that Φ∗ω fulfills

(2.15) i∗ω=Π∗Φ∗ω,

where ω is the canonical symplectic form on T ∗Mreg. Since ω = −d%, with % the tautological one-form

on T ∗Mreg, the desired equation (2.15) would follow, by applying the exterior derivative d to bothsides and using compatibility of pullbacks with d, from the more basic relation

(2.16) i∗%=Π∗Φ∗%,

where % is the tautological one-form on T ∗Mreg. We will now show that (2.16) is in fact true. Since

by definition ˜∂π α−1 Π = ∂π α−1, we have

Φ Π = β ∂π α−1 : T ∗Mreg ∩ Ωreg → T ∗Mreg.

Taking into account that Π∗Φ∗ = (ΦΠ)∗, what we have to show is

(2.17) %|η(v)=((β ∂π α−1)∗%)|η(v) ∀ η ∈ T ∗Mreg ∩ Ωreg, v ∈ Tη(T ∗Mreg ∩ Ωreg).

Let τ : T ∗Mreg → Mreg and τ : T ∗Mreg → Mreg be the cotangent bundles. By definition of thetautological one-forms on the two bundles we have

%|η(v) = η(∂τ |η(v)), η ∈ T ∗Mreg, v ∈ TηT ∗Mreg,(2.18)

%|η(v) = η(∂τ |η(v)), η ∈ T ∗Mreg, v ∈ TηT ∗Mreg.(2.19)

Using the definition of the pullback of differential forms and inserting definition (2.19) into the righthand side of (2.17) we obtain for η ∈ T ∗Mreg, v ∈ Tη(T ∗Mreg ∩ Ωreg)

((β ∂π α−1)∗%)|η(v) = %|β∂πα−1(η)(∂(β ∂π α−1)v)

= (β ∂π α−1)(η)(∂τ |β∂πα−1(η)∂(β ∂π α−1)v).(2.20)

Denote the Riemannian metrics on Mreg and Mreg by g and g, respectively, so that α(v) = g|x(v, ·) for

v ∈ TxMreg, x ∈Mreg and β(v) = g|x(v, ·) for v ∈ TxMreg, x ∈ Mreg. Inserting the definition of β into(2.20), we obtain for η ∈ T ∗Mreg, v ∈ Tη(T ∗Mreg ∩ Ωreg)

(2.21) ((β ∂π α−1)∗%)|η(v) = g|σ∂πα−1(η)

((∂π α−1)(η), ∂τ |β∂πα−1(η)∂(β ∂π α−1)v)

),

where we wrote σ : TMreg → Mreg for the tangent bundle. Consider now the commutative diagram


TTMreg TTMreg

TMregTT ∗Mreg TMreg TT ∗Mreg

T ∗Mreg Mreg Mreg T ∗Mreg

∂(∂π)

∂σ = can. ∂σ = can.∂β∂α

∂τ = can.

can.

∂π

σ = can. σ = can.β

∂τ = can.

can.α

τ = can. π

τ = can.

In the diagram, “can.” denotes the canonical projection of a tangent or cotangent bundle. Theother maps have been introduced in the paragraph above. That the triangles and the square in thebottom row of the diagram commute is a consequence of the relevant pairs of maps being morphismsof vector bundles. The triangles and the square in the top row of the diagram are just the derivativesof the corresponding triangles and the square in the bottom row. Therefore, they commute. From thecommutativity of the top row of the diagram we get

(2.22) ∂τ ∂β ∂(∂π) ∂α−1 = ∂π ∂τ.

We can simplify (2.21) using (2.22), which leads to

((β ∂π α−1)∗%)|η(v) = g|σ∂πα−1(η)

(∂π α−1(η), ∂π ∂τ(v)

).

By definition of the metric g, this can be rewritten as

((β ∂π α−1)∗%)|η(v) = gσα−1(η)

(α−1(η), ∂τ(v)

).

Moreover, by definition of α−1, this simplifies into

(2.23) ((β ∂π α−1)∗%)|η(v) = η (∂τ(v)) .

In view of (2.18), we have shown (2.17), and so we have proved (2.16). By the discussion at thebeginning of the proof and in the lines above (2.16), we are done.

2.4. Measure spaces and group actions. In what follows, we give an overview of the spaces andmeasures that will be relevant in the sequel. As before, let M be a connected closed Riemannianmanifold of dimension n, carrying an isometric effective action of a compact connected Lie group G,and define

C := (η, g) ∈ Ω×G, g · η = η.As Ω, the space C is not a manifold in general. We consider therefore the space Reg C of all regularpoints in C, that is, all points that have a neighbourhood which is a smooth manifold. Reg C is asmooth, non-compact submanifold of T ∗M ×G, and it is not difficult to see that

Reg C = (η, g) ∈ Ω×G, g · η = η, η ∈ Ωreg,

see e.g. [20, (17)]. We list now the measure spaces that will become relevant in the upcoming sections.


(1) (G, dg). Here dg is the normalized Haar measure on G. If dim G > 0, dg is equivalent to thenormalized Riemannian measure on G associated to any choice of a left-invariant Riemannianmetric on G. If dim G = 0, dg is the normalized counting measure.

(2) (M,dM). Here dM is the Riemannian measure associated to the Riemannian metric on M .With respect to some local coordinates x1, . . . , xn, the measure is given by the volume

density dM(x) = |√

det g(x) dx1 ∧ . . . ∧ dxn|, where g(x) denotes the matrix representing theRiemannian metric in coordinates.

(3) (Mreg, dM). Here we denote the restriction of the measure dM on M to the open subsetMreg ⊂M again by dM .

(4) (T ∗M,d(T ∗M)). Here d(T ∗M) is the measure defined by the symplectic volume form ωn/n!,ω being the canonical symplectic form on T ∗M . If one regards T ∗M as a Riemannian manifoldendowed with the natural Sasaki metric, then the Riemannian measure with respect to theSasaki metric and the measure d(T ∗M) agree, see [15, page 537].

(5) (Ωreg, dΩreg). Here we regard Ωreg ⊂ T ∗M as a Riemannian submanifold with Riemannianmetric induced by the Sasaki metric on T ∗M . The measure dΩreg is the associated Riemannianmeasure.

(6) (Reg C, d(Reg C)). Here we regard Reg C ⊂ T ∗M×G as a Riemannian manifold with Riemann-ian metric induced by the product metric of the Sasaki metric on T ∗M and some left-invariantRiemannian metric on G. The measure d(Reg C) is the associated Riemannian measure.

(7) (Ωreg, dΩreg). Here dΩreg is the measure defined by the volume form ω(n−κ)/(n− κ)!, where ω

is the symplectic form on Ωreg from Theorem 2.7.

(8) (Mreg, dMreg). Here dMreg is the Riemannian measure associated to the Riemannian metric ginduced by the metric g on M .

(9) (M/G, d(M/G)). Here d(M/G) := π∗dM is the pushforward of dM along the canonicalprojection π : M →M/G. This means that, for f ∈ C(M/G), we have

ˆ

M/G

f(O) d(M/G)(O) =

ˆ

M

f π(x) dM(x).

(10) (T ∗Mreg, d(T ∗Mreg)). Here T ∗Mreg is the cotangent bundle of Mreg. The measure d(T ∗Mreg) is

defined by the volume form obtained from the canonical symplectic form on T ∗Mreg, similarlyas in the case of T ∗M .

(11) (S∗Mreg, d(S∗Mreg)). Here S∗Mreg is the unit sphere bundle over Mreg with respect to the

Riemannian metric g induced by the metric g on M . The measure d(S∗Mreg) is the Liouvillemeasure. It agrees with the hypersurface measure obtained in Lemma 2.19 from the proper

function T ∗Mreg → R, η 7→ ‖η‖g at the regular value 1.

(12) (Σc, dµc). Here we set Σc := p−1(c) for given c ∈ R, where p is a smooth proper function

Ωreg → R and c a regular value of p. By Lemma 2.19, the measure dµc is the unique measure

on Σc with

(2.24) limδ→0

1

δ

ˆ

p−1([c−δ/2,c+δ/2])

f(η) d Ωreg(η) =

ˆ

Σc

f(η) dµc(η) ∀ f ∈ C∞(Ωreg).

(13) (G · x, dµG·x). Here x ∈ M . If dim G · x > 0, we regard the orbit Ox = G · x ⊂ M asa Riemannian manifold with Riemannian metric induced by the one on M . The measuredµG·x is the associated Riemannian measure. If dim G · x = 0, G · x must be finite, as it iscompact, and then dµG·x is the counting measure given by

´G·x f(x)dµG·x :=

∑p∈G·x f(p) for

f ∈ C(G · x). Since the orbit projection Mreg → Mreg defines a fiber bundle by [3, Theorem3.3], the orbit measures dµG·x for varying x ∈ Mreg are the unique measures on the orbits in


Mreg such that we have

(2.25)

ˆ

Mreg

f(x) dM(x) =

ˆ

Mreg

ˆ

G·x

f(x′) dµG·x(x′) dMreg(G · x) ∀ f ∈ C(Mreg).

Note that we have vol(G · x) > 0 for all orbits G · x in M , by definition of the measure dµG·x.An important property of the orbit measure is its relation to the normalized Haar measure onG. Namely, for any orbit G · x and any continuous function f : G · x→ C, we have

(2.26)

ˆ

G·x

f(x′) dµG·x(x′) = vol(G · x)

ˆ

G

f(g · x) dg.

To see why (2.26) holds, recall that for each x ∈Mreg, there is a G-equivariant diffeomorphismΦ : G · x → G/H. Then Φ∗dµG·x is a G-invariant, finite measure on G/H. Now consider theprojection Π : G → G/H. Then Π∗dg is also a G-invariant, finite measure on G/H. Hence,Φ∗dµG·x = C · Π∗dg for some constant C which is precisely given by vol(G · x), since Π∗dg isnormalized. Observing that

´Gf(g · x) dg =

´G/H

f(gHx)Π∗dg, (2.26) follows.

(14) (Mreg,dM

volO ). Here dMvolO (x) = dM(x)

volOx for given x ∈Mreg, and the volume of Ox = G·x is defined

using the orbit measure dµG·x defined above. We may call dMvolO the orbit normalized measure

on Mreg. Due to (2.25) and (2.26), there is the following relation between the measures dMvolO

and dMreg. For G-invariant f ∈ C(Mreg), which induces f ∈ C(Mreg), we have

(2.27)

ˆ

Mreg

f(G · x)dMreg(G · x) =

ˆ

Mreg

f(x)dM(x)

vol(G · x).

(15) (G · η, dµG·η). Here η ∈ T ∗M . If dim G · η > 0, we regard the orbit G · η ⊂ T ∗M as aRiemannian manifold with Riemannian metric induced by the Sasaki metric on T ∗M . Themeasure dµG·η is the associated Riemannian measure. If dim G · η = 0, G · η must be finite,and then dµG·η is the counting measure. As in (2.25), and taking into account Lemma 2.11,the measures dµG·η for varying η ∈ Ωreg are the unique measures on the orbits in Ωreg suchthat we have

(2.28)

ˆ

Ωreg

f(η) d(Ωreg)(η) =

ˆ

Ωreg

ˆ

G·η

f(η′) dµG·η(η′) dΩreg(G · η) ∀ f ∈ C(Ωreg).

Note that we have vol(G ·η) > 0 for all orbits G ·η in T ∗M , by definition of the measure dµG·η.

(16) (Ωreg,dΩreg

volO ). HeredΩreg

volO (η) =dΩreg(η)volOη for η ∈ Ωreg, and the volume of Oη = G · η is defined

using the orbit measure dµG·η defined above. We may calldΩreg

volO the orbit normalized measureon Ωreg.

(17) (M/G, d(M/G)vol ). Here d(M/G)

vol (O) = d(M/G)(O)volO , where the volume of Ox = G · x is defined as

above. We may call d(M/G)vol the orbit normalized measure on M/G.

Next, we collect a few important facts related to the spaces and measures introduced above.

Lemma 2.11. The measure dΩreg agrees with the pushforward measure Π∗ dΩreg, where Π : Ωreg →Ωreg = Ωreg/G is the orbit projection.

Proof. By [2, Theorem 4.6] all metrics on Ωreg which are associated to the symplectic form ω by analmost complex structure define the same Riemannian measure. Hence, it suffices to show that the

Riemannian metric on Ωreg induced by the G-invariant Riemannian metric on Ωreg is associated to ωby an almost complex structure. Now, the Sasaki metric gS on T ∗M is associated to the canonicalsymplectic form ω on T ∗M by an almost complex structure J : TT ∗M → TT ∗M . Consequently,the Riemannian metric ι∗gS on Ωreg is associated to the symplectic form ι∗ω by the almost complexstructure ι∗J , where ι : Ωreg → T ∗M is the inclusion. Since both ι∗gS and ι∗ω are G-invariant, ι∗J :


TΩreg → TΩreg is G-invariant, and therefore induces an almost complex structure ι∗J : T Ωreg → T Ωreg

which associates i∗gS with ω.

Lemma 2.12. M − Mreg is a null set in (M,dM), and Ωreg − (T ∗Mreg ∩ Ωreg) is a null set in(Ωreg, dΩreg).

Proof. The proof is completely analogous to the proof of [5, Lemma 3 on p. 13].

Corollary 2.13. M/G − Mreg is a null set in (M/G, d(M/G)), and Ωreg − (T ∗Mreg ∩ Ωreg)/G is a

null set in (Ωreg, dΩreg).

Proof. The first claim is true by definition of the measure d(M/G) and Lemma 2.12. Concerning thesecond claim, note that

(Ωreg − (T ∗Mreg ∩ Ωreg))/G = Ωreg − (T ∗Mreg ∩ Ωreg)/G.

Consequently, (2.28) and Lemma 2.12 together yield

vol(

Ωreg − (T ∗Mreg ∩ Ωreg)/G)

=

ˆ

Ωreg−(T∗Mreg∩Ωreg)

1

vol (G · η)dΩreg(η) = 0.

Lemma 2.14. The function vol(O)|Mreg : Mreg → R, x 7→ vol(G · x), is smooth. Moreover, if thedimension of the principal orbits is at least 1, the function vol(O)|Mreg can be extended by zero to a

continuous function vol(O) : M → R.

Proof. See [19, Proposition 1].

Remark 2.15. The function vol(O) : M → R from the previous lemma is in general different from thefunction vol(O) : M → R, x 7→ vol(G · x). The latter function is by definition nowhere zero and notcontinuous if there are some orbits of dimension 0 and some of dimension > 0.

Lemma 2.16. The orbit normalized measure on M/G fulfills

d(M/G)

vol|Mreg

= dMreg,d(M/G)

vol|M/G−Mreg

≡ 0.

Proof. Considering (2.25), (2.26), and the first statement of Corollary 2.13, the claimed relations areobvious.

Corollary 2.17. The following two measures on (T ∗Mreg ∩ Ωreg)/G agree:

(1) the measure j∗dΩreg, where j is the inclusion j : (T ∗Mreg ∩ Ωreg)/G → Ωreg and dΩreg the

symplectic volume form on Ωreg;

(2) the measure Φ∗d(T ∗Mreg), where Φ : (T ∗Mreg ∩ Ωreg)/G → T ∗Mreg is the canonical symplec-

tomorphism from Lemma 2.9 and d(T ∗Mreg) the symplectic volume form on T ∗Mreg.

Proof. The measures dΩreg and d(T ∗Mreg) are defined by the volume forms ωn−κ/(n − κ)! and

ωn−κ/(n − κ)!, respectively, which implies that the measures j∗dΩreg and Φ∗d(T ∗Mreg) are definedby the volume forms j∗ωn−κ/(n − κ)! and Φ∗ωn−κ/(n − κ)!, respectively. Using compatibility of thewedge product with pullbacks and Lemma 2.9 we obtain

j∗ωn−κ = (j∗ω)n−κ = (Φ∗ω)n−κ = Φ∗(ωn−κ).


Lemma 2.18. As above, denote by τ : T ∗M → M the cotangent bundle projection, and by τ :

(T ∗Mreg ∩Ωreg)/G→ Mreg the smooth map induced by the G-equivariant projection τ |T∗Mreg∩Ωreg . Let

f ∈ C(M) be G-invariant, inducing f ∈ C(M/G), and let p : T ∗M → R be the G-invariant map given

by p(η) = ‖η‖2π(η), inducing p ∈ C∞(Ωreg). Then we have for all a, b ∈ R with 0 < a < b

(2.29)

p−1([a,b])∩((T∗Mreg∩Ωreg)/G)

f(τ(G · η)) dΩreg(G · η) =

Mreg

f(G · x) dMreg(G · x).

Proof. Using Corollary 2.17 we obtainˆ


f(τ(G · η)) dΩreg(G · η)

=

ˆ

Φ(p−1([a,b])∩((T∗Mreg∩Ωreg)/G))

f(τ(Φ−1(q, ξ))) d(T ∗Mreg)(q, ξ).

(2.30)

Now, let q1, . . . , qn−κ, ξ1, . . . , ξn−κ be local Darboux coordinates in some chart domain U of T ∗Mreg.

Then the volume form d(T ∗Mreg) reads

d(T ∗Mreg)(q, ξ) ≡ 1

(n− κ)!dq1 ∧ . . . ∧ dqn−κ ∧ dξ1 ∧ . . . ∧ dξn−κ, (q, ξ) ∈ U.

As before, denote by τ : T ∗Mreg → Mreg the canonical projection and use q1, . . . , qn−κ as coordinates

on τ(U) ⊂ Mreg. Then the local Riemannian volume density is given by

dMreg(q) = |√

det g(q) dq1 ∧ . . . ∧ dqn−κ|, q ∈ τ(U),

where g(q) is the Riemannian metric at the point q, regarded as a symmetric bilinear form on T ∗q Mreg.Let A ⊂ U be measurable and Ξ : A→ C be an integrable function. Thenˆ

A

Ξ(q, ξ)d(T ∗Mreg)(q, ξ) =1

(n− κ)!

ˆ

A

Ξ(q, ξ) dq1 ∧ . . . ∧ dqn−κ ∧ dξ1 ∧ . . . ∧ dξn−κ

=1

(n− κ)!

ˆ

τ(A)

1√det g(q)

ˆ

T∗q Mreg∩A

Ξ(q, ξ) |dξ1 ∧ . . . ∧ dξn−κ|

dMreg(q).

(2.31)

Fix some q ∈ Mreg. The coordinates ξ1, . . . , ξn−κ correspond to an isomorphism T ∗q Mreg∼= Rn−κ

which maps dξi to the i-th standard basis vector ei. Under that identification the symmetric bilinear

form g(q) on T ∗q Mreg is represented by a real symmetric (n− κ)× (n− κ)-matrix of signature n− κ,

which we will also denote by g(q). Furthermore, under the identification T ∗q Mreg∼= Rn−κ the inner

integral in (2.31) is the Lebesgue integral on Rn−κ. Now, basic linear algebra tells us that we can finda real (n − κ) × (n − κ)-matrix Q(q) such that Q(q)ᵀg(q)Q(q) = I, where I is the identity matrix.That implies

det(Q(q)ᵀ)det(g(q))det(Q(q)) = det(I) = 1.

Since det(Q(q)ᵀ) = det(Q(q)) holds, we conclude that det(Q(q))2 = 1/det g(q), which implies inparticular det Q(q) 6= 0, so that Q(q) is invertible. The relation Q(q)ᵀg(q)Q(q) = I means thatwith respect to the new basis Q(q)e1, . . . , Q(q)en−κ of Rn−κ, g(q) is the identity matrix. Denoting

the automorphism T ∗q Mreg → T ∗q Mreg corresponding to Q(q) also by Q(q), we obtain a new ba-

sis Q(q)dξ1, . . . , Q(q)dξn−κ of T ∗q Mreg with respect to which g(q) is the standard Euclidean innerproduct. Write dζi := Q(q)dξi for the new i-th basis element. By the transformation formula for the


Lebesgue integral, the inner integral in (2.31) transforms under the linear substitution dξi = Q(q)−1dζias

(2.32)

ˆ

T∗q Mreg∩A

Ξ(q, ξ) |dξ1 ∧ . . . ∧ dξn−κ| =ˆ

Q(q)(T∗q Mreg∩A)

Ξ(q, ζ)|det Q(q)−1| dζ1 · · · dζn−κ.

However, the observed equality det(Q(q))2 = 1/det(g(q)) implies |det Q(q)−1| =√

det g(q), so that(2.31) and (2.32) together yield

(2.33)

ˆ

A

Ξ(q, ξ)d(T ∗Mreg)(q, ξ) =1

(n− κ)!

ˆ

τ(A)

ˆ

Q(q)(T∗q Mreg∩A)

Ξ(q, ζ) dζ1 · · · dζn−κ︸︷︷︸=:dζ

dMreg(q).

Now, we apply the general result (2.33) to the concrete integral (2.30). By the definition of Φ in (2.14)we have τ Φ−1 = τ , obtaining

(2.34) (n− κ)!

ˆ

U∩Φ(p−1([a,b])∩((T∗Mreg∩Ωreg)/G))

f(τ(Φ−1(q, ξ))) d(T ∗Mreg)(q, ξ)

=

ˆ

τ(U)∩τ [Φ(p−1([a,b])∩((T∗Mreg∩Ωreg)/G))]

ˆ

Q(q)[T∗q Mreg∩U∩Φ(p−1([a,b])∩((T∗Mreg∩Ωreg)/G))]

f(τ(q, ζ)) dζ d(Mreg)(q).

From the definitions of Φ, p, and the Riemannian metric on Mreg one easily sees that Mreg =τ [Φ(p−1([a, b]) ∩ ((T ∗Mreg ∩ Ωreg)/G))] as well as

T ∗q Mreg ∩ Φ(p−1([a, b]) ∩ ((T ∗Mreg ∩ Ωreg)/G)) = ξ ∈ T ∗q Mreg : a ≤ g(q)(ξ, ξ) ≤ b.

Using this and τ(q, ξ) = q, (2.34) simplifies toˆ

τ(U)

ˆ

Q(q)(U∩a≤g(q)(ξ,ξ)≤b)

f(q) dζ d(Mreg)(q)

=

ˆ

τ(U)

f(q) volT∗q Mreg,dζ

(Q(q)(ξ ∈ T ∗q Mreg ∩ U : a ≤ g(q)(ξ, ξ) ≤ b)

)d(Mreg)(q).(2.35)

Taking into account how Q(q) was defined, we proceed by observing that

Q(q)(ξ ∈ T ∗q Mreg : a ≤ g(q)(ξ, ξ) ≤ b) = Q(q)ξ ∈ T ∗q Mreg : a ≤ g(q)(Q(q)ξ,Q(q)ξ) ≤ b

= ζ ∈ T ∗q Mreg, a ≤n−κ∑i=1

ζ2i ≤ b,

from which we conclude

volT∗q Mreg,dζ

(Q(q)(ξ ∈ T ∗q Mreg : a ≤ g(q)(ξ, ξ) ≤ b)

)= Vn−κ(a, b) ∀ q ∈ Mreg,

where we wrote Vn−κ(a, b) for the Euclidean volume of Bn−κ√b− Bn−κ√

a, Bn−κr denoting the standard

ball of radius r around 0 in Rn−κ. In particular, the volume occuring in (2.35) is independent of q.

The result above holds for every coordinate neighbourhood U ⊂ T ∗Mreg. Hence, using a finite open

cover of the compact set Φ(p−1([a, b])∩ ((T ∗Mreg∩Ωreg)/G)) ⊂ T ∗Mreg by coordinate neighbourhoodsand an associated partition of unity, we can establish the global result

(2.36)

ˆ

Φ(p−1([a,b])∩((T∗Mreg∩Ωreg)/G))

f(τ(Φ−1(q, ξ))) d(T ∗Mreg)(q, ξ) =Vn−κ(a, b)

(n− κ)!

ˆ

Mreg

f(q) d(Mreg)(q).


In view of (2.30)we have shownˆ


f(τ(G · η)) dΩreg(G · η) =Vn−κ(a, b)

(n− κ)!

ˆ

Mreg

f(q) d(Mreg)(q).

The last equality holds for arbitrary G-invariant f ∈ C(M), so that in particularˆ


dΩreg(G · η) =Vn−κ(a, b)

(n− κ)!

ˆ

Mreg

d(Mreg)(q).

Taking averages finally yields the assertion.

Lemma 2.19. Let X be a smooth manifold with a smooth measure dX. Consider c ∈ R, and let f :X → R be a smooth, proper function for which c is a regular value. For each δ > 0, let Iδ ⊂ [c−δ, c+δ]be a non-empty interval. Then, for all a ∈ C∞(X) the limit

(2.37) limδ→0

1

vol R(Iδ)

ˆ

f−1(Iδ)

a(x) dX(x) =:

ˆ

f−1(c)

a(x) dµc(x)

exists, and uniquely defines a measure dµc on the hypersurface f−1(c). Furthermore, in the limitδ → 0, we have the estimate

(2.38)1

vol R(Iδ)

ˆ

f−1(Iδ)

a(x) dX(x)−ˆ

f−1(c)

a(x) dµc(x) = O(δ).

Proof. Since f is continuously differentiable and proper, there are α > 0 and γ > 0 such that |∂f | ≥ γon f−1([c − α, c + α]). Consequently, there is a β > 0 such that the function f : X → (c − β, c + β)is a surjective submersion. Ehresmann’s theorem [10] now says that there exists a 0 < % < β and adiffeomorphism

Ψ : f−1((c− %, c+ %)) ' (−%, %)× f−1(c).Under this diffeomorphism, the measure dX splits into a product measure dµ(−%,%) dµf−1(c). Usingthat dX is a smooth measure and absorbing a possible constant into dµf−1(c), we have dµ(−%,%) =dt+O(%), where t denotes the coordinate in the open interval (−%, %). In this way we get

ˆ

f−1((c−%,c+%))

a(x) dX(x) =

ˆ

f−1(c)

%ˆ

−%

a(Φ−1(c+ t, x))( dt+O(%)) dµf−1(c)(x).

Since a is smooth, we have a(Φ−1(c + t, x)) = a(Φ−1(c, x)) + O(|t|) uniformly for t ∈ (−%, %) andx ∈ f−1(c). Moreover, Φ−1(c, x) = x for all x ∈ f−1(c). Therefore, we obtainˆ

f−1(I%)

a(x) dX(x) =

ˆ

f−1(c)

ˆ

I%−c

(a(Φ−1(c, x)) +O(|t|)

)( dt+O(%)) dµf−1(c)(x)

=

ˆ

f−1(c)

(vol(I%) +O(%)) (a(x) +O(%)) dµf−1(c)(x)

= vol(I%)

ˆ

f−1(c)

a(x) dµf−1(c)(x) +O(%)

.(2.39)

In the last equality we used that f−1(c) is compact, hence of finite volume, since f is proper. Now, ifb ∈ C∞(f−1(c)), then b := bπΨ ∈ C∞(f−1((c−%, c+%))), where π : (−%, %)×f−1(c)→ f−1(c)


is the canonical projection. We can choose a function ϕ ∈ C∞c (f−1((c − %, c + %))) with ϕ ≡ 1 in aneighbourhood of f−1(c) to define

(2.40)

ˆ

f−1(c)

b(x) dµc(x) := lim%→0

1

2%

ˆ

f−1((c−%,c+%))

ϕ(x)b(x) dX(x).

The result (2.39) shows that the definition above is independent of the choice of ϕ, and that dµc ≡dµf−1(c) is uniquely characterized by (2.40). Finally, the measure dµc extends from smooth functionsto more general functions by the usual density arguments.

As a corollary, we get a refined version of Lemma 2.18 that applies to hypersurfaces of Ωreg.

Lemma 2.20. In the situation of Lemma 2.18 we have for all G-invariant f ∈ C(M) and c ∈ (0,∞)

(2.41)

p−1(c)

f(τ(G · η)) dµc(G · η) =

Mreg

f(G · x) dMreg(G · x),

where τ : Ωreg → Mreg is the smooth map induced by the G-equivariant projection τ |Ωreg.

Proof. This is a direct consequence of Corollary 2.13, Lemma 2.18, and Lemma 2.19.

Finally, we observe that Corollary 2.17 allows us to replace integrals over particular hypersurfaces

of Ωreg by integrals over the cosphere bundle on Mreg.

Lemma 2.21. Let p : T ∗M → R be the G-invariant map given by p(η) = ‖η‖2π(η), inducing p ∈C∞(Ωreg). Then the symplectomorphism (T ∗Mreg ∩ Ωreg)/G ' T ∗Mreg from Lemma 2.9 induces ameasure-preserving diffeomorphism

(p−1(1), dµ1) ' (S∗Mreg, d(S∗Mreg))

up to null sets of measure spaces.

Proof. By definition of the Riemannian metric on Mreg, the symplectomorphism (T ∗Mreg∩Ωreg)/G 'T ∗Mreg maps (T ∗Mreg/G) ∩ p−1(1) onto S∗Mreg. The claim follows now from Corollary 2.13,Corollary 2.17, and the fact that the hypersurface measure obtained in Lemma 2.19 is unique if theinitial smooth measure is fixed.

2.5. Oscillatory integrals with G-action dependent phase functions. As it will turn out in theensuing sections, our results rely on the description of the asymptotic behavior of certain oscillatoryintegrals that were already examined in [20] while studying the spectrum of an invariant elliptic

operator. Thus, consider a chart γ : M ⊃ U'→ V ⊂ Rn on M , and for a point z ∈ T ∗U write

z = (x, ξ), using the fact that T ∗U is trivial for suitable U . Let aµ ∈ C∞c (U × T ∗U × G) be anamplitude that might eventually depend on a parameter µ ∈ R>0, and consider the phase function

Φ(x, ξ, g) := 〈γ(x)− γ(g · x), ξ〉 ,

where 〈·, ·〉 is the Euclidean scalar product on Rn. The central question in [20] is to describe theasymptotic behavior of oscillatory integrals of the form

(2.42) I(µ) =

ˆ

T∗U

ˆ

G

eiµΦ(x,ξ,g)aµ(g · x, x, ξ, g) dg d (T ∗U) (x, ξ),

and the main result is the following


Theorem 2.22. [20, Theorem 9.1] Let Λ be the maximal number of elements of a totally ordered subsetof the set of isotropy types of the G-action on M . Then, as µ→∞,∣∣∣I(µ)−

(2π

µ

)κ ˆ

Reg Cγ

aµ(g · x, x, ξ, g)∣∣det Φ′′(x, ξ, g)N(x,ξ,g)Reg Cγ∣∣1/2 d(Reg C)(x, ξ, g)

∣∣∣≤ Cvol (supp aµ) sup

l≤2κ+3

∥∥Dlaµ∥∥∞ µ−κ−1 log(µ)Λ−1,

(2.43)

where Reg Cγ = (Reg C) ∩ (T ∗U ×G) and Dl is a differential operator of order l. The expressionΦ′′(x, ξ, g)N(x,ξ,g)Reg Cγ denotes the restriction of the Hessian of Φ to the normal space of Reg Cγ inside

T ∗U ×G at the point (x, ξ, g). In particular, the integral in (2.43) exists.

The proof is based on the stationary and non-stationary phase principles, and uses resolution of

singularities. In particular, the precise form of the remainder estimate follows from the correspondingestimate in the generalized stationary phase theorem, see [20], Theorem 4.1. This precise form willallow us to give remainder estimates also in the case when the amplitude depends on µ. Finally, letus note the following

Lemma 2.23. Let b ∈ C∞c (Ω) and χ ∈ G. Thenˆ

Reg Cγ

χ(g)b(x, ξ)∣∣det Φ′′(x, ξ, g)N(x,ξ,g)Reg Cγ∣∣1/2 d(Reg C)(x, ξ, g) =

[πχ|H : 1

] ˆΩreg

b(x, ξ)d(Ωreg)(x, ξ)

vol (G · (x, ξ)),

where[πχ|H : 1

]denotes the multiplicity of the trivial representation in the representation of H given

by the restriction of the irreducible G-representation πχ to H.

Proof. By using a partition of unity, the proof essentially reduces to the one of [5, Lemma 7], whichinvolves only local calculations. Furthermore, b ∈ C∞c (Ωreg) is required there. However, similarly as in[20, Lemma 9.3], one can use Fatou’s Lemma to show that it suffices to require only b ∈ C∞c (Ω).

3. An equivariant trace formula

In this section, we shall prove a trace formula for Schrodinger operators in the equivariant settingwhich will be crucial for all what follows. Let the notation and setup be as in the previous sections.Thus, let M be a connected closed Riemannian manifold with an effective and isometric action of acompact connected Lie group G, and consider the Schrodinger operator

P (h) = −h2∆ + V, V ∈ C∞(M,R), h ∈ R>0,

V being a G-invariant potential, together with its unique self-adjoint extension

(3.1) P (h) : H2(M)→ L2(M).

Its principal symbol is given by the h-independent symbol function p = ‖·‖2T∗M + V τ ∈ C∞(T ∗M),

and as p is G-invariant, it defines a reduced symbol function p ∈ C∞(Ωreg) on the reduced space Ωreg.Next, recall from (2.6) the Peter-Weyl decomposition

L2(M) =⊕χ∈G

L2χ(M).

In the next section, we shall examine the semiclassical asymptotics of the restricted operator

P|χ(h) = P (h)|L2χ(M) : L2

χ(M)→ L2χ(M),

and prove an equivariant version of the semiclassical Weyl law [27, Theorem 14.11]. Our results will bea consequence of the following trace formula, which is a generalization of [25, Theorem 3] to compactG-manifolds in the case of Schrodinger operators. To state it, recall that by the functional calculus forself-adjoint operators we can define for any % ∈ C∞c (R) the operator %(P (h)), which by Theorem 2.4


is a trace class operator in Ψ−∞h (M). Further, any B ∈ Ψ0h(M) defines a bounded operator in L2(M),

compare [27, Theorem 14.2]. We then have

Theorem 3.1 (Equivariant semiclassical trace formula for Schrodinger operators). Let % ∈C∞c (R) and B ∈ Ψ0

h(M) be a G-equivariant pseudodifferential operator with principal symbol σ(B) = [b]given by a h-independent symbol function b. Then (% (P (h)) B)χ is a trace class operator, and its

trace satisfies the asymptotic estimate4∣∣∣∣ (2πh)n−κ

tr (% (P (h)) B)χ − dχ[πχ|H : 1

] ˆΩreg

(% p)(x, ξ) b(x, ξ) d(Ωreg)(x, ξ)

vol (G · (x, ξ))

∣∣∣∣≤ C sup

j≤2κ+3

∥∥%(j)∥∥∞h log

(h−1

)Λ−1,

where %(j) denotes the j-th derivative of %. If b is G-invariant with reduced symbol function b, the

integral in the formula above simplifies to´

Ωreg(% p) b d(Ωreg).

Proof of Theorem 3.1. Step 1. By Theorem 2.3, the operator P (h) has only finitely many eigenvaluesE(h)1, . . . , E(h)N(h) in supp %, and the corresponding eigenspaces are all finite-dimensional. By thespectral theorem,

(3.2) % (P (h)) =

N(h)∑j=1

% (Ej(h)) Πj ,

where Πj denotes the spectral projection onto the eigenspace Eig (P (h), Ej(h)) of P (h) belonging tothe eigenvalue Ej(h). Hence, % (P (h)) is a finite sum of projections onto finite-dimensional spacesand, consequently, a finite rank operator, and therefore of trace class. It follows immediately that% (P (h)) B Tχ = (% (P (h)) B)χ is trace class, too.

Step 2. To compute the trace of (% (P (h)) B)χ, recall that P (h)− z is invertible for z ∈ C− R,

see [27, Lemma 14.6], and that by the Helffer-Sjostrand formula [27, Theorem 14.8] one has

%(P (h)) =1

iπ

ˆC∂z %(z)(P (h)− z)−1 dz, z = s+ it ∈ C,

where dz denotes the Lebesgue measure on C and % : C→ C the almost analytic extension of % definedin [27, Theorem 3.6], while ∂z = (∂s + i∂t)/2. In what follows, we construct an approximation for

(P (h) − z)−1 in the framework of ordinary pseudodifferential operators following the construction ofthe parametrix of a hypoelliptic operator in [23, Proof of Theorem 5.1]. The class of such operators ofdegree m will be denoted by Ψm(M). Notice that our strategy follows the proof of the correspondingstatement in the non-equivariant setting [27, Theorem 14.10], the major difference being that we have

to consider approximations to (P (h) − z)−1 up to order [(κ + 1)/2]. Consider thus a finite atlas

(Uα, γα)α∈A for M with charts γα : M ⊃ Uα'→ Vα ⊂ Rn, and let ϕαα∈A be a partition of unity

subordinate to Uαα∈A. In the chart (Uα, γα), the action of P (h) − z on a function f ∈ C∞(Uα) isgiven by

(P (h)− z)f(x) = Aα(f γ−1α )(y) := Pα(f γ−1

α )(y)− z

:=−h2√g(y)

n∑j=1

∂

∂ yj

(√g

n∑i=1

∂(f γ−1α )

∂ yigij

)(y) + (V γ−1

α )(y)− z,

where we wrote y = γα(x), while gij : Vα → R are the local coefficients of the metric, g = det gij ,and (gij) denotes the inverse matrix of (gij). As a differential operator on Vα, the local operator

4Note that % p is compactly supported.


Aα ∈ Ψ2(Vα) is a properly supported pseudodifferential operator with a well-defined total symbol

σAα(y, η) := −h2n∑

i,j=1

ηiηjgij(y) + (V γ−1

α )(y)︸︷︷︸=:pα(y,η)

+−h2√g(y)

n∑i,j=1

∂

∂ yj(√ggij

)(y) ηi − z.

In particular Aα is elliptic, and therefore an hypoelliptic operator of class H2,21,0 (Vα) [23, Proposition

5.1]. Assume now Im z 6= 0, and let q0α ∈ HS−2,−2

1,0 (Vα × Rn) be a hypoelliptic symbol satisfying

q0α(y, η) = (pα(y, η)− z)−1 and Q0

α ∈ H−2,−21,0 (Vα) a properly supported operator such that σQ0

α− q0

α ∈S−∞(Vα × Rn). Then

Q0α Aα = 1−R1

α, R1α ∈ Ψ−1(Vα),

since by the composition formula [23, Theorem 3.4] one has

(3.3) σQ0αAα(y, η) ∼ σAα(y, η)

pα(y, η)− z+∑|α|≥1

1

α!(∂αη (pα − z)−1)(y, η)((−i∂y)ασAα)(y, η).

Iteration yields with

Rkα = (R1α)k ∈ Ψ−k(Vα), Qkα = Q0

α Rkα ∈ Ψ−k−2(Vα),

the relationN∑k=0

Qkα Aα = 1−RN+1α ,

compare also [27, Proof of Theorem 14.9]. In particular, one sees that R1α ∈ hΨ−1

h (Vα). In this way,we obtain a parametrix for the local operator Aα. By considering a collection of functions ϕαα∈Awith ϕα ∈ C∞c (M, [0, 1]), supp ϕα ⊂ Uα and ϕα ≡ 1 in a neighborhood of supp ϕα, one can construct

a global parametrix for P (h) − z, see [23, Theorem 5.1] for details, and we denote the correspondingglobal operators on M by

Rk =∑α

Φα Rkα Φα, Qk =∑α

Φα Qkα Φα,

where Φα and Φα are the operators of multiplication with φα and φα, respectively, so that

N∑k=0

Qk (P (h)− z) = 1−RN+1.

Now, the kernels of Q0α and Op(q0

α) differ only outside a neighborhood of the diagonal [23, Proposition3.3]. Therefore, the Helffer-Sjostrand-formula implies that up to terms of order h∞

(3.4) %(Pα) = Op(% pα) +

N∑k=1

Qkα + RN+1α ,

since a direct calculation shows that

1

iπ

ˆC∂z %(z)Op(q0

α(y, η)) dz = Op(% pα),

while

Qkα =1

iπ

ˆC∂z %(z)Qkα(z) dz, RN+1

α =1

iπ

ˆC∂z %(z)RN+1

α (z) A−1α dz.

Similarly, up to terms of order h∞,

(3.5) %(P (h)) = Oph(% p) +

N∑k=1

Qk + RN+1,

where Qk and RN+1 are global operators corresponding to the operators Qkα and RN+1α , respectively.


Step 3. We examine now the h-dependence of the introduced operators. Thus, let rkα(z) be thesymbol of the local operator Rkα. Then (3.3) implies that for each N ∈ N, N ≥ k, there is an expansion

rkα(z) =

N∑l=k

h2l

l∑j=1

ckα,l,j(pα − z)−j+OS−k−N−1(Vα)(h

2(N+1)), Im z 6= 0,

with explicitly given coefficients ckα,l,j ∈ C∞(Vα × Rn) that are independent of z and h. With similar

arguments as above, we obtain for the local symbol qkα(z) of the operator Qkα

qkα(z) =

N∑l=k

h2l

l+1∑j=1

dkα,l,j(pα − z)−j+OS−k−2−N−1(Vα)(h

2(N+1)), Im z 6= 0,

where, again, the coefficients dkα,l,j ∈ C∞(Vα×Rn) are independent of z and h. The complex number z

is just a parameter, that is, a constant with respect to the occurring derivatives. Note that (pα− z)−j ,viewed as a function in z from C − Im−1(0) to C, is analytic due to the ellipticity of pα. As aconsequence, we get, by the linearity of the symbol calculus, for the symbol qkα of the local operator

Qkα for each N ≥ k the expression

qkα(x, ξ) =1

iπ

ˆC∂z %(z)

N∑l=k

h2l

l+1∑j=1

dkα,l,j(x, ξ)(pα(x, ξ)− z)−j dz +OS−k−2−N−1(Vα)(h

2(N+1))

=1

iπ

ˆC∂z

%(z)

N∑l=k

h2l

l+1∑j=1

dkα,l,j(x, ξ)(pα(x, ξ)− z)−j dz +OS−k−2−N−1(Vα)(h

2(N+1)).

As in [27, proof of Theorem 14.8], we can evaluate the complex integrals explicitly using Stokes’theorem and the Cauchy integral formula. Thus,

1

iπ

ˆC∂z

%(z)

N∑l=k

h2l

l+1∑j=1

dkα,l,j(x, ξ)(pα(x, ξ)− z)−j dz

=1

iπlimε→0

ˆC−Bε(pα(x,ξ))

∂z

%(z)

N∑l=k

h2l

l+1∑j=1


=1

2iπlimε→0

ˆ∂Bε(pα(x,ξ))

%(z)

N∑l=k

h2l

l+1∑j=1


=1

2iπ

N∑l=k

h2ll+1∑j=1

dkα,l,j(x, ξ) limε→0

ˆ∂Bε(pα(x,ξ))

((% pα)(x, ξ) +O(ε)

)(pα(x, ξ)− z)−j dz

=

N∑l=k

h2ldkα,l,1(x, ξ)(% pα)(x, ξ),

where Bε(t) ⊂ C denotes the disk with center t and radius ε. Setting dkα,l := dkα,l,1 we conclude thatfor each N ≥ k

(3.6) qkα = % pαN∑l=k

h2ldkα,l +OS−k−2−N−1(Vα)(h2N+2), dkα,l ∈ C∞(Vα × Rn).

In particular, since % pα has compact support in ξ, one sees that qkα is rapidly falling in ξ. Thus,

for each k ∈ N, the operator Qk is an element of Ψ−∞(M). Regarding the operator RN+1, note thatby the symmetry of P (h) one has ‖(P (h)− z)u‖L2 ≥ |Im z| ‖u‖L2 uniformly in h for u ∈ C∞(M) [27,


Proof of Lemma 14.6]. On the other hand, RN+1 is a bounded, compact operator in L2(M), compare[23, Theorem 6.5]. With the expansion for rkα(z) we therefore see that∥∥RN+1 (P (h)− z)−1

∥∥L2→L2 = O(h2N+2|Im z|−k)

for some k ∈ N. Since |∂z %(z)| ≤ Cl|Im z|l for any l ∈ N, we conclude that

‖RN+1‖L2→L2 = O(h2N+2).

The same reasoning applies to the contributions to Qk coming from the remainder terms of orderh2N+2 in (3.6). Let us now examine the composition of %(P (h)) with B. If Bloc ∈ Ψ0(Vα) denotes aproperly supported operator, (3.4) yields

%(Pα) Bloc =

N∑k=0

Qkα Bloc + RN+1α Bloc,

where

Qkα Bloc =1

iπ

ˆC∂z %(z)Qkα(z) Bloc dz, RN+1

α Bloc =1

iπ

ˆC∂z %(z)RN+1

α (z) A−1α Bloc dz.

By the composition formula, the symbol of Qkα(z) Bloc has an analogous expansions to that of qkα,and we can repeat all our previous considerations. In particular,∥∥RN+1 (P (h)− z)−1 B

∥∥L2→L2 = O(h2N+2|Im z|−k),

B being a bounded operator in L2(M). Collecting everything together we have established that up toterms of order h∞ we have

(3.7) %(P (h)) B = Oph((% p) b) +

N∑k=1

h2kSk + TN+1

for each N , where the TN+1 are bounded operators satisfying

‖TN+1‖L2→L2 = O(h2N+2),

and the Sk ∈ Ψ−∞(M) are pseudodifferential operators on M with local symbols of compact supportand independent of h and %. The operator Oph((% p) b) is an element of Ψ−∞(M), and by Theorems2.2 and 2.4 we have %(P (h)) B ∈ Ψ−∞(M), so that TN+1 ∈ Ψ−∞(M). Thus, for arbitrary N ∈ Nand f ∈ C∞(M) we have that up to terms of order h2(N+1)

[Φα % (P (h)) B Φαf ](x) = ϕα(x)1

(2πh)n

ˆ

T∗Uα

eih 〈γα(x)−γα(y),ξ〉sα(x, ξ)ϕα(y)f(y) d (T ∗Uα) (y, ξ),

(3.8)

where

(3.9) sα = (% p)(b+

N∑k=1

h2kskα

), skα ∈ C∞(Vα × Rn),

each skα being independent of h and %, and compacty supported in ξ.

Step 4. We can now compute the trace of the reduction of the localized operator Φα% (P (h))BΦαto the χ-isotypic component. Using (2.7) and (3.8) we obtain for f ∈ C∞(M) up to terms of orderh2(N+1)[

Tχ Φα %(P (h)) B Φαf](x) =

dχ(2πh)n

ˆ

G

ˆ

T∗Uα

χ(g)eih 〈γα(g−1x)−γα(y),ξ〉

· (% p)(g−1x, ξ)(b(g−1x, ξ) +

N∑k=1

h2kskα(g−1x, ξ))ϕα(g−1x)ϕα(y)f(y) d (T ∗Uα) (y, ξ) dg.

(3.10)


Note that the integrands are compactly supported and that the remainder operator has a smoothSchwartz kernel. Thus, if Kχ

α(h)(x, y) denotes the Schwartz kernel of the operator Tχ Φα % (P (h)) B Φα, we infer from this that

tr[Tχ Φα % (P (h)) B Φα

]=

ˆ

M

Kχα(h)(x, x) dM(x)

=dχ

(2πh)n

ˆ

M

ˆ

G

ˆ

Rn

χ(g)eih 〈γα(g−1x)−γα(x),ξ〉ϕα(g−1x)ϕα(x)

· %(p(g−1x, ξ)

)(b(g−1x, ξ) +

N∑k=1

h2kskα(g−1x, ξ)

)dξ dg dM(x) +O

(h2(N+1)

)=

dχ(2πh)n

ˆ

G

ˆ

T∗Uα

eih 〈γα(x)−γα(g·x),ξ〉Jα(x, g)χ(g)ϕα(x)ϕα(g · x)

· % (p(x, ξ))

(b(x, ξ) +

N∑k=1

h2kskα(x, ξ)

)d (T ∗Uα) (x, ξ) dg +O

(h2(N+1)

),

(3.11)

where Jα(x, g) is the Jacobian of the substitution x = g · x′, and N ∈ N is arbitrary. We are now

prepared to use our last major tool, Theorem 2.22. For this, let uχ,Nh,α ∈ C∞c (Uα × T ∗Uα ×G) be givenby

(3.12) uχ,Nh,α (g · x, x, ξ, g) = Jα(x, g)χ(g)% (p(x, ξ))(b(x, ξ) +

N∑k=1

h2kskα(x, ξ))ϕα(x)ϕα(g · x).

Then (3.11) and Theorem 2.22 imply for each N ∈ N the estimate∣∣∣∣(2πh)ntr[Tχ Φα % (P (h)) B Φα

]− dχ (2πh)

κˆ

Reg C

uχ,Nh,α (g · x, x, ξ, g)∣∣det Φ′′(x, ξ, g)N(x,ξ,g)Reg Cα∣∣1/2 d(Reg C)(x, ξ, g)

︸︷︷︸=:Aα

∣∣∣∣≤ Cα,N

(sup

l≤2κ+3

∥∥Dluχ,Nh,α∥∥∞h

κ+1 log(h−1

)Λ−1+ h2(N+1)

),

(3.13)

where Reg Cα = ((x, ξ), g) ∈ (Ω ∩ T ∗Uα) × G, g(x, ξ) = (x, ξ), x ∈ M(H), Dl is a differentialoperator of order l, and Φ′′(x, ξ, g)N(x,ξ,g)Reg Cα denotes the restriction of the Hessian of Φ(x, ξ, g) =

〈γα(x)− γα(g · x), ξ〉 to the normal space of Reg Cα inside T ∗Uα ×G at the point (x, ξ, g). Note thatthe domain of integration contains only such g and x for which we have g · x = x, so that the integralsimplifies to

(3.14) Aα =

ˆ

Reg C

χ(g)% (p(x, ξ))(b(x, ξ) +

∑Nk=1 h

2kskα(x, ξ))ϕα(x)∣∣det Φ′′(x, ξ, g)N(x,ξ,g)Reg Cα

∣∣1/2 d(Reg C)(x, ξ, g).

Here we used that Jα(x, g) = 1 in the domain of integration, since the substitution x′ = g · x is theidentity when g · x = x, and that ϕα ≡ 1 on supp ϕα. By Lemma 2.23 this simplifies further to

(3.15) Aα =[πχ|H : 1

] ˆΩreg

% (p(x, ξ))(b(x, ξ) +

N∑k=1

h2kskα(x, ξ))ϕα(x)

d(Ωreg)(x, ξ)

vol (G · (x, ξ)).

Step 5. To finally calculate the trace of (% (P (h)) B)χ, note that by definition supp (1− ϕα) ∩supp ϕα = ∅. As already noted, a fundamental property of a semiclassical pseudodifferential operator


is that outside the diagional its kernel is smooth and the supremum norm of the kernel is rapidlydecreasing in h. Therefore,

(% (P (h)) B)χ Φα = Tχ Φα % (P (h)) B Φα + Tχ Rα∞,(3.16)

where Rα∞ ∈ h∞Ψ−∞h (M). The remainder operator Tχ Rα∞ in (3.16) is trace class since it has asmooth kernel, and its trace is of order h∞. Consequently, for each N ∈ N we have

tr (% (P (h)) B)χ = tr (% (P (h)) B)χ ∑α∈A

Φα =∑α∈A

tr (% (P (h)) B)χ Φα

=∑α∈A

tr Tχ Φα % (P (h)) B Φα +O (h∞) .

With (3.13) this leads to∣∣∣∣ (2πh)ntr (% (P (h)) B)χ − dχ (2πh)

κ∑α∈A

Aα

∣∣∣∣≤∑α∈A

Cα,N

(sup

l≤2κ+3

∥∥∥Dluχ,Nh,α

∥∥∥∞hκ+1 log

(h−1

)Λ−1+ h2(N+1)

).

(3.17)

Since the functions skα do not depend on any derivatives of %, we have

supl≤2κ+3

∥∥∥Dluχ,Nh,α

∥∥∥∞≤ c1 sup

l≤2κ+3

∥∥Dl(% p)∥∥∞ ≤ c2 sup

l≤2κ+3

∥∥∥%(l)∥∥∥∞

(3.18)

for suitable constants ci > 0. From (3.17) and (3.18) we conclude that∣∣∣∣ (2πh)n

tr (% (P (h)) B)χ − dχ (2πh)κ∑α∈A

Aα

∣∣∣∣≤ CN

(sup

l≤2κ+3

∥∥∥%(l)∥∥∥∞hκ+1 log

(h−1

)Λ−1+ h2(N+1)

).

Finally, (3.15) implies that for each N ∈ N

∑α∈A

Aα =[πχ|H : 1

] ˆΩreg

% (p(x, ξ)) b(x, ξ)

(∑α∈A

ϕα(x)

)d(Ωreg)(x, ξ)

vol (G · (x, ξ))

+[πχ|H : 1

] N∑k=1

h2k∑α∈A

ˆ

Ωreg

% (p(x, ξ)) skα(x, ξ)ϕα(x)d(Ωreg)(x, ξ)

vol (G · (x, ξ)).

Absorbing the second term on the right hand side into the remainder estimate we finally obtain withN = [(κ+ 1)/2] as h→ 0∣∣∣∣ (2πh)

n−κtr (% (P (h)) B)χ − dχ

[πχ|H : 1

] ˆ

Reg Ω

% (p(x, ξ)) b(x, ξ)d(Ωreg)(x, ξ)

vol (G · (x, ξ))

∣∣∣∣≤ C sup

l≤2κ+3

∥∥∥%(l)∥∥∥∞h log

(h−1

)Λ−1.

That the integral simplifies to the claimed expression if b is G-invariant follows immediately from the

fact that by (2.28) the measured(Ωreg)volO projects onto the measure d(Ωreg) when passing to the quotient

space.


4. Generalized equivariant semiclassical Weyl law

We are now in the position to state and prove a generalized semiclassical Weyl law for Schrodingeroperators in the equivariant setting. For this, fix a Hilbert basis uj(h)j∈N of L2(M) such thateach uj(h) is an eigenfunction of P (h), and denote the eigenvalue associated to uj(h) by Ej(h). Forsimplicity, we shall also write uj(h) = uj and Ej(h) = Ej in the following, but it is essential to keepin mind that these quantities depend on h.

Theorem 4.1 (Generalized equivariant semiclassical Weyl law). Let B ∈ Ψ0h(M) be a G-

equivariant semiclassical pseudodifferential operator with principal symbol σ(B) = [b], and assume thatb ∈ S0(M) is G-invariant and independent of h. Let c ∈ R be a regular value of the reduced symbol

function p. Then we have for each β ∈(

0, 12(κ+2)

)and χ ∈ G


j ∈ N, uj(h) ∈ L2χ(M),

Ej(h) ∈ [c, c+ hβ ]


= dχ [πχ|H : 1]

ˆΣc

b dµc +O(h

12(κ+2)

−β(log h−1)Λ−1),

(4.1)

where [πχ|H : 1] denotes the multiplicity of the trivial representation in the H-representation πχ|H .

Proof. The proof is an adaptation of the proof of [27, Theorem 15.3] to our situation, but with asharper energy localization. To begin, let h > 0 and 0 < ε < 1

6 . Choose fε, gε ∈ C∞c (R, [0, 1]) such

that supp fε ⊂ [− 12 + ε, 1

2 − ε], fε ≡ 1 on [− 12 + 3ε, 1

2 − 3ε], supp gε ⊂ [− 12 − 3ε, 1

2 + 3ε], gε ≡ 1 on

[− 12 − ε,

12 + ε], and

(4.2) | ∂jy fε(y)| ≤ Cj ε−j , | ∂jy gε(y)| ≤ Cj ε−j ,

compare [13, Theorem 1.4.1 and (1.4.2)]. Put c(h) := ch−β + 12 , so that x 7→ h−βx − c(h) defines

a diffeomorphism from [−1/2, 1/2] to [c, c + hβ ], and set fε,h−β (x) := fε(h−βx − c(h)), gε,h−β (x) :=

gε(h−βx − c(h)). Let Πχ be the projection onto the span of uj ∈ L2

χ(M) : Ej ∈ [c, c + hβ ]. Then,for sufficiently small ε > 0,

fε,h−β (P (h))χ Πχ = Πχ fε,h−β (P (h))χ = fε,h−β (P (h))χ,

gε,h−β (P (h))χ Πχ = Πχ gε,h−β (P (h))χ = Πχ.(4.3)

Note that the operators fε,h−β (P (h)), gε,h−β (P (h)), Πχ and arbitrary, multiple compositions of theseoperators with Tχ or B are finite rank operators. For that elementary reason, all operators we considerin the following are trace class. In particular, by (4.3) we have∑

j ∈ N, uj ∈ L2χ(M)

Ej ∈ [c, c+ hβ ]

〈Buj , uj〉L2(M) = tr Πχ B Πχ = tr Πχ Bχ Πχ

= tr fε,h−β (P (h))χ Bχ + tr Πχ gε,h−β (P (h))χ (1− fε,h−β (P (h))χ

)Bχ Πχ

= tr(fε,h−β (P (h)) B

)χ

+ tr Πχ (gε,h−β (P (h))

(1− fε,h−β (P (h))

)B)χΠχ︸︷︷︸

=Rε,h−β

.(4.4)

Next, we estimate Rε,h−β using the trace norm ‖·‖L1(M). It is defined for a compact operator A ∈B(L2(M)

)by ‖A‖L1(M) =

∑k∈N sk, where

s2k

k∈N are the eigenvalues of the self-adjoint operator

A∗A. Now, if L ∈ L1(M) and M ∈ B(L2(M)

), then ‖LM‖L1(M) ≤ ‖L‖L1(M) ‖M‖B(L2(M)), see e. g.


[27, p. 337]. By the functional calculus this implies

|Rε,h−β | ≤∥∥∥Πχ

(gε,h−β (P (h))

(1− fε,h−β (P (h))

)B)χΠχ

∥∥∥L1(M)

(4.5)

≤∥∥∥(gε,h−β (P (h))

(1− fε,h−β (P (h))

))χ

∥∥∥L1(M)

‖B‖B(L2(M))(4.6)

=∥∥vε,h−β (P (h))χ

∥∥L1(M)

‖B‖B(L2(M)) ,(4.7)

where we set vε,h−β = gε,h−β (1− fε,h−β ) ∈ C∞c (R, [0, 1]). In particular, vε,h−β is non-negative. By thespectral theorem, vε,h−β (P (h)) is a positive operator. Tχ is a projection, hence positive as well. Itfollows that vε,h−β (P (h))χ is positive as the composition of positive operators. For a positive operator,the trace norm is identical to the trace. Therefore (4.7) implies

(4.8) |Rε,h−β | ≤ ‖B‖B(L2(M)) tr vε,h−β (P (h))χ.

From our knowledge about the supports of fε and gε, we conclude that

(4.9) supp vε,h−β ⊂ [c− 3εhβ , c+ 3εhβ ] ∪ [c+ hβ − 3εhβ , c+ hβ + 3εhβ ].

Now, by Theorem 3.1 with B = id L2(M) and (4.2) we conclude∣∣∣ (2πh)n−κ

tr vε,h−β (P (h))χ − dχ[πχ|H : 1

] ˆΩreg

(vε,h−β p)(G · (x, ξ)) d(Ωreg)(G · (x, ξ))∣∣∣

≤ Cε−2κ−3h1−β(2κ+3)(log h−1

)Λ−1.

(4.10)

On the other hand, applying Theorem 3.1 to the first summand on the right hand side of (4.4) yields∣∣∣ (2πh)n−κ

tr(fε,h−β (P (h)) B

)χ− dχ

[πχ|H : 1

] ˆΩreg

(fε,h−β p)(G · (x, ξ))b(G · (x, ξ))d(Ωreg)(G · (x, ξ))∣∣∣

≤ Cε−2κ−3h1−β(2κ+3)(log h−1

)Λ−1.

Combining this with (4.4) leads to

(2πh)n−κ ∑

j ∈ N, uj ∈ L2χ(M)

Ej ∈ [c, c+ hβ ]

〈Buj , uj〉L2(M)

= dχ[πχ|H : 1

] ˆ

Ωreg

(fε,h−β p)(G · (x, ξ))b(G · (x, ξ))d(Ωreg)(G · (x, ξ))

+ (2πh)n−κRε,h−β +O(ε−2κ−3h1−β(2κ+3)(log h−1

)Λ−1).

(4.11)

Furthermore, by (4.8), and (4.10),

|(2πh)n−κRε,h−β | ≤ ‖B‖B(L2(M)) dχ [πχ|H : 1]

ˆ

Ωreg

(vε,h−β p)(G · (x, ξ)) d(Ωreg)(G · (x, ξ))

+O(ε−2κ−3h1−β(2κ+3)(log h−1

)Λ−1).(4.12)

Now, let a, b ∈ R. Then p−1([a, b]) ⊂ T ∗M is closed and bounded, hence compact. On the other hand,p−1([a, b]) is a closed subset of p−1([a, b])/G. The latter space carries the quotient topology, and inparticular is compact. It follows that p−1([a, b]) is a compact subspace of T ∗M/G. Since Ωreg ⊂ T ∗Mcarries the subspace topology, also Ωreg ⊂ T ∗M/G carries the subspace topology. Therefore, p−1([a, b])


is in fact a compact subspace of Ωreg. Thus, we have established that the reduced symbol function pis a proper map. Next, note that by Lemma 2.19 we haveˆ

Ωreg

(vε,h−β p)(G · (x, ξ)) d(Ωreg)(G · (x, ξ)) = O(εhβ).

Similarly, Lemma 2.19 givesˆ

Ωreg

(fε,h−β p)(G · (x, ξ))b(G · (x, ξ))d(Ωreg)(G · (x, ξ))

= vol ([c+ εhβ , c+ hβ − εhβ ])

(ˆp−1(c)

b(G · (x, ξ)) dµc(G · (x, ξ)) +O(εhβ)

).

Using the last two estimates together with (4.12) in (4.11) yields


j ∈ N, uj ∈ L2χ(M)

Ej ∈ [c, c+ hβ ]

〈Buj , uj〉L2(M) − dχ[πχ|H : 1

] ˆ

p−1(c)

b(G · (x, ξ)) dµc(G · (x, ξ))

= O(ε) +O(ε−2κ−3h1−β(2κ+4)(log h−1)Λ−1

).

If we now put ε = h1

2(κ+2)−β , the assertion of the theorem follows.

As consequence of the previous theorem we obtain in particular

Theorem 4.2 (Equivariant semiclassical Weyl law). Let χ ∈ G, β ∈(0, 1

2(κ+2)

)and let c ∈ R be

a regular value of p. Then

(4.13)

(2π)n−κhn−κ−β∑j ∈ N,

Ej(h) ∈ [c, c+ hβ ]

multjχ(h) = [πχ|H : 1] vol µcΣc +O(h

12(κ+2)

−β(log h−1)Λ−1),

where Λ is as in Theorem 2.22, multjχ(h) denotes the multiplicity of the irreducible representation πχin the eigenspace Eig (P (h), Ej(h)), and [πχ|H : 1] denotes the multiplicity of the trivial representationin the H-representation πχ|H .

We close this section with the following elementary

Corollary 4.3. Let c ∈ R be as in Theorem 4.2 and β ∈(0, 1

2(κ+2)

). Then, for each χ ∈ G there is a

hχ > 0 such that for all h ∈ (0, hχ)

(4.14)⋃

j ∈ N,Ej(h) ∈ [c, c+ hβ ]

Eig (P (h), Ej(h)) ∩ L2χ(M) 6= 0.

Proof. Since c is a regular value of p, Σc ≡ p−1(c) is a non-degenerate hypersurface of Ωreg, which

implies that volµcΣc > 0. Consequently, the leading term on the right hand side of (4.13) is non-zero.If the claim did not hold, we could find a sequence (hk)k∈N ⊂ R>0, converging to zero, such that theleft hand side of (4.13) would be zero for all k ∈ N, a contradiction.


5. Equivariant classical ergodicity

This section is devoted to the study of classical ergodicity in the presence of symmetries on basisof the general theory of symplectic reduction. Consider thus a connected, symplectic, paracompactmanifold (X, ω) with a global Hamiltonian action of a Lie group G, and let J : X → g∗ be thecorresponding momentum map. Let µ be a value of J, K ⊂ G an isotropy group of the G-action on X,

and η ∈ J−1(µ). With the notation as in Theorem 2.7, let now c ∈ R, and put Σ(K)µ (c) := (p

(K)µ )−1(c).

Let g be a Riemannian metric on Ω(K)µ and J : T Ω

(K)µ → T Ω

(K)µ the almost complex structure

determined by ω(K)µ and g, so that (Ω

(K)µ ,J , g) becomes an almost Hermitian manifold. We then make

the following

Assumption 1. c is a regular value of p(K)µ .

Note that this assumption is implied by the condition that for all ξ ∈ J−1(µ) ∩ Gµ ·XηK ∩ Σc one

has

Hp(ξ) /∈ gµ · ξ.

Indeed, assume that there exists some [ξ] ∈ Σ(K)µ (c) such that grad p

(K)µ ([ξ]) = 0. Since

ω(K)µ (s-grad p(K)

µ ,X) = dp(K)µ (X) = g(grad p(K)

µ ,X),

we infer that Hp(K)µ

([ξ]) = s-grad p(K)µ ([ξ]) = 0, which means that [ξ] ∈ Σ

(K)µ (c) is a stationary point

for the reduced flow, so that ϕµt ([ξ]) = [ξ] for all t ∈ R. By the fourth assertion in Theorem 2.7, thisis equivalent to

π(K)µ ϕt ι(K)

µ (ξ′) = ϕµt ([ξ]) ∀ t ∈ R, ξ′ ∈ Gµ · ξ,

which in turn is equivalent to ϕtι(K)µ (ξ′) ∈ Gµ·ξ′. Thus, there exists a Gµ-orbit in J−1(µ)∩Gµ·Xη

K∩Σcwhich is invariant under ϕt. In particular one has Hp(ξ

′) ∈ gµ · ξ′ for all ξ′ ∈ Gµ · ξ.Now, the assumption ensures that Σ

(K)µ (c) is a smooth submanifold of Ω

(K)µ . Equipping Ω

(K)µ with

the smooth measure defined by the unique symplectic form on Ω(K)µ described in Theorem 2.7, Lemma

2.19 says that there is a unique hypersurface measure ν(K)µ on Σ

(K)µ (c). Moreover, ν

(K)µ is invariant

under the reduced flow ϕµt , since the latter constitutes a symplectomorphism due to Cartan’s homotopyformula. Next, we make the following

Definition 5.1. The reduced flow ϕµt is called ergodic on Σ(K)µ (c) if for any measurable set E ⊂

Σ(K)µ (c) with ϕµt (E) = E, one has

ν(K)µ (E) = 0 or ν(K)

µ (E) = ν(K)µ (Σ(K)

µ (c)).

We can now formulate the reduced mean ergodic theorem.

Theorem 5.1 (Equivariant or reduced mean ergodic theorem). Let Assumption 1 above be ful-

filled, and suppose that the reduced flow ϕµt is ergodic on Σ(K)µ (c). Then, for each f ∈ L2

(Σ

(K)µ (c), dν

(K)µ

)we have

〈f〉TT→∞→ 1

ν(K)µ (Σ

(K)µ (c))

ˆ

Σ(K)µ (c)

f dν(K)µ

with respect to the norm topology of L2(

Σ(K)µ (c), dν

(K)µ

), where

〈f〉T ([µ]) :=1

T

T

0

f (ϕµt ([µ])) dt, [µ] ∈ Σ(K)µ (c).

Proof. The proof is completely analogous to the proof of [27, Theorem 15.1].


In all what follows, we shall be interested mainly in the case where X = T ∗M is the cotangentbundle of a G-manifold M , µ = 0, and K = H is given by a principal isotropy group. In the rest ofthe paper we shall therefore use the simpler notation

Ωreg = Ω(H)0 , ϕt = ϕ0

t , Σc = Σ(H)0 (c), dµc = dν

(H)0 , p = p

(H)0 .

As a special case of Theorem 5.1 we then obtain the following equivariant version of [27, Theorem15.1].

Theorem 5.2. Let c ∈ R be a regular value of the reduced symbol function p. Denote by dµc the hyper-

surface measure on Σc ⊂ Ωreg which is induced by the smooth measure on Ωreg defined by the symplectic

form. Suppose that the reduced flow ϕt is ergodic on(

Σc, dµc

). Then for each f ∈ L2

(Σc, dµc

),

limT→∞

ˆ

Σc

〈f〉T − ˆ

Σc

f dµc

2

dµc = 0,

where 〈f〉T (G · (x, ξ)) := 1T

´ T0f (ϕt(G · (x, ξ))) dt, G · (x, ξ) ∈ Σc.

6. Equivariant quantum ergodicity

We commence now with our study of equivariant quantum ergodicity. For this we shall examinefirst the relation between classical and quantum time evolution in the equivariant setting.

6.1. Equivariant quantum time evolution. An important theorem which relates classical andquantum time evolution in the non-equivariant case is the weak Egorov theorem [27, Theorem 15.2].In contrast to the semiclassical Weyl law, we will see in the next section that we do not need a specialequivariant version of the weak Egorov theorem to prove an equivariant quantum ergodicity theorem.To explain this, let the notation be as in the previous sections, and consider the Schrodinger operatorP (h) given by the unique self-adjoint extension of the differential operator (3.1) with symbol function pregarded as a Hamiltonian function on (T ∗M,ω). Let p be the corresponding reduced symbol function,

regarded as a Hamiltonian function on (Ωreg, ω), and denote the corresponding Hamiltonian flows byϕt and ϕt, respectively, compare Theorem 2.7. Next, define the time evolution operator

Fh(t) : L2(M)→ L2(M), Fh(t) := e−itP (h)/h, t ∈ R,

which by Stone’s theorem is a well-defined bounded operator, and similarly introduce for each χ ∈ Gthe operator

Fhχ (t) : L2(M)→ L2(M), Fhχ (t) := e−itPχ(h)/h.

Both time evolution operators are given by their defining Neumann series, from which one infers thatthe G-equivariance of P (h) implies that F (t) and Tχ commute. Furthermore, a computation showsthat

(6.1) Fhχ (t) = ΠL2χ(M)⊥ +

(Fh(t)

)χ,

where ΠL2χ(M)⊥ denotes the projection onto the orthogonal complement of L2

χ(M) in L2(M). Next, let

A ∈ Ψ−∞h (M) be G-equivariant. Then

A(t) := Fh(t)−1AFh(t)


represents the time evolution with respect to Fh(t). To find an appropriate expression for a G-reducedtime evolution of A, we consider first all possible combinations

χχA(t) :=

(Fh(t)−1

)χAχ(Fh(t)−1

)χ, Aχ(t) := Fh(t)−1AχF

h(t),

Aχ(t) := Fhχ (t)−1AFhχ (t), Aχχ(t) := Fhχ (t)−1AχFhχ (t),

χA(t) :=(Fh(t)−1

)χA(Fh(t)−1

)χ, χA(t) := A(t)χ.

One then easily verifies the following

Lemma 6.1. We have

Aχ(t) = Aχχ(t) = χA(t) = χA(t) = χχA(t) = A(t)χ, Aχ(t) = AΠL2

χ(M)⊥ +Aχ(t).

In particular, all operators agree on L2χ(M).

As a consequence of the preceding lemma, we can use the usual time evolution operator Fh(t) also

in the G-equivariant setting. That observation explains why we will not need a special equivariantversion of the weak Egorov theorem [27, Theorem 15.2] in the proof of Theorem 6.4. However, we needsome elementary lemmas as preparation.

Lemma 6.2 (Time evolution of equivariant operators). Let T > 0 be fixed, and A ∈ Ψ−∞h (M) aG-equivariant semiclassical pseudodifferential operator. Assume that σ(A) = [a], where a ∈ S−∞(M)is a G-invariant symbol function, and define for each t ∈ [0, T ]

[A](t) := Oph ([a ϕt]) ∈ Ψ−∞h (M)/(hΨ−∞h (M)

).

Then, each representant A(t) ∈ Ψ−∞h (M) of the equivalence class [A](t) is, modulo hΨ−∞h (M), aG-equivariant operator.

Proof. Since a is G-invariant, and ϕt commutes with the G-action on T ∗M by Theorem 2.7, a ϕt isG-invariant. However, the quantization of a G-invariant symbol function of order −∞ is G-equivariantmodulo hΨ−∞h (M) by Theorem 2.1, and the assertion follows. Note that R(t) := A(t) − A(t) is a

G-equivariant operator modulo hΨ−∞h (M).

Note that if i : Ωreg → T ∗M denotes the inclusion and π : Ωreg → Ωreg the projection onto theG-orbit space, Theorem 2.7 4.) says that π ϕt i = ϕt π. Since

a ϕt i = (a ϕt) π, a i = s π,

we get

a ϕt π=a π ϕt i=a i ϕt i = a ϕt i= (a ϕt) π,

where we used that i ϕt i = ϕt i holds as ϕt leaves Ωreg invariant. Since π is surjective, we haveshown

Lemma 6.3 (Time evolution and reduction). Let m ∈ R and a ∈ Sm(M) be a G-invariant symbolfunction. Let ϕt be the flow on T ∗M associated to the Hamiltonian p. Let ϕt be the reduced flow on

Ωreg associated to p. Then time evolution and reduction commute,

(a ϕt) = a ϕt.


6.2. Equivariant quantum ergodicity I. We are now ready to formulate our first quantum ergodictheorem in the equivariant setting.

Theorem 6.4 (Integrated equivariant quantum ergodicity for Schrodinger operators). Letuj(h)j∈N be a Hilbert basis of L2(M) of eigenfunctions of the Schrodinger operator P (h) with symbolfunction p, and denote the eigenvalue associated to uj(h) by Ej(h). Let c ∈ R be a regular value of

the reduced symbol function p, and suppose that the reduced flow ϕt is ergodic on Σc := p−1(c). LetA ∈ Ψ0

h(M) be G-equivariant and a ∈ S0(M) a G-invariant, h-independent symbol function such that

σ(A) = [a]. Then, for each χ ∈ G and β ∈(0, 1

2(κ+2)

),

(6.2) limh→0

hn−κ−β∑

j ∈ N, uj(h) ∈ L2χ(M),

Ej(h) ∈ [c, c+ hβ ]

∣∣∣∣〈Auj(h), uj(h)〉L2(M) −

Σc

a dµc

∣∣∣∣2 = 0.

Proof. As before, we shall adapt the existing proofs of quantum ergodicity to the equivariant situation,following mainly [27, Theorem 15.4] and also [9, Theorem 5 in Appendix D]. For simplicity, we shallagain write uj(h) = uj and Ej(h) = Ej . Let % ∈ C∞c (R, [0, 1]) be such that % ≡ 1 in a neighbourhoodof c. Without loss of generality we may assume for the rest of the proof that h is small enough so that% ≡ 1 on [c, c+ hβ ]. Set

(6.3) B := %(P (h)) (A− α id L2(M)

), α :=

Σc

a(G · (x, ξ)) dµc(G · (x, ξ)).

Note that by Theorems 2.4 and 2.2 we have B ∈ Ψ−∞h (M). Furthermore, by Theorem 2.2

σ(B) = (% σ(P (h)))σ(A− α id L2(M)

)= [(% p) (a− α 1M )] ∈ S−∞(M)/hS−∞(M).

The reduced symbol function fulfills

(6.4) ((% p) (a− α 1M )) = (% p) (a− α 1M ) = (% p)(a− α 1Ωreg

),

where 1Ωregdenotes the constant function on Ωreg with value 1. Next, we define

(6.5) ε(h) := (2π)n−κhn−κ−β∑

j ∈ N, uj ∈ L2χ(M)

Ej ∈ [c, c+ hβ ]

∣∣∣〈Buj , uj〉L2(M)

∣∣∣2 .By the spectral theorem, %(P (h))uj = uj for Ej ∈ [c, c + hβ ], since % ≡ 1 on [c, c + hβ ]. Taking intoaccount the self-adjointness of %(P (h)) one sees that

(6.6) 〈Buj , uj〉L2(M) = 〈Auj , uj〉L2(M) − α.

Consequently, we would be done with the proof if we can show that

(6.7) limh→0

ε(h)=0.

Now, in order to make use of classical ergodicity, one notes that the expectation value

〈Buj , uj〉L2(M) =⟨Be−itEj/huj , e

−itEj/huj

⟩L2(M)

=⟨Be−itP (h)/huj , e

−itP (h)/huj

⟩L2(M)

= 〈B(t)uj , uj〉L2(M) ∀ t ∈ [0, T ]

is actually time-independent. This implies

〈Buj , uj〉L2(M) = 〈〈B〉T uj , uj〉L2(M),

where 〈B〉T = 1T

´ T0B(t)dt ∈ Ψ−∞h (M). Taking into account ‖uj‖2L2(M) = 1 and the Cauchy-Schwarz

inequality one arrives at

(6.8)∣∣∣〈Buj , uj〉L2(M)

∣∣∣2 ≤ ‖〈B〉T uj‖2L2(M).


We therefore conclude from (6.5) and (6.8) that

(6.9) ε(h) ≤ (2π)n−κhn−κ−β∑

j ∈ N, uj ∈ L2χ(M),

Ej ∈ [c, c+ hβ ]

〈〈B∗〉T 〈B〉T uj , uj〉L2(M).

Next, let B(t) be a representant of the equivalence class Oph (σ(B) ϕt). Lemma 6.2 implies that B(t)is G-equivariant modulo hΨ−∞h (M) for all t ∈ [0, T ], and by the weak Egorov theorem [27, Theorem15.2] one has ∥∥B(t)−B(t)

∥∥B(L2(M))

= O(h) uniformly for t ∈ [0, T ],

which implies

(6.10) 〈B〉T =⟨B⟩T

+OTB(L2(M))(h).

Without loss of generality we will assume from now on that B(t) is G-equivariant, a property it has upto the same order in h as the remainder occurring in Egorov’s theorem, and this will be the only resultabout B(t) that we use. As a consequence, the averaged operator

⟨B⟩T∈ Ψ−∞h (M) is G-equivariant,

and from the definition of B we get

σ(⟨B⟩T

)=

1

T

T

0

b ϕt dt

,where we wrote b := (% p) (a− α 1M ), so that σ(B) = [b]. Since b is G-invariant, we have

(6.11)

1

T

T

0

b ϕt dt

˜=

1

T

T

0

(b ϕt) dt =1

T

T

0

b ϕt dt = 〈b〉T ,

where we took into account the linearity of the -operation and Lemma 6.3. The symbol map is a∗-algebra homomorphism from Ψ−∞h (M) to S−∞(M)/hS−∞(M), with involution given by the adjointoperation and pointwise complex conjugation on representants, respectively. That leads to

σ(⟨B∗⟩

T

⟨B⟩T

)=

∣∣∣∣∣∣ 1

T

T

0

b ϕt dt

∣∣∣∣∣∣2 .

Since the symbol function∣∣∣ 1T

´ T0b ϕt dt

∣∣∣2 is certainly G-invariant and independent of h, we can apply

the equivariant generalized Weyl law, Theorem 4.1, which together with (6.11) yields


j ∈ N, uj ∈ L2χ(M),

Ej ∈ [c, c+ hβ ]

⟨⟨B∗⟩T

⟨B⟩Tuj , uj

⟩L2(M)

= dχ[πχ|H : 1]

ˆΣc

|〈b〉T |2 dµc +O(h1

2(κ+2)−β(log h−1)Λ−1).

(6.12)

From (6.4) we see that over Σc ≡ p−1(c) we have b|Σc = a|Σc − α · 1Σc=: bc. With (6.9), (6.10) and

(6.12) we deduce

ε(h) ≤ dχ[πχ|H : 1]

ˆΣc

|〈bc〉T |2 dµc +O(h1

2(κ+2)−β(log h−1)Λ−1)

+ (2π)n−κhn−κ−β#j ∈ N, Ej ∈ [c, c+ hβ ], uj ∈ L2χ(M) ·O(h).


By the equivariant semiclassical Weyl law, Theorem 4.2, the factor in front of the O(h)-estimate isconvergent and therefore bounded as h→ 0. Thus,

(6.13) lim suph→0

ε(h) ≤ dχ [πχ|H : 1]

ˆΣc

|〈bc〉T |2 dµc.

This is now the point where classical ergodicity is used. Since bc fulfillsffl

Σcbc dµc = 0, Theorem 5.2

yields limT→∞´

Σc|〈bc〉T |2dµc = 0. Since the left hand side of (6.13) is independent of T , it follows

that it must be zero, yielding (6.7).

Remark 6.5. Note that one could have still exhibited the Weyl remainder estimate in (6.13). But sincethe rate of convergence in Theorem 5.2 is not known in general, it is not possible to give a remainderestimate in Theorem 6.4 with the methods employed here.

We would like to close this section by noting that a weaker version of Theorem 6.4 can be provenwith a less sharp energy localization in an interval [r, s] with r < s by the same methods employedhere. In fact, under the additional assumption that the mean value α introduced in (6.3) is the samefor all c ∈ [r, s] and all considered c are regular values of p, the reduced flow being ergodic on each of

the contemplated hypersurfaces Σc, one can show that

limh→0

hn−κ∑

j ∈ N, uj(h) ∈ L2χ(M),

Ej(h) ∈ [r, s]

∣∣∣ 〈Auj(h), uj(h)〉L2(M) −

p−1([r,s])

a dΩreg

∣∣∣2 = 0.

The proof of this relies on a corresponding semiclassical Weyl law

(2πh)n−κ∑

j ∈ N, uj(h) ∈ L2χ(M),

Ej(h) ∈ [r, s]


= dχ [πχ|H : 1]

ˆ

p−1([r,s])

b dΩreg +O(h

12(κ+2) (log h−1)Λ−1

),

for the interval [r, s], which is proven analogously to Theorem 4.1. The point is that for this weakerstatement no remainder estimate in Weyl’s law is necessary, since the rate of convergence hn−κ isthe one of the leading term in Weyl’s law. Thus, in principle, this weaker result could have also beenobtained using heat kernel methods as in [8] or [4]. Nevertheless, for the stronger version of equivariantquantum ergodicity proven in Theorem 6.4, remainder estimates in Weyl’s law, and in particular theresults of [20], are necessary due to the lower rate of convergence hn−κ−β .

6.3. Equivariant quantum ergodicity II. In what follows, we shall use our previous results to provean equivariant quantum ergodicity theorem for Schrodinger operators. Again, remainder estimates inWeyl’s law, and in particular the results of [20], are necessary.

Theorem 6.6 (Equivariant quantum ergodicity for Schrodinger operators). Choose a Hilbertbasis uj(h)j∈N of L2(M) such that each uj(h) is an eigenfunction of P (h) with eigenvalue Ej(h).Let c ∈ R be a regular value of the reduced symbol function p, and suppose that the reduced flow ϕt is

ergodic on Σc := p−1(c). Let χ ∈ G, β ∈(0, 1

2(κ+2)

)be fixed, and set

Jχ(h) := j ∈ N : Ej(h) ∈ [c, c+ hβ ], uj(h) ∈ L2χ(M).

Then there is a h0 > 0 such that for each h ∈ (0, h0] we have a subset Λχ(h) ⊂ Jχ(h) satisfying

(6.14) limh→0

#Λχ(h)

#Jχ(h)= 1


such that for each G-equivariant A ∈ Ψ0h(M) and G-invariant a ∈ S0(M) with σ(A) = [a] the following

holds. For all ε > 0 there is a hε ∈ (0, h0] such that

(6.15)

∣∣∣∣〈Auj(h), uj(h)〉L2(M) −

Σc

a dµc

∣∣∣∣ < ε ∀ j ∈ Λχ(h), ∀h ∈ (0, hε].

Remark 6.7. Note that the limit in (6.15) does not contain any representation theoretic quantity,contrasting with the form of the leading term in the equivariant Weyl law of Theorem 4.1. Nevertheless,this is consistent with the expectation that there should not be too many quantum limits.

Proof. For simplicity, we shall again write uj(h) = uj and Ej(h) = Ej in the following. By Corollary4.3, we can choose a h0 := hχ > 0 such that Jχ(h) 6= ∅ for all h ∈ (0, h0), and in what follows, we shallsuppose that h ∈ (0, h0). For any G-invariant function s on M set

α(s) :=

Σc

s dµc.

Let τ ∈ C∞c (R, [0, 1]) be such that τ ≡ 1 in a neighbourhood of c. Without loss of generality, we

assume for the rest of the proof that h0 is small enough so that τ ≡ 1 on [c, c + hβ0 ]. Now, for anyoperator A as in the statement of the theorem, set

B := A− α(a) τ (P (h)) .

From Theorem 2.4 we know that the principal symbol of B is given by σ(B) = [b] with b := a−α(a) τp.Clearly, α(b) = 0, since τ p ≡ 1 on Σc. Let us now assume that the statement of the theorem holdsfor all operators A with α(a) = 0. Then, there is a sequence of subsets Λχ(h) of density 1 such thatfor all ε > 0 there is a hε ∈ (0, h0) such that

(6.16)∣∣∣〈Buj , uj〉L2(M)

∣∣∣ < ε ∀h ∈ (0, hε), ∀ j ∈ Λχ(h).

Due to the choice of the function τ we have τ (P (h)) (uj) = uj for all uj with Ej ∈ [c, c + hβ ].Consequently, (6.16) implies that for all ε > 0 there is hε ∈ (0, h0) such that∣∣∣〈Auj , uj〉L2(M) − α(a)

∣∣∣ < ε ∀h ∈ (0, hε), ∀ j ∈ Λχ(h),

and we obtain the statement of the theorem for general A. We are therefore left with the task ofproving (6.16) for arbitrary operators B with α(b) = 0, and shall proceed in a similar fashion to parts1 - 5 of the proof of [27, Theorem 15.5], pointing out only the main arguments. By Theorem 6.4 wehave for fixed B

hn−κ−β∑

j∈Jχ(h)

∣∣∣〈Buj , uj〉L2(M)

∣∣∣2 =: r(h)→ 0

as h→ 0. One then defines the B-dependent subsets Λχ(h) = Jχ(h)−j ∈ Jχ(h) : |〈Buj , uj〉L2(M)|2 ≥√r(h), and verifies that (6.16) is fulfilled for these particular Λχ(h) and B by taking into account

Theorem 4.2. Consider now a family Akk∈N of G-invariant pseudodifferential operators in Ψ0h(M)

whose principal symbols are given by G-invariant symbol functions. By our previous considerations,for each k there is a sequence of subsets Λχk (h) ⊂ Jχ(h) such that (6.14) and (6.15) hold for eachparticular Ak and Λχk (h). One then shows that for sufficiently small h there is a sequence of subsetsΛχ∞(h) ⊂ Jχ(h) of density 1 such that Λχk (h) ⊂ Λχ∞(h). Hence, the theorem is true for countablefamilies of operators. To obtain it in general, we shall construct below a sequence of G-invariantoperators

AGkk∈N whose symbols σ(AGk ) = [aGk ] are given by G-invariant symbol functions aGk , and

which is dense in5

PG := A ∈ Ψ−∞h (M) : σ(A) = [a], A and a are G-invariant

5Note that we do not need Zworski’s technical condition that the value of the integralfflΣca dµc must stay the same

when varying c in some interval, which slightly simplifies the proof.


in the sense that for given A ∈ Ψ−∞h (M) and ε > 0 there exists a k such that

‖A−Ak‖L2→L2 < ε,

Σc

˜(a− aGk ) dµc < ε

for sufficiently small h. To construct the sequenceAGkk∈N, note that by Theorem 2.1 quantizations

of G-invariant symbol functions are G-equivariant up to order h. Let therefore a and b be two suchsymbols. If A and B are the corresponding quantizations, one has

‖A−B‖L2→L2 ≤ ‖a− b‖L∞(M) + C√h,

Σc

(a− b) dµc ≤ C‖a− b‖L∞(T∗M).

Consequently, we only need to find a sequence in S−∞(M) of G-invariant symbol functions aGk k∈N,such that for each symbol function aG whose equivalence class in S−∞(M)/ (hS−∞(M)) is the principalsymbol of a pseudodifferential operator in PG, we have for each ε > 0 an index k ∈ N satisfying

(6.17)∥∥aG − aGk ∥∥L∞(M)

< ε.

To define such a sequence, note that C∞c (T ∗M) is dense in the Banach space C0(T ∗M) ⊃ S−∞(M) ofcontinuous functions vanishing at infinity. Since C∞c (T ∗M) is separable, we can choose a dense subsetakk∈N ⊂ C∞c (T ∗M) and take the averages

(6.18) aGk (η) :=

G

ak(g · η) dg, η ∈ T ∗M, k ∈ N.

Clearly, aGk ∈ S−∞(M). Let now aG ∈ S−∞(M) be G-invariant and ε > 0 be given. By choosingk ∈ N such that

∥∥aG − ak∥∥∞ ≤ ε we get for each µ ∈ T ∗M

∣∣aG(µ)− aGk (µ)∣∣ =

∣∣∣∣∣∣

G

[aG(µ)− ak(g · µ)] dg

∣∣∣∣∣∣ ≤

G

∣∣aG(g · µ)− ak(g · µ)∣∣ dg

≤ sup∣∣aG(µ)− ak(µ)

∣∣ : µ ∈ T ∗M

=∥∥aG − ak∥∥∞ ≤ ε.

Thus,∥∥aG − aGk ∥∥∞ ≤ ε, which proves the theorem for operators A in Ψ−∞h (M). Finally, if A ∈ Ψ0

h(M)

is a general G-invariant operator, we can multiply A with the smoothing operator %(P (h)), where% ∈ C∞c (R) equals 1 near c. This completes the proof of the theorem.

Using the following lemma, we can extend the result of Theorem 6.6 from single characters χ ∈ Gto arbitrary finite subsets of G.

Lemma 6.8. Let δ ∈ R be fixed. For each k ∈ N, let(akn)n∈N ,

(bkn)n∈N be sequences of real numbers

such that bkn > 0 for all n, k ∈ N and such that limn→∞ akn/bkn = δ for all k ∈ N. Then

(6.19) limn→∞

∑Nk=1 a

kn∑N

k=1 bkn

= δ ∀ N ∈ N.

Proof. We prove this by induction over N . For N = 1 the result is true by assumption. So supposethat (6.19) holds for some N , and set

aNn :=

N∑k=1

akn, bNn :=

N∑k=1

bkn.

Then we have bNn > 0 for all n ∈ N and limn→∞aNnbNn

= δ. We need to show limn→∞aNn +aN+1

n

bNn +bN+1n

=δ. Since

limn→∞aN+1n

bN+1n

= δ, it follows that

(6.20) limn→∞

1

2

(aNn

bNn+aN+1n

bN+1n

)= δ.


However,∣∣∣∣∣ aNn + aN+1n

bNn + bN+1n

− 1

2

(aNn

bNn+aN+1n

bN+1n

)∣∣∣∣∣ =

∣∣∣∣∣∣∣aNn b

Nn b

N+1n + aN+1

n bNn bN+1n − aNn

(bN+1n

)2 − aN+1n

(bNn

)2

2bNn bN+1n

(bNn + bN+1

n

)∣∣∣∣∣∣∣

=

∣∣∣∣∣∣aNnbNn· 1bN+1n

+aN+1n

bN+1n· 1bNn− aNn

bNn· 1bNn− aN+1

n

bN+1n· 1bN+1n

2(

1bN+1n

+ 1bNn

)∣∣∣∣∣∣ =

1

2

∣∣∣∣∣ aNnbNn − aN+1n

bN+1n

∣∣∣∣∣∣∣∣∣∣∣

1bN+1n− 1

bNn1

bN+1n

+ 1bNn

∣∣∣∣∣∣≤ 1

2

∣∣∣∣∣ aNnbNn − aN+1n

bN+1n

∣∣∣∣∣1

bN+1n

+ 1bNn

1bN+1n

+ 1bNn

=1

2

∣∣∣∣∣∣∣∣∣∣aNn

bNn︸︷︷︸n→∞−→ δ

− aN+1n

bN+1n︸︷︷︸n→∞−→ δ

∣∣∣∣∣∣∣∣∣∣n→∞−→ 0.

By relation (6.20), we are done.

Corollary 6.9. Let G ⊂ G be a finite subset, and define

JG(h) := j ∈ N; Ej(h) ∈ [c, c+ hβ ], uj(h) ∈ L2χ(M), χ ∈ G.

Then, there is a h0 > 0 such that for each h ∈ (0, h0] there exists a subset ΛG(h) ⊂ JG(h) such that

(6.21) limh→0

#ΛG(h)

#JG(h)= 1,

and for A ∈ Ψ0h(M) as in Theorem 6.6, (6.15) is true for the sets ΛG(h).

Proof. Write G = χ1, . . . , χN. Then Theorem 6.6 gives us for each k ∈ 1, . . . , N a number hk > 0and a subset Λχk(h) ⊂ Jχk(h) such that (6.15) is true for each set Λχk(h) and

(6.22) limh→0

#Λχk(h)

#Jχk(h)= 1 ∀ k ∈ 1, . . . , N.

Set h0 := minh1, . . . , hN and for h ∈ (0, h0) define ΛG(h) :=⋃Nk=1 Λχk(h). Then

(6.23) #ΛG(h) =

N∑k=0

#Λχk(h), #JG(h) =

N∑k=0

#Jχk(h).

Let (h(n))n∈N be a sequence of strictly positive real numbers which converges to zero and is bounded

from above by h0. Due to (6.22), we can apply Lemma 6.8 to akn := #Λχk(h(n)) and bkn := #Jχk(h(n)).That yields

(6.24) limn→∞

#ΛG(h(n))

#JG(h(n))= 1.

As R is a first countable space, it suffices to consider sequential convergence of h. Therefore, (6.24)implies (6.21). Because (6.15) is true for each set Λχk(h), it is also true for the finite union ΛG(h) ofthose sets, and the assertion follows.

7. Equivariant quantum limits for the Laplace-Beltrami operator

We shall now apply the semiclassical results from the previous section to study the distributionof eigenfunctions of the Laplace-Beltrami operator on a compact G-manifold M as the eigenvaluesbecome large. In what follows, let ∆ be the unique self-adjoint extension of the Laplace-Beltramioperator ∆ on M , and choose an orthonormal basis ujj∈N of L2(M) of eigenfunctions of −∆ with

corresponding eigenvalues Ejj∈N, repeated according to their multiplicity. Consider further the

Schrodinger operator P (h) = −h2∆ + V with V ≡ 0 and principal symbol defined by the symbol

function p = ‖·‖2T∗M . Then the self-adjoint extension of P (h) is given by P (h) = −h2∆. Each uj


is an eigenfunction of P (h) for each h > 0 with eigenvalue Ej(h) = h2Ej . Furthermore, under theidentification T ∗M ' TM given by the Riemannian metric, the Hamiltonian flow ϕt induced by pcorresponds to the geodesic flow of M . In that context, we call ϕt the reduced flow. Since V ≡ 0, the

dynamics of the reduced geodesic flow are equivalent on any two hypersurfaces Σc and Σc′ . In thefollowing, we therefore choose c = 1 without loss of generality. That means we will call the reducedgeodesic flow ergodic if it is ergodic on p−1(1).

We are now prepared to state and prove an equivariant version of the classical Shnirelman-Zelditch-Colin-de-Verdiere quantum ergodicity theorem[22, 26, 6].

Theorem 7.1 (Equivariant quantum limits for the Laplacian). Let G be a compact connectedLie group, and M a closed connected Riemannian manifold on which G acts effectively by isometries.

Assume that the reduced geodesic flow is ergodic. Choose a finite subset G ⊂ G and let uGj j∈N be an

orthonormal basis of L2G(M) =

⋃χ∈G L2

χ(M) of eigenfunctions of −∆. Then, there is a subsequence

uGjkk∈N of density 1 in uGj j∈N such that for all G-invariant functions s ∈ C∞(S∗M) one has6

(7.1)⟨Op(s)uGjk , u

Gjk

⟩L2(M)

−→ S∗Mreg

s d(S∗Mreg) as k →∞,

where s : S∗Mreg → C is the function corresponding to s : Σ1 → C under the diffeomorphism Σ1 'S∗Mreg up to a null set7, d(S∗Mreg) is the Liouville measure on the unit sphere bundle, and theexpression of density 1 means

limm→∞

#k, jk ≤ mm

= 1.

Proof. First, note that p(G · (x, ξ)) = ‖ξ‖2T∗xM for (x, ξ) ∈ Ω, so that each c > 0 is a regular value of p.

By Ehresmann’s theorem [10], there is an open tubular neighbourhood N of S∗M in T ∗M with radiusδ, such that N is diffeomorphic to (−δ, δ)× S∗M . Hence, we can use a cutoff function f ∈ C∞c (−δ, δ)with f ≡ 1 in a neighbourhood of 0 to extend s to a function s ∈ C∞c (T ∗M) with s|S∗M = s. Moreover,we can assume s to be G-invariant by replacing it with the function obtained by averaging over theG-orbits. Then Op(s) is a pseudodifferential operator in Ψ−∞(M) with principal symbol σ(Op(s)) =[s]. Thus, we can apply Corollary 6.9 to P (h) = −h2∆. As in Theorem 6.6, fix some β > 0, and letEGj j∈N be the eigenvalues associated to the eigenfunctions uGj j∈N. Set

JG(h) = j ∈ N; h2EGj ∈ [1, 1 + hβ ] =

j ∈ N; EGj ∈

[1

h2,

1

h2+

1

h2−β

].

By Corollary 6.9, there is a number h0 > 0 together with subsets ΛG(h) ⊂ JG(h), where h ∈ (0, h0],satisfying

(7.2) limh→0

#ΛG(h)

#JG(h)= 1,

and for each G-invariant a ∈ C∞(M) and arbitrary ε > 0 there is a hε ∈ (0, h0] such that

(7.3)

∣∣∣∣⟨Op(s)uGjk , uGjk

⟩L2(M)

−

Σ1

s dµ1

∣∣∣∣ < ε ∀ j ∈ ΛG(h), ∀h ∈ (0, hε].

Now, due to the discreteness of the set EGj j∈N in R, it is possible to find a strictly decreasing sequence

hii∈N ⊂ (0, h0] including h0 such that

(7.4) JG(hi) ∩ JG(hi′) = ∅ for i 6= i′, JG =

j ∈ N, EGj ≥

1

h20

,

6See Footnote 1.7see Lemma 2.21.


where we set

JG :=

∞⋃i=1

JG(hi), ΛG :=

∞⋃i=1

ΛG(hi).

Writing jkk∈N := ΛG we deduce from (7.2) that

(7.5) limi→∞

#k : 1

h2i≤ EGjk ≤

1h2i

+ 1

h2−βi

#j : 1

h2i≤ EGj ≤ 1

h2i

+ 1

h2−βi

= 1.

We now have the following

Lemma 7.2. Let (an)n∈N and (bn)n∈N be sequences of real numbers such that 0 < an ≤ bn for all n,and lim infn→∞ bn > 0, limn→∞

anbn

= 1. Then

limN→∞

∑Nn=1 an∑Nn=1 bn

= 1.

Proof. Let ε > 0 be arbitrary and choose Nε ∈ N such that anbn≥ 1 − ε for each n ≥ Nε. Then one

computes for N > Nε∑Nn=1 an∑Nn=1 bn

=

∑Nε−1n=1 an +

∑Nn=Nε

an∑Nn=1 bn

≥∑Nε−1n=1 an∑Nn=1 bn

+ (1− ε)∑Nn=Nε

bn∑Nε−1n=1 bn +

∑Nn=Nε

bn

=

∑Nε−1n=1 an∑Nn=1 bn︸︷︷︸

→0 as N→∞

+(1− ε) 1∑Nε−1n=1 bn∑Nn=Nε

bn︸︷︷︸→0 as N→∞

+1

−→ 1− ε

as N →∞. We therefore conclude

1 ≥ lim supN→∞


≥ lim infN→∞


≥ 1− ε,

and the lemma follows.

With the previous lemma we deduce from (7.5) the equality

limN→∞

∑Ni=1 #

k : 1

h2i≤ EGjk ≤

1h2i

+ 1

h2−βi

∑Ni=1 #

j : 1

h2i≤ EGj ≤ 1

h2i

+ 1

h2−βi

= 1.

In view of (7.4), this is equivalent to

limh→0

#k : 1

h20≤ EGjk ≤

1h2 + 1

h2−β

#j : 1

h20≤ EGj ≤ 1

h2 + 1h2−β

= 1,

and directly implies

limE→∞

#k : EGjk ≤ E

#j : EGj ≤ E

= 1, limj→∞

# k : jk ≤ jj

= 1,

where we took into account that EGj j∈N and EGjkk∈N are unbounded increasing sequences. Now,

s = s, and by Lemma 2.21 we have Σ1

s dµ1 =

S∗Mreg

s d(S∗Mreg).

From (7.3) we therefore conclude that the sequence uGjkk∈N associated to EGjkk∈N fulfills (7.1). Thiscompletes the proof of Theorem 7.1.


Projecting from S∗Mreg onto Mreg we now obtain

Corollary 7.3 (Equivariant equidistribution of eigenfunctions of the Laplacian). In thesituation of Theorem 7.1, let f ∈ C(M) be G-equivariant. Then

(7.6)

ˆ

M

f(x)|uGjk(x)|2 dM(x) −→

Mreg

f(x)dM(x)

vol(G · x)as k →∞.

Proof. First, we show that the assertion holds for smooth functions. Let f ∈ C∞(M) be G-invariant,and define A ∈ Ψ0

h(M) to be pointwise multiplication with f . Then σ(A) = [f ]. Furthermore, Lemma2.20 together with (2.27) imply

Σ1

f(G · (x, ξ)) dµ1(G · (x, ξ)) =

Mreg

f(G · x) dMreg(G · x) =

Mreg

f(x)dM(x)

vol(G · x),

and the assertion follows for smooth functions directly from Theorem 7.1 and Lemma 2.21. It remainsto show (7.6) for G-invariant functions which are only continuous. Thus, let r ∈ C(M) be G-invariant,and choose some ε > 0. By the density of C∞(M), we can find a function s ∈ C∞(M) such that‖r − s‖∞ < ε. Setting sG(x) :=

´Gs(g · x) dg defines a smooth G-invariant function for which (7.6)

holds. Since r is G-invariant, one easily sees that∥∥r − sG∥∥∞ < ε is fulfilled. Furthermore, for all j ∈ N

we have ∣∣∣ˆM

r(x)|uGj (x)|2 dM(x)−ˆ

M

sG(x)|uGj (x)|2 dM(x)∣∣∣ ≤ ˆ

M

|r(x)− sG(x)||uGj (x)|2 dM(x)

≤∥∥r − sG∥∥∞ ∥∥uGj ∥∥L2

< ε,

as well as ∣∣∣∣∣∣∣

Mreg

r(G · x) dMreg(G · x)−

Mreg

sG(G · x) dMreg(G · x)

∣∣∣∣∣∣∣ ≤∥∥r − sG∥∥∞ < ε.

Since ε was arbitrary, (7.6) must hold for r, completing the proof of the corollary.

Corollary 7.3 can be interpreted as a statement about weak convergence of measures8 on the topo-logical Hausdorff space M/G. To do so, define for an orbit-wise integrable function a : M → C theorbital integral

〈a〉G (x) :=

ˆ

G

a(g−1x) dg.

Clearly, 〈a〉G is a G-invariant function. We then have the following

Corollary 7.4. In the situation of Corollary 7.3,

(7.7)⟨|uGjk |

2⟩Gd(M/G)

k→∞−→(

vol d(M/G)vol

M/G)−1 d(M/G)

vol, as k →∞,

with respect to the weak convergence of measures on M/G.

8Here a sequence of measures µj on a metric space X is said to converge weakly to a measure µ, if for all bounded

and continuous functions f on X one hasˆXf dµj −→

ˆXf dµ as j →∞.


Proof. Let f ∈ C(M/G) and let π : M → M/G be the canonical projection. Then f lifts to theG-invariant function f := f π ∈ C(M) which by Lemma 2.12 fulfillsˆ

M

f(x)|uGjk(x)|2 dM(x) =

ˆ

Mreg

f(x)|uGjk(x)|2 dM(x) =

ˆ

Mreg

ˆ

G·x

f(x′)|uGjk(x′)|2 dµG·x(x′) dMreg(G · x)

=

ˆ

Mreg

f(G · x)

ˆ

G·x

|uGjk(x′)|2 dµG·x(x′) dMreg(G · x)

=

ˆ

Mreg

f(G · x)vol(G · x)

ˆ

G

|uGjk(g−1x)|2 dg dMreg(G · x) by (2.26)

=

ˆ

Mreg

f(G · x)⟨|uGjk |

2⟩Gd(M/G)(G · x), by Lemma 2.16.

Moreover, we havefflMreg

f(x) dM(x)vol(G·x) =

fflMreg

f(G · x) dMreg(G · x) =fflM/G

f(O)d(M/G)(O)vol O by Lemma

2.16 and Corollary 2.13. The claim now follows from Corollary 7.3.

We can understand the orbital integral in (7.7) still better using elementary representation theory.

Lemma 7.5. Let V ⊂ L2(M) be an irreducible G-module of class χ ∈ G. Let further v1, . . . , vdχdenote an L2-orthonormal basis of V , and a ∈ V ∩ C∞(M) have L2-norm equal to 1. Then, for anyx ∈M ,

(7.8)⟨|a|2⟩G

(x) =

ˆ

G

|a(g−1x)|2 dg = d−1χ

dχ∑k=1

|vk(x)|2.

In particular, the function

ΘV : M → R, x 7→ d−1χ

dχ∑k=1

|vk(x)|2,

is a G-invariant element of C∞(M) that is independent of the choice of orthonormal basis, and theleft hand side of (7.8) is independent of the choice of a.

Proof. Since the left hand side of (7.8) is clearly G-invariant, smooth, and independent of the choice

of orthonormal basis, it suffices to prove (7.8). Now, a =∑dχj=1 ajvj with aj ∈ C,

∑dχj=1 |aj |2 = 1, and

(Lga)(x) = a(g−1x) =

dχ∑j=1

ajvj(g−1x) =

dχ∑j,k=1

ajcjk(g)vk(x), g ∈ G, x ∈M,

where cjk1≤j,k≤dχ denote the matrix coefficients of the G-representation on V . This yields

ˆ

G

|a(g−1x)|2 dg =

ˆ

G

a(g−1x)a(g−1x) dg =

ˆ

G

dχ∑j,k=1

ajcjk(g)vk(x)

dχ∑l,m=1

alclm(g)vm(x)

dg,

and we obtain (7.8) by taking into account the Schur orthogonality relations [16, Corollary 1.10]ˆ

G

cjk(g)clm(g) dg = d−1χ δjlδkm.

We are now in the position to restate the equivariant equidistribution theorem in representation-theoretic terms.


Theorem 7.6 (Representation-theoretic equidistribution theorem). Let G be a compact con-nected Lie group, and M a closed connected Riemannian manifold on which G acts by isometries.

Assume that the reduced geodesic flow is ergodic and choose χ ∈ G. By the Peter-Weyl theorem andthe spectral theorem, choose an orthogonal decomposition L2

χ(M) =⊕

i∈N Vχi into irreducible unitary

G-modules of class χ such that each V χi is contained in some eigenspace of the Laplace-Beltrami op-erator. As in Lemma 7.5, assign to each V χi the G-invariant function Θχ

i := ΘV χi: M → [0,∞), and

regard it as a function on M/G. Then there is a subsequence V χikk∈N of density 1 in V χi i∈N suchthat

(7.9) Θχikd(M/G)

k→∞−→(

vol d(M/G)vol

M/G)−1 d(M/G)

vol, as k →∞,

with respect to the weak convergence of measures on M/G. In other words, for all f ∈ C(M/G) onehas ˆ

M/G

f Θχikd(M/G)

k→∞−→

M/G

f(O)d(M/G)(O)

vol O.

Proof. Let uχj j∈N be an orthonormal basis of L2χ(M) of eigenfunctions of −∆ chosen such that

V χi = span uχj : j ∈ Jχi , where Jχi := i dχ, . . . , (i+ 1)dχ − 1. By taking G = χ in Corollary 7.4,

there is a subsequence uχjkk∈N of density 1 in uχj j∈N such that we have the weak convergence

(7.10)⟨|uχjk |

2⟩Gd(M/G) −→

(vol d(M/G)

volM/G

)−1 d(M/G)

vol, as k →∞,

and by Lemma 7.5,

(7.11)⟨|uχjk |

2⟩G

= Θχi if uχjk ∈ V

χi , k ∈ N.

Let now ikk∈N be the sequence of those indices i occurring in (7.11), without repetitions. Then

dχ#k, ik ≤ m ≥ #k, jk ≤ dχm.

Passing to the limit m→∞ we obtain

1 ≥ lim supm→∞

#k, ik ≤ mm

≥ lim infm→∞

#k, ik ≤ mm

≥ limm→∞

#k, jk ≤ dχmdχm

= 1,

where the final equality holds because uχjkk∈N has density 1 in uχj j∈N. This concludes the proof ofthe theorem.

Note that Theorem 7.6 is a statement about limits of representations, or multiplicities, in thesense that it assigns to each irreducible χ-isotypic G-module in L2(M) a measure on M/G, and thenconsiders the limiting measure. Remarkably, no more explicit mention is made to eigenfunctions ofthe Laplacian. Of course, one can also formulate a version of Theorem 7.6 which explicitly involvesthe eigenfunctions of the Laplace-Beltrami operator in the style of Shnirelman’s original formulationby taking for Θχ

i the function

Θχi : M → R, x 7→ d−1

χ

∑j∈Jχi

|uχj (x)|2,

where the sum runs over the eigenfunctions of −∆ spanning the G-module V χi .

8. An example

As illustration of the obtained results, we shall contemplate the case of the two-dimensional sphereM = S2 ' SO(3)/SO(2), acted upon by the group G = SO(2) of rotations around the symmetry axisof S2 ⊂ R3 through the points N = (0, 0, 1) and S = (0, 0,−1).


8.1. Equivariant Weyl law. Endow S2 with the standard metric, and identify T ∗M and TM viathis metric. Consider the G-invariant Hamiltonian p(x, ξ) = ‖ξ‖2T∗xS2 , and denote the corresponding

Laplace-Beltrami operator, as well as its self-adjoint extension in L2(S2) by ∆. The eigenvalues of−∆ are given by the numbers l(l + 1), l = 0, 1, 2, 3 . . . , and the corresponding eigenspaces El are ofdimension 2l + 1, and spanned by the spherical harmonics

(8.1) Yl,m(φ, θ) =

√2l + 1

4π

(l −m)!

(l +m)!Pl,m(cos θ)eimφ, 0 ≤ φ < 2π, 0 ≤ θ < π,

where m ∈ Z, |m| ≤ l, and Pl,m are the associated Legendre polynomials

(8.2) Pl,m(x) =(−1)m

2ll!

(1− x2

)m2dl+m

dxl+m(x2 − 1

)l.

The irreducible representations of SO(2) ' S1 are all 1-dimensional, and given by the charactersχk(eiφ) = eikφ, k ∈ Z. Thus, each subspace C · Ylm corresponds to an irreducible representation ofSO(2), and each irreducible representation χk with |k| ≤ l occurs in the eigenspace El with multi-plicity 1. The situation is illustrated by the following table. The columns of the table represent theeigenspaces, whereas the k-th row represents the isotypic component corresponding to χk.

m\l 0 1 2 3 · · ·... . .

.

3 Y3,3 · · ·2 Y2,2 Y3,2 · · ·1 Y1,1 Y2,1 Y3,1 · · ·0 Y0,0 Y1,0 Y2,0 Y3,0 · · ·−1 Y1,−1 Y2,−1 Y3,−1 · · ·−2 Y2,−2 Y3,−2 · · ·−3 Y3,−3 · · ·...

. . .

We therefore obtain for the equivariant counting function Nχk(λ) for λ 1 the estimate

Nχk(λ) = dimχk∑

l(l+1)≤λ

mult(χk, El) =∑

l(l+1)≤λ,|k|≤l

1 ≈∑

|k|≤l≤√λ

1 ≈√λ− |k|,

which agrees with the equivariant Weyl law proved in [20, Theorem 9.5]. From this one recovers theclassical Weyl law

N(λ) =∑

l(l+1)≤λ

dimEl =∑k∈Z

Nχk(λ) ≈∑|k|≤√λ

(√λ− |k|) ≈ (2

√λ+ 1)

√λ− 2

√λ(√λ+ 1)

2= λ.

8.2. Ergodicity of the reduced geodesic flow. We examine next the underlying dynamical system,

in particular the geodesic flow ϕt on T ∗M and the reduced geodesic flow ϕt on Ωreg. Let c > 0 and

put Σc := p−1(c) and Σc := p−1(c), where p ∈ C∞(Ωreg) is the function induced by p|Ωreg. Both

ϕt and ϕt leave Σc and Σc invariant, respectively, and the pointwise norms of ∂p as well as ∂p are

bounded away from zero on Σc and Σc. Now, it is a well-known fact that ϕt is not ergodic on Σc, sincedue to the presence of a continuum of closed geodesics on S2 there exist flow-invariant proper subsets

of Σc of measure greater than zero. To examine whether the reduced flow is ergodic on Σc, note thatwith the identification T ∗M ' TM one has

Ω = J−1(0) '⊔x∈S2

Tx(G · x)⊥,


so that

Ωreg '( ⋃x∈S2

reg

x × Tx(G · x)⊥)∪(N ×

(TN (S2)\0

) )∪(S ×

(TS(S2)\0

) ),

Ωreg '((−1, 1)× R

)∪(1 × (0,∞)

)∪(−1 × (0,∞)

)' R2\(0, 1), (0,−1),

where S2reg = S2\ N,S and S2

reg/G ' (−1, 1). The diffeomorphism Ωreg ' R2\(0, 1), (0,−1) is

illustrated in Figures 8.1 and 8.2. Under this diffeomorphism, the hypersurface Σc corresponds to an

ellipse with radii determined by c, as illustrated in Figure 8.2. Let now G · (x, ξ) ∈ Σc. The reducedgeodesic flow ϕt(G ·(x, ξ)) ≡ G ·ϕt(x, ξ) through G ·(x, ξ) corresponds to a periodic curve on the ellipse

Σc. Since ξ ∈ Tx(G · x)⊥, the geodesic flow ϕt transports (x, ξ) around circles in T ∗S2 that project to

meridians through N and S. Consequently, the only subsets of Σc which are invariant under ϕt are

the whole ellipse and the empty set. Therefore, we conclude that the reduced flow ϕt on Σc is ergodicfor arbitrary c > 0.

Figure 8.1. The space T ∗NS2\0 (red) and

three cotangent spaces (blue) with arrowsthat represent elements of Ωreg. The threecircles in each plane (brown, teal, green) cor-respond to the intersection of the plane withΣc for three different values of c.

Figure 8.2. Under the projection Ωreg →Ωreg, T ∗NS

2\0 and T ∗SS2\0 collapse to

open half-lines (red) and for every x ∈S2\N,S, T ∗xS2 ∩ Ωreg collapses to a line(blue). The ellipses (brown, teal, green) de-

pict Σc for three different values of c.

8.3. Equivariant equidistribution of spherical harmonics. Having verified that the reducedgeodesic flow of S2 with respect to our SO(2)-action is ergodic, we can apply Theorem 7.6. Since eachirreducible representation of S1 is 1-dimensional, Theorem 7.6 yields in the present case that

(1) for each m and l the function S2 → R, x 7→ |Yl,m(x)|2, is SO(2)-invariant;


(2) for each m ∈ Z ' SO(2) we get a subsequence Ylk,mk∈N of density 1 in Yl,ml∈N such thatfor all f ∈ C(S2/SO(2))ˆ

S2/SO(2)

f(SO(2) · x)|Ylk,m(x)|2 d(S2/SO(2))(SO(2) · x)

k→∞−→ V −1

ˆ

S2/SO(2)

f(SO(2) · x)d(S2/SO(2))(SO(2) · x)

vol (SO(2) · x),

(8.3)

where V =´S2/SO(2)

(vol (SO(2) · x))−1

d(S2/SO(2))(SO(2) · x).

Of course, we want to write (8.3) much more explicitly. That can be done easily using sphericalcoordinates. Up to a set of measure zero with respect to dS2, we have S2 ' (φ, θ) : 0 < φ < 2π, 0 <θ < π ⊂ R2 and dS2(φ, θ) ≡ sin θ dφ dθ. An orbit SO(2) · (φ, θ) is of the form (φ′, θ) : 0 < φ′ < 2π,up to a set of measure zero with respect to the induced orbit measure dµSO(2)·(φ,θ) ≡ sin θ dφ, and weobtain

vol (SO(2) · (φ, θ)) =

2πˆ

0

sin θ dφ = 2π sin θ.

Furthermore, S2/SO(2) is homeomorphic to the closed interval [0, π] ⊂ R with standard Euclideanmetric, and the pushforward measure on S2/SO(2) is given by d(S2/SO(2))(θ) ≡ 2π sin θ dθ, where weidentified SO(2)·(φ, θ) and θ. Summing up, (8.3) says in more concrete terms that for each f ∈ C([0, π])

(8.4) 2π

π

0

f(θ) sin(θ) ˜|Ylk,m|(θ)2 dθk→∞−→ 1

π

π

0

f(θ) dθ,

where ˜|Ylk,m|(θ) := |Ylk,m|(φ0, θ) for some arbitrary φ0 ∈ (0, 2π).

8.4. Agreement with known results. The first result from Theorem 7.6, namely that the functionS2 → R, x 7→ |Yl,m(x)|2, is SO(2)-invariant, is immediately clear since

|Yl,m(φ, θ)| = |const. · Pl,m(cos θ)eimφ| = const. · |Pl,m(cos θ)|,

so that |Yl,m(φ, θ)| is φ-independent. Let us now verify that the main result (8.4) is in agreementwith the classical theory of asymptotic properties of spherical harmonics. In fact, we will not only seethat one can deduce (8.4) with classical methods, but also that one does not even need to pass to asubsequence of density 1. Indeed, one has the following

Proposition 8.1. For fixed m, the full sequence (Yl,m)l∈N fulfills (8.4) as l→∞.

Proof. Let us begin by recalling the following classical result about the asymptotic behavior of Legendrepolynomials [12, page 303]. For fixed m ∈ Z and each small ε > 0 one has

(8.5)1

lmPl,m(cos θ) =

(2

lπ sin θ

)1/2

cos

((l +

1

2

)θ − π

4+mπ

2

)+O

(l−3/2

)as l→∞ uniformly in θ ∈ (ε, π − ε). From (8.1) and (8.5) we therefore obtain

|Yl,m|(θ)2 =

∣∣∣∣∣√

2l + 1

4π

(l −m)!

(l +m)!Pl,m(cos θ)

∣∣∣∣∣2

=2l + 1

4π

(l −m)!

(l +m)!l2m

∣∣∣∣ 1

lmPl,m(cos θ)

∣∣∣∣2

=2l + 1

4π

(l −m)!

(l +m)!l2m

∣∣∣∣∣(

2

lπ sin θ

)1/2

cos

((l +

1

2

)θ − π

4+mπ

2

)+O

(l−3/2

)∣∣∣∣∣2

=2l + 1

4π

(l −m)!

(l +m)!l2m

(2

lπ sin θcos2

((l +

1

2

)θ − π

4+mπ

2

)+O

(l−2))

.


The asymptotic relation

(8.6) (l −m)!/(l +m)! ∼ l−2m as l→∞

implies that (l−m)!(l+m)! l

2m is bounded in l, so we can use the simple relation 2l+1l = 2 +O(l−1) to obtain

(8.7) |Yl,m|(θ)2 =(l −m)!

(l +m)!l2m

1

π2 sin θcos2

((l +

1

2

)θ − π

4+mπ

2

)+O

(l−1),

uniformly for θ ∈ (ε, π − ε) and each small ε > 0. Now let f ∈ C([0, π],R) and choose ε > 0. Due tothe uniform estimate (8.7) and boundedness of the integration domain we get

2π

π−εˆ

ε

f(θ)|Yl,m|(θ)2 sin θ dθ

= 2π

π−εˆ

ε

f(θ)(l −m)!

(l +m)!l2m

1

π2 sin(θ)cos2

((l +

1

2

)θ − π

4+mπ

2

)sin(θ) dθ +O

(l−1)

=2

π

(l −m)!

(l +m)!l2m

π−εˆ

ε

f(θ) cos2

((l +

1

2

)θ − π

4+mπ

2

)dθ +O

(l−1).

(8.8)

The oscillatory integral in (8.8) has the limit

(8.9) liml→∞

π−εˆ

ε

f(θ) cos2

((l +

1

2

)θ − π

4+mπ

2

)dθ = lim

l→∞

π−εˆ

ε

f(θ) cos2(lθ) dθ =1

2

π−εˆ

ε

f(θ) dθ,

where the final equality is true because liml→∞´ π−εε

f(θ) cos2(lθ) dθ = liml→∞´ π−εε

f(θ) sin2(lθ) dθ

and sin2 + cos2 = 1. Using (8.9) and (8.6) we conclude from (8.8) for each small ε > 0 that

(8.10) liml→∞

2π

π−εˆ

ε

f(θ)|Yl,m|(θ)2 sin(θ) dθ =1

π

π−εˆ

ε

f(θ) dθ.

On the other hand, (8.6) and (8.7) imply the pointwise asymptotic property

lim supl→∞

|Yl,m|(θ)2 = lim supl→∞

(l −m)!

(l +m)!l2m

1

π2 sin θcos2

((l +

1

2

)θ − π

4+mπ

2

)=

1

π2 sin θliml→∞

((l −m)!

(l +m)!l2m)

lim supl→∞

cos2

((l +

1

2

)θ − π

4+mπ

2

)≤ 1

π2 sin θ∀ θ ∈ (0, π).

Now, Fatou’s Lemma implies that for each small ε > 0 one has the estimate

lim supl→∞

2π

εˆ

0

f(θ)|Yl,m|(θ)2 sin θ dθ ≤ 2π

εˆ

0

lim supl→∞

|f(θ)||Yl,m|(θ)2 sin θ dθ

≤ 2π

εˆ

0

|f(θ)| 1

π2 sin θsin θ dθ ≤ 2ε

π‖f‖∞ ,


and analogously for the integral over (π − ε, π). In a similar way one computes

lim infl→∞

2π

εˆ

0

f(θ)|Yl,m|(θ)2 sin θ dθ ≥ 2π

εˆ

0

lim infl→∞


≥ −2π

εˆ

0

|f(θ)| lim supl→∞

|Yl,m|(θ)2 sin θ dθ ≥ −2π

εˆ

0

|f(θ)| 1

π2 sin θsin θ dθ

≥ −2ε

π‖f‖∞

and again analogously for the integral over (π − ε, π). Together with (8.10) this allows us to concludethat

lim supl→∞

2π

π

0

f(θ)|Yl,m|(θ)2 sin θ dθ ≤ lim supl→∞

2π

εˆ

0


+ liml→∞

2π

π−εˆ

ε

f(θ)|Yl,m|(θ)2 sin θ dθ + lim supl→∞

2π

π

π−ε


≤ 2ε

π‖f‖∞ +

1

π

π−εˆ

ε

f(θ) dθ +2ε

π‖f‖∞ =

1

π

π−εˆ

ε

f(θ) dθ +4ε

π‖f‖∞

(8.11)

and similarly

(8.12) lim infl→∞

2π

π

0

f(θ)|Yl,m|(θ)2 sin θ dθ ≥ 1

π

π−εˆ

ε

f(θ) dθ − 4ε

π‖f‖∞ .

The left hand sides of (8.11) and (8.12) are independent of ε, so that passing to the limit ε → 0 weobtain

(8.13) liml→∞

2π

π

0

f(θ)|Yl,m|(θ)2 sin(θ) dθ =1

π

π

0

f(θ) dθ.

References

[1] L. B. Bergery and J.-P. Bourguignon, Laplacians and Riemannian submersions with totally geodesic fibres, IllinoisJournal of Mathematics 26 (1982), no. 2, 181–200.

[2] D. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, vol. 203, Birkhauser,Basel, 2010.

[3] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972, Pure and AppliedMathematics, Vol. 46.

[4] J. Bruning and E. Heintze, Representations of compact Lie groups and elliptic operators, Inv. math. 50 (1979),169–203.

[5] R. Cassanas and P. Ramacher, Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II.The compact group case, J. Funct. Anal. 256 (2009), 91–128.

[6] Y. Colin de Verdiere, Ergodicite et fonctions propres du Laplacien, Communications in Mathematical Physics 102(1985), no. 3, 497–502.

[7] M. Dimassi and J. Sjostrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society LectureNote Series, vol. 268, Cambridge University Press, 1999.

[8] H. Donnelly, G-spaces, the asymptotic splitting of L2(M) into irreducibles, Math. Ann. 237 (1978), 23–40.

[9] S. Dyatlov and C. Guillarmou, Limites microlocales des ondes planes et les fonctions d’Eisenstein., Ann. Sci. Ec.Norm. Super. (4) 47 (2014), no. 2, 371–448.

[10] C. Ehresmann, Les connexions infinitesimales dans un espace fibre differentiable, Colloque de topologie − espaces

fibres, George Thone, Bruxelles, 1950, pp. 29–55.


[11] F. Faure, N. Nonnenmacher, and S. de Bievre, Scarred eigenstates for quantum cat maps of minimal periods, Comm.

Math. Phys. 239 (2003), 449–492.[12] E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Cambridge University Press, 1931.

[13] L. Hormander, The analysis of linear partial differential operators, vol. I, Springer–Verlag, Berlin, Heidelberg, New

York, 1983.[14] , The analysis of linear partial differential operators, vol. III, Springer–Verlag, Berlin, Heidelberg, New York,

1985.

[15] A. Katok, Smooth ergodic theory and its applications, Proceedings of the AMS Summer Research Institute onSmooth Ergodic Theory and Its Applications, University of Washington, Seattle, 1999.

[16] A. Knapp, Representation theory of semisimple groups. An overview based on examples, Princeton MathematicsSeries, vol. 36, Princeton University Press, 1986.

[17] E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. 163 (2006), 165–

219.[18] J. P. Ortega and T. S. Ratiu, Momentum maps and Hamiltonian reduction, Progress in Mathematics, vol. 222,

Birkhauser Boston Inc., Boston, MA, 2004.

[19] T. Pacini, Mean curvature flow, orbits, moment maps, Transactions of the AMS 355 (2003), no. 8, 3343–3357.[20] P. Ramacher, Singular equivariant asymptotics and Weyl’s law. On the distribution of eigenvalues of an invariant

elliptic operator, to be published in Crelle’s Journal, DOI 10.1515/crelle-2014-0008, 2014.

[21] Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys.161 (1994), 195–213.

[22] A. Shnirelman, Ergodic properties of eigenfunctions, Uspekhi Mat. Nauka 29 (1974), 181–182.

[23] M. A. Shubin, Pseudodifferential operators and spectral theory, 2nd edition, Springer–Verlag, Berlin, Heidelberg,New York, 2001.

[24] B. Watson, Manifold maps commuting with the Laplacian, Journal of Differential Geometry 8 (1973), no. 1, 85–94.[25] T. Weich, Equivariant spectral asymptotics for h-pseudodifferential operators, dissertation at Marburg University,

2014.

[26] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Mathematical Journal 55(1987), no. 4, 919–941.

[27] M. Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society,

Providence, 2012.

Philipps-Universitat Marburg, Fachbereich Mathematik und Informatik, Hans-Meerwein-Str., 35032 Mar-burg, Germany

E-mail address, Benjamin Kuster: [email protected]

E-mail address, Pablo Ramacher: [email protected]

mailto:[email protected]

mailto:[email protected]

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