Equivariant local scaling asymptotics for smoothed Töplitz spectral projectors
aa r X i v : . [ m a t h . S G ] A p r Equivariant local scaling asymptoticsfor smoothed T¨oplitz spectral pro jectors
Roberto Paoletti ∗ Abstract
Let X be the unit circle bundle of a positive line bundle on a Hodgemanifold. We study the local scaling asymptotics of the smoothedspectral projectors associated to a first order elliptic T¨oplitz operator T on X , possibly in the presence of Hamiltonian symmetries. Theresulting expansion is then used to give a local derivation of an equi-variant Weyl law. It is not required that T be invariant under thestructure circle action, that is, T needn’t be a Berezin-T¨oplitz opera-tor. Let (
M, J, ω ) be a compact complex d -dimensional Hodge manifold, and let( A, h ) be a positive holomorphic line bundle on M , such that the uniquecompatible covariant derivative on A has curvature Θ = − i ω . Let X ⊆ A ∨ be the unit circle bundle in the dual line bundle, with the induced connection1-form α . Thus ( X, α ) is a contact manifold and d α = 2 π ∗ ( ω ), where π : X → M is the bundle projection. We shall consider on M and X the volumeforms d V M =: 1 d ! ω ∧ d d V X =: 12 π α ∧ π ∗ (d V M ) , and the associated densities | d V M | , | d V X | .With these choices, one may consider the Hilbert spaces L (cid:0) M, A ⊗ k (cid:1) ofsquare summable sections of powers of A , and the Hilbert space L ( X ) ofsquare summable complex functions on X . The structure circle action on X determines an equivariant splitting into isotypes L ( X ) ∼ = M k L ( X ) k . ∗ Address:
Dipartimento di Matematica e Applicazioni, Universit`a degli Studi di Mi-lano Bicocca, Via R. Cozzi 53, 20125 Milano, Italy; e-mail : [email protected]
1s is well-known, there is for every k ∈ N a natural unitary isomorphism L ( X ) k ∼ = L (cid:0) M, A ⊗ k (cid:1) , under which the Hilbert direct sum of the spaces of holomorphic sections H (cid:0) M, A ⊗ k (cid:1) ⊆ L (cid:0) M, A ⊗ k (cid:1) , ( k = 0 , , , . . . ) , corresponds to the Hardy space H ( X ) ⊆ L ( X ) (see [BtG], [BSZ] and [SZ]for a detailed discussion). The orthogonal projector Π : L ( X ) → H ( X ) isknown in the literature as the Szeg¨o projector, and its distributional kernelas the Szeg¨o kernel, of X .A T¨oplitz operator of degree k (in the sense of [BtG]) is a composition T = Π ◦ Q ◦ Π, where Q is a pseudo-differential operator of degree k on X ,regarded as an endomorphism of H ( X ). By the theory in [BtG], we mayassume without loss that [Π , Q ] = 0, so that T is the restriction of Q to H ( X ).A T¨oplitz operator T has a well-defined principal symbol σ T , given bythe restriction of the principal symbol of Q to the closed symplectic conesprayed by α : Σ =: (cid:8) ( x, r α x ) : x ∈ X, r > (cid:9) ⊆ T X \ { } ;thus σ T : Σ → C is independent of the particular choice of Q [BtG]. Onecalls T elliptic if σ T is the restriction of an elliptic symbol; if T is elliptic,one may assume in addition to the above that Q is also elliptic.We shall say that T is self-adjoint to mean that it is formally self-adjointwith respect to the L -product on X associated to d V X . In this case, Q itselfmay be assumed to be self-adjoint, and σ T is real-valued.For instance, given h ∈ C ∞ ( X ), one obtains a zeroth-order self-adjointT¨oplitz operator T ( h ) by taking Q = M h , the multiplication operator by h .Then ς T ( h ) = h , so that T ( h ) is elliptic precisely when h is nowhere vanishing.Clearly, T ( h ) is self-adjoint if and only if h is real-valued. Definition 1.1.
Let T be a T¨oplitz operator as above. The reduced symbol of T is the function ς T : X → C defined by ς T ( x ) =: σ T ( x, α x ) ( x ∈ X ) . So let T be a first order self-adjoint first order T¨oplitz operator on X with positive reduced symbol ς T >
0. Let λ ≤ λ ≤ · · · be the eigenvaluesof T , repeated according to multiplicity, so that λ j ↑ + ∞ . Let ( e j ) be any2omplete orthonormal system of H ( X ), with e j eigensection associated to λ j .Thus, for any eigenvalue η ∈ R the C ∞ function on X × X given by P η ( x, y ) =: X j : λ j = η e j ( x ) · e j ( y ) ( x, y ∈ X )is the Schwartz kernel of the L -orthogonal projector onto the eigenspace H ( X, T ) η ⊆ H ( X ) of T associated to η . If I ⊆ R is any bounded interval,we may view it as a spectral band and similarly consider the correspondingspectral projector P I =: P η ∈ I P η ; this is a smoothing operator, with kernel P I ( x, y ) =: X j : λ j ∈ I e j ( x ) · e j ( y ) ( x, y ∈ X ) . If H ( X, T ) I is the range of P I , its dimension is the number of λ j ∈ I . Thusif λ ∈ R the trace P λ + I is the number of eigenvalues of T within the spectralband I λ = λ + I , drifting to infinity as λ → + ∞ ; locally on X × X , P λ + I ( x, y )encapsulates the asymptotic concentration behavior of the eigensections of T pertaining to the band I λ traveling to infinity.In practice, rather than dealing directly with the P I ’s, after [H] and [DG]one considers the approximations obtained by replacing the characteristicfunction of I by a C ∞ function γ : R → R of rapid decrease. Thus onedefines P γ ( x, y ) =: X j γ ( λ j ) e j ( x ) · e j ( y ) ( x, y ∈ X ) . Then P γ is again a smoothing operator, given by a smoothed average of the P η ’s (see the discussion in [GrSj]). The analogue of P λ + I is then given by P γ λ , where γ λ = γ ( · − λ ).A convenient description of these smoothly averaged spectral projectorsis as ‘smoothed T¨oplitz wave operators’, as follows. For τ ∈ R , let U T ( τ ) =: e iτT = Π ◦ e iτQ ◦ Π; thus U T ( τ ) is the unitary endomorphism of H ( X ) givenby the restriction of e iτQ , and has distributional kernel U T ( τ )( x, y ) =: X j e iτλ j e j ( x ) · e j ( y ) ( x, y ∈ X ) . (1)For any χ ∈ S ( R ) (function of rapid decrease) the averaged operator S χ =: Z + ∞−∞ χ ( τ ) U T ( τ ) d τ (2)is a smoothing operator, with Schwartz kernel S χ ( x, y ) =: X j b χ ( − λ j ) e j ( x ) · e j ( y ) ( x, y ∈ X ) . (3)3hus in the previous notation S χ = P γ , with γ = b χ ( −· ) (here b χ is the Fouriertransform of χ ). If χ is replaced by χ · e − iλ ( · ) , we get S χ · e − iλ ( · ) ( x, y ) = X j b χ ( λ − λ j ) e j ( x ) · e j ( y ) ( x, y ∈ X ) . (4)That is, S χ · e − iλ ( · ) = P γ λ . In particular,trace (cid:0) S χ · e − iλ ( · ) (cid:1) = X j b χ ( λ − λ j );an asymptotic estimate on the latter trace leads, by a Tauberian argument,to a Weyl law for T . In [P2] a pointwise asymptotic estimate on the diagonalrestriction S χ · e − iλ ( · ) ( x, x ) was given for λ → + ∞ , leading (in this specialsetting) to a local proof of the Weyl law for T¨oplitz operators in [BtG]. Inthe present paper, we shall look at the near diagonal scaling asymptoticsof S χ · e − iλ ( · ) . We shall also consider similar asymptotics for the equivariantversions of these operators, arising in the presence of quantizable Hamiltonianactions on ( M, J, ω ).Suppose that G is a connected compact Lie group of real dimension e ,and that µ M : G × M → M is a holomorphic and Hamiltonian action, withmoment map Φ : M → g ∨ , where g is the Lie algebra of G . Also, assume that µ can be linearized to a metric preserving holomorphic action of G on ( A, h ),so that by restriction we obtain an action of G on X , µ X : G × X → X . Thiscan always be done infinitesimally: if ξ ∈ g , let ξ M ∈ X ( M ) be the vectorfield on M induced by ξ under µ , and let Φ ξ =: h Φ , ξ i be the ξ -componentof Φ; then ξ X =: ξ ♯M − Φ ξ ∂∂θ (5)is a contact vector field on X , lifting ξ M under d π . Here υ ♯ ∈ X ( X ) is thehorizontal lift of υ ∈ X ( M ), and ∂/∂θ ∈ X ( X ) is the infinitesimal generatorof the structure S -action on X . Thus the obstruction to the existence of aglobal lifting is of topological nature.In view of the compatibility assumptions on µ M , G acts on X under µ X as a group of contactomorphims and leaves the Hardy space invariant; hencethere is a naturally induced unitary representation e µ : G → U (cid:0) H ( X ) (cid:1) , where e µ g ( f ) =: f ◦ µ Xg − . Let the set { ̟ } label the collection of all irreducible char-acters χ ̟ of G , associated to the (finite dimensional) unitary representations (cid:0) ρ ̟ , V ( ̟ ) (cid:1) .By the Peter-Weyl theorem, if ρ : G → U ( H ) is a unitary representa-tion of G on any Hilbert space H , then there is a natural equivariant and4rthogonal Hilbert space direct sum decomposition H = M ̟ H ( ̟ ) , where H ( ̟ ) ⊆ H is a closed subspace, unitarily and equivariantly isomorphicto a Hilbert space direct sum of copies of V ( ̟ ) . Correspondingly, in our casewe obtain a Hilbert space direct sum decomposition H ( X ) = M ̟ H ( X ) ( ̟ ) . Now suppose that T is a G -invariant self-adjoint T¨oplitz operator on X , with positive symbol ς T >
0. Then for every eigenvalue η ∈ Spec( T ) theeigenspace H ( X, T ) η ⊆ H ( X ) is a finite-dimensional unitary G -representation,and so it also admits an equivariant direct sum decomposition into isotypicalcomponents: H ( X, T ) η = M ̟ H ( X, T ) ( ̟ ) η ;here H ( X, T ) ( ̟ ) η = H ( X, T ) η ∩ H ( X ) ( ̟ ) is, for any fixed η , the null space foralmost every ̟ . Changing point of view, every H ( X ) ( ̟ ) is invariant under T , and so it splits equivariantly as a Hilber space direct sum H ( X ) ( ̟ ) = M η H ( X, T ) ( ̟ ) η . (6)Let T ( ̟ ) : H ( X ) ( ̟ ) → H ( X ) ( ̟ ) be restriction of T , and let λ ( ̟ )1 ≤ λ ( ̟ )2 ≤· · · be the eigenvalues of T ( ̟ ) , repeated according to multiplicity. Let ( e ( ̟ ) j )be any complete orthonormal system of H ( X ) ( ̟ ) such that T ( ̟ ) ( e ( ̟ ) j ) = λ ( ̟ ) j e ( ̟ ) j . Then the equivariant analogue of (1) is its restriction to H ( X ) ( ̟ ) : U ( ̟ ) T ( τ )( x, y ) =: X j e iτλ ( ̟ ) j e ( ̟ ) j ( x ) · e ( ̟ ) j ( y ) ( x, y ∈ X ) . (7)Similarly, we may consider its traveling smoothly averaged version, S ( ̟ ) χ · e − iλ ( · ) =: Z + ∞−∞ χ ( τ ) e − iλτ U ( ̟ ) T ( τ ) d τ, (8)which is again a smoothing operator, with Schwartz kernel S ( ̟ ) χ · e − iλ ( · ) ( x, y ) =: X j b χ (cid:0) λ − λ ( ̟ ) j (cid:1) e ( ̟ ) j ( x ) · e ( ̟ ) j ( y ) (9)5n particular, trace (cid:16) S ( ̟ ) χ · e − iλ ( · ) (cid:17) = X j b χ (cid:16) λ − λ ( ̟ ) j (cid:17) , Thus the local asymptotics of S ( ̟ ) χ · e − iλ ( · ) ( · , · ) captures the collective con-centration behavior of the eigensections e ( ̟ ) j ’s, while its trace controls theasymptotic distribution of the eigenvalues λ ( ̟ ) j ’s.In the following, let us adopt the following notation: M ′ =: Φ − ( ) ⊆ M, X ′ =: π − ( M ′ ) ⊆ X. Furthermore, it is convenient to make a more specific choice for a cut-offfunction.
Definition 1.2.
Fix ǫ >
0. A good ǫ -cut-off is a function χ ∈ C ∞ (cid:0) ( − ǫ, ǫ ) (cid:1) suchthat1. χ ≥ χ (0) = 1;2. b χ ≥ § χ (0) = 1is simply a convenient normalization. Our first result is the following: Theorem 1.1.
Let T be a G -invariant self-adjoint first oder T¨oplitz operatoron X with ς T > , and let ̟ be an irreducible character of G . Suppose ǫ > is small enough and χ is a good ǫ -cut-off. Then the following holds.1. Uniformly in x, y ∈ X , for λ → −∞ we have S ( ̟ ) χ · e − iλ ( · ) ( x, y ) = O (cid:0) λ −∞ (cid:1) .
2. For any
C, ǫ > , uniformly for min n dist X ( x, X ′ ) , dist X ( y, X ′ ) o ≥ C λ − we have for λ → + ∞ : S ( ̟ ) χ · e − iλ ( · ) ( x, y ) = O (cid:0) λ −∞ (cid:1) . Here λ − might be replaced by λ a − for any a >
0. We set a = 1 /
24 tofix ideas and because it is a convenient choice in the following.6e shall next consider the asymptotics of S ( ̟ ) χ · e − iλ ( · ) ( x, y ) for x, y → X ′ , anddist X ( x, y ) →
0. By the same approach, one might more generally considerthe case dist X (cid:0) x, µ g ( y ) (cid:1) → g ∈ G , but for the sake of simplicitywe shall restrict ourselves to near-diagonal asymptotics. More specifically,we shall consider the asymptotics of S ( ̟ ) χ · e − iλ ( · ) ( x ′ , x ′′ ) for x ′ , x ′′ → x ∈ X ′ ata controlled rate. We shall think of x ′ and x ′′ as obtained from x by smalldisplacements along tangent directions, and the scaling asymptotics will becontrolled by the geometry of these directions with respect to X ′ and the G -orbit. It is then in order to label the various components of the displacementsthat go into the statement.The connection α determines a direct sum decomposition of the tangentbundle of X , T X = V ⊕ H , where V =: ker(d π ) = span (cid:0) ∂/∂θ (cid:1) is the verticaltangent bundle, and H =: ker( α ) is the horizontal distribution. If υ ∈ T x X ,we shall write accordingly υ = ( υ ′ , −→ υ ), where υ ′ ∈ V x and −→ υ ∈ H x ∼ = T m M ,where m = π ( x ).For m ∈ M ′ , let N m =: T m M ′⊥ ⊆ T m M be the normal space to M ′ in M at m , and consider the orthogonal direct sum decomposition in the tangentbundle of M along M ′ : T m M =: T m M hor ⊕ T m M ver ⊕ T m M trasv , (10)where: T m M hor =: T m ( G · m ) ⊥ ∩ T m M ′ , T m M ver =: T m ( G · m ) , T m M trasv =: N m . Here G · m ⊆ M ′ is the G -orbit through m , and T m ( G · m ) its tangent space.Furthermore, T m M hor is a complex vector subspace of T m M , while T m M ver and T m M trasv are totally real subspaces, related by T m M ver = J m (cid:0) T m M trasv (cid:1) ,and J m is the complex structure at m ∈ M .When m ∈ M ′ we shall correspondingly write −→ υ ∈ T m M as −→ υ = −→ υ h + −→ υ v + −→ υ t , (11)where −→ υ h ∈ T m M hor , −→ υ v ∈ T m M ver , −→ υ t ∈ T m M trasv .The components of the displacements from x to x ′ and from x to x ′′ willcontrol the scaling asymptotics by certain ‘universal exponents’, dependingon the symplectic and Euclidean structures at m , and given by quadraticfunctions Q T hor , Q T tv in a pair (cid:0) −→ υ , −→ υ ′ (cid:1) ∈ T m M × T m M , that we now define. Definition 1.3.
Let (
V, J V ) is a complex vector space, and suppose that h V = g V − iω V is a positive definite Hermitian product on it; thus g V is a J V -invariant Euclidean product and ω V = g (cid:0) J V ( · ) , · (cid:1) is a a symplectic structure.7et us define after [SZ] ψ V : V × V → C by ψ V ( v , v ′ ) = − i ω V ( v , v ′ ) − k v − v ′ k V , where k · k V is the norm for g V . If ( V, J V ) is C d with its standard Hermitianproduct, we shall write ψ C d = ψ .We obtain ψ M : T M ⊕ T M → C given by ψ M ( −→ υ , −→ υ ) =: ψ T m M ( −→ υ , −→ υ )if m ∈ M and −→ υ j ∈ T m M . Definition 1.4.
Let us define maps (also depending on the T¨oplitz operator T ) Q T h , Q vt , Q T vt : T M ⊕ T M | M ′ → C as follows. Suppose m ∈ M ′ and −→ υ j ∈ T m M . Then: Q T h ( −→ υ , −→ υ ) =: 1 ς T ( x ) ψ M ( −→ υ , −→ υ ) (12)= ψ M p ς T ( x ) −→ υ , p ς T ( x ) −→ υ ! .Q vt ( −→ υ , −→ υ ) = i h ω m (cid:0) −→ υ , −→ υ (cid:1) − ω m (cid:0) −→ υ , −→ υ (cid:1)i − (cid:16) k−→ υ k m + k−→ υ k m (cid:17) ,Q T vt ( −→ υ , −→ υ ) =: 1 ς T ( x ) Q vt ( −→ υ , −→ υ )= Q vt p ς T ( x ) −→ υ , p ς T ( x ) −→ υ ! . (13) Remark . A notational warning is in order. Tangent vectors get decom-posed into vertical and horizontal components in two stages. First, we write υ = (cid:0) υ ′ , −→ υ (cid:1) ∈ T x X with respect to the connection α ; thus −→ υ may be viewedin a natural manner as an element of T m M if m = π ( x ). Secondly, if m ∈ M ′ then −→ υ ∈ T m M may itself be decomposed as in (11), and here horizontal-ity refers to the decomposition of −→ υ − −→ υ t ∈ T m M ′ with the respect to thenatural connection of principal G -bundle M ′ → M .8he scaling asymptotics in this paper will be expressed is a system ofHeisenberg local coordinates (HLC for short) for X centered at x [SZ], thatwe shall denote by: γ x : ( θ, w ) ∈ ( − π, π ) × B d ( , δ ) x + ( θ, w ) ∈ X ;here B d ( , δ ) ⊆ R d is the open ball centered at the origin of radius δ > § θ is an angular coordinate along the circlefiber, and w is a local coordinate on M with good metric properties; in addi-tion, the unitary local section of A ∨ given in local coordinates by w ( w , x + w =: x + (0 , w ).Given the choice of HLC centered at x ∈ X , there are induced unitaryisomorphisms T x X ∼ = R ⊕ R d and T m M ∼ = R d ∼ = C d , which will be implicitin the following; accordingly any υ ∈ T x X will be written as a pair υ =( θ, w ) ∈ R × R d ; if υ ∈ T x X is small enough, we shall then write x + υ withthis identification. We shall also write Q T h ( w , w ) and Q T vt ( w , w ) for and(12) and (13), respectively, if υ j = ( θ j , w j ) in local coordinates.Under this unitary isomorphism T m M ∼ = R d , the direct sum decomposi-tion (10) corresponds to the one R d ∼ = R d − e )hor ⊕ R e ver ⊕ R e trasv , (14)where we label each Euclidean summand according to the correspondingcomponent.We need some further ingredients that go into the scaling asymptotics of S ( ̟ ) χ · e − iλ ( · ) . Definition 1.5. If ∈ g ∨ is a regular value of Φ, then G acts locally freely on M ′ , hence a fortiori on X ′ . Thus any m ∈ M ′ has finite stabilizer G Mm ⊆ G ,and if x ∈ π − ( m ) its stabilizer is a normal subgroup G Xx E G Mm . Since theaction of G on A (and A ∨ ) is fiberwise linear, G Xx = G Xy if x, y ∈ π − ( m );thus we shall write G Xm for G Xx when m = π ( x ).Let m =: (cid:2) G Mm : G Xm (cid:3) for m ∈ M ′ . Definition 1.6.
Let G · m ∼ = G/G Mm ⊆ M be the G -orbit of m ∈ M ′ .The effective volume V M eff ( m ) at m is the volume of G · m for the inducedRiemannian structure of M [BrG].Similarly, if x ∈ X ′ the effective volume V X eff ( x ) is the volume of G · x ∼ = G/G Xx ⊆ X . As V X eff is obviously S -invariant, with a slight abuse of languagewe shall view it as a function on M ′ . 9 emark . Since the G -action on X ′ is horizontal with respect to α , if m = π ( x ) ∈ M ′ then the projection G · x → G · m is a local Riemannianisometry and a covering of degree m ; therefore, V X eff ( m ) = m · V M eff ( m ) = ⇒ (cid:12)(cid:12) G Mm (cid:12)(cid:12) · V M eff ( m ) = (cid:12)(cid:12) G Xm (cid:12)(cid:12) · V X eff ( m ) . Definition 1.7. If x ∈ X ′ , m = π ( x ), and ̟ an irreducible character of G ,let us define A T̟ : M ′ = Φ − ( ) → R A T̟ ( x ) =: 2 e/ dim( V ̟ ) V X eff ( m ) · ς T ( x ) − ( d +1 − e/ . For any g ∈ G Mm , the differential d m µ Mg : T m M → T m M is a unitaryautomorphism of T m M ; hence its Jacobian matrix A g in HLC centered at x (with m = π ( x )) is unitary.In the following, ∼ will mean ‘has the same asymptotics as’. Theorem 1.2.
Suppose that ∈ g ∨ is a regular value of Φ , and let T be a G -invariant first order self-adjoint T¨oplitz operator with ς T > .For any sufficiently small ǫ > any good ǫ -cut-off χ , the following holds.Fix x ∈ X ′ and adopt HLC on X centered at x ; set m =: π ( x ) .Then, uniformly for υ j = ( θ j , w j ) ∈ T x X with k υ j k ≤ C λ / , j = 1 , ,as λ → + ∞ we have S ( ̟ ) χ · e − iλ ( · ) (cid:18) x + υ √ λ , x + υ √ λ (cid:19) ∼ π · A T̟ ( x ) · e i √ λ ( θ − θ ) /ς T ( x )+ Q T tv ( w , w ) · (cid:18) λπ (cid:19) d − e/ · (cid:12)(cid:12) G Xm (cid:12)(cid:12) X g ∈ G Xm χ ̟ ( g ) · e Q T h (cid:0) w ,A g w (cid:1) · S g ( λ, x, υ , υ ) , where each factor S g ( λ, x, υ , υ ) satisfies an asymptotic expansion of the form S g ( λ, x, υ , υ ) ∼ X l ≥ λ − l/ F gl ( x, υ , υ ) ,F gl ( x, υ , υ ) being a polynomial in υ a = ( θ a , w a ) , a = 1 , , of total degree ≤ l (also depending on T ). Let us consider the special case where υ = υ = (0 , w ), where w = w t ∈ N m = T m M trasv ∼ = R e trasv in the notation of (10) and (14).10 efinition 1.8. Let us define a function a Φ ,̟ : M ′ → R by setting a Φ ,̟ ( m ) =: 1 (cid:12)(cid:12) G Xm (cid:12)(cid:12) · X g ∈ G Xm χ ̟ ( g ) = D χ ̟ | G Xm , E L ( G Xm ) . (15)Note that a Φ ,̟ is real, as by unitarity χ ̟ ( g ) = χ ̟ (cid:0) g − (cid:1) , for any g ∈ G .Also, there is a dense open subset of M ′′ ⊆ M ′ on which the conjugacy classof G Xm is constant [GGK], hence a Φ ,̟ is constant on M ′′ ; we shall denote by a gen (Φ , ̟ ) ∈ R the constant value it takes on M ′′ .For example, if e µ is generically free on X ′ , then a gen (Φ , ̟ ) = dim( V ̟ ). Corollary 1.1.
Under the same assumptions of the Theorem, uniformly in w = w t ∈ N m with k w k ≤ C λ / we have S ( ̟ ) χ · e − iλ ( · ) (cid:18) x + w √ λ , x + w √ λ (cid:19) ∼ π · A T̟ ( x ) a Φ ,̟ ( m ) (cid:18) λπ (cid:19) d − e/ · exp (cid:18) − ς T ( x ) k w k (cid:19) · " X j ≥ λ − j/ F j ( x, w ) , where F j is a polynomial in w of degree ≤ j , of the same parity as j (alsodepending on T ). The Theorem yields information about the asymptotic concentration be-havior of the equivariant eigenfunctions of T ; from this, one obtains by inte-gration an estimate on the asymptotic distribution of the eigenvalues λ ( ̟ ) j .Let us define: Γ(Φ , ς T ) =: Z X ′ V X eff ( m ) ς T ( x ) − ( d − e +1) d V X ′ ( x ) , (16)where m = π ( x ), and d V X ′ is the induced volume form on X ′ . Corollary 1.2.
Under the same assumptions, there is an asymptotic expan-sion X j b χ (cid:0) λ − λ ( ̟ ) j (cid:1) ∼ π dim( V ̟ ) a gen (Φ , ̟ ) Γ(Φ , ̟ ) (cid:18) λπ (cid:19) d − e · " X j ≥ λ − j E j , ̟ -equivariant counting function of T , defined as N ( ̟ ) T ( λ ) =: ♯ n j : λ ( ̟ ) j ≤ λ o . (17) Corollary 1.3. As λ → + ∞ , one has N ( ̟ ) T ( λ ) = πd − e + 1 · dim( V ̟ ) a gen (Φ , ̟ ) Γ(Φ , ̟ ) (cid:18) λπ (cid:19) d − e +1 + O (cid:0) λ d − e (cid:1) . In the spirit of [H] and [BtG], the leading asymptotics of N ( ̟ ) T ( λ ) may berelated to an appropriate symplectic volume.Just to fix ideas, let us make the symplifying assumption that µ M is freeon M ′ , leaving it to the interested reader to consider the general case. Thenthe symplectic quotient c M =: M ′ /G , with the induced symplectic structure b ω , is a Hodge manifold in a natural manner [GSt]. Furthermore, b X =: X ′ /G is the unit circle bundle on c M for the positive line bundle b A induced by A on c M by passage to the quotient. Let b α be the connection form of the latter,and denote by b Σ the cone in T ∗ b X sprayed by b α . Then b Σ = Σ ′ /G , where Σ ′ is the restriction of Σ to X ′ .In addition, being e µ -invariant, ς T descends to a C ∞ function on b X , andsimilarly σ T descends to a C ∞ homogeneous function b σ T on b Σ. Corollary 1.4.
Let b Σ ⊆ b Σ be the locus where b σ T ≤ . Then, under theprevious assumptions, as λ → + ∞ , one has N ( ̟ ) T ( λ ) = dim( V ̟ ) vol (cid:16)b Σ (cid:17) (cid:18) λ π (cid:19) d − e +1 + O (cid:0) λ d − e (cid:1) , As in the case of [P2], the arguments in this paper combine the classicalapproach to trace formulae and Weyl laws for pseudodifferential operators[DG], [H], [GrSj] with the microlocal theory of the Szeg¨o kernel [F], [BtSj],and especially its description as an FIO with complex phase function in thelatter article. This follows the philosophy in [SZ] and [Z], where the theory of[BtSj] is specialized to the case of of algebro-geometric Szeg¨o kernels, and infact we shall extensively build on ideas ad techniques from the latter papers.In order to include symmetries in this picture, and describe how the localcontribution to the equivariant trace formula asymptotically concentratesnear the zero locus of the moment map, we shall furthermore adapt theapproach and techniques in [P1], [P3], [P4].12
Preliminaries
We collect here some well-known basic facts about smoothed averages of waveoperators of the form e iτQ , and their T¨oplitz counterparts ([GrSj], [BtG])Let T be a G -invariant first order self-adjoint T¨oplitz operator with ς T >
0. By the theory of [BtG], there exists a G -invariant first order elliptic self-adjoint pseudo-differential operator of classical type Q on X , such that[Π , Q ] = 0 , σ Q > , T = Π ◦ Q ◦ Π . (18)In fact, such a Q exists by Lemma 12.1 of [BtG], and averaging over G yieldsinvariance. In particular, T is the restriction of Q to H ( X ).Let U ( τ ) =: e iτQ , U T ( τ ) =: e iτT ; thus U T ( τ ) = Π ◦ U ( τ ) ◦ Π = U ( τ ) ◦ Π.Let β ≤ β ≤ · · · be the eigenvalues of Q , repeated according to multpicity,and let ( f j ) be a complete orthonormal system of L ( X ) with Q ( f j ) = β j f j ;then the same holds of (cid:0) f gj (cid:1) for any g ∈ G , where f g = f ◦ µ Xg − . It followsthat the distributional kernel U ( τ ) ∈ D ′ ( X × X ) satisfies U ( τ )( x, y ) = U ( τ ) (cid:0) µ Xg ( x ) , µ Xg ( y ) (cid:1) for any g ∈ G . The same considerations hold with U T ( τ ) in place of U ( τ ).If χ ∈ S ( R ), the averaged operator U χ =: Z + ∞−∞ χ ( τ ) U ( τ ) d τ (19)is C ∞ , with Schwartz kernel the series U χ ( x, y ) = X j b χ ( − β j ) f j ( x ) · f j ( y ) , which converges uniformly and absolutely in C ∞ ( X × X ) [GrSj]. The se-quences ( λ j )’s and (cid:0) λ ( ̟ ) j (cid:1) in (2) and (9) are subsequences of ( β j ), and wemay assume without loss that the same holds for the corresponding eigen-functions. Similar conclusions therefore hold for S χ in (2) and (3), and itsequivariant drifting counterpart (8) and (9).Furthermore, U ( τ ) is an FIO associated to the Hamiltonian flow φ T ∗ Xτ of σ Q on T ∗ X [DG], [GrSj]. More precisely, for τ ∼ X × X we have U ( τ ) = V ( τ ) + R ( τ ), where R ( τ ) is asmoothing operator, while in local coordinates V ( τ ) has the form V ( τ )( x, y ) = 1(2 π ) d +1 Z R d +1 e i [ ϕ ( τ,x,η ) −h y,η i ] a ( τ, x, y, η ) d η, (20)13here the phase and amplitude are as follows. First, ϕ ( τ, · , · ) is the generatingfunction of φ T ∗ Xτ , and therefore it satisfies the Hamilton-Jacobi equation.Since φ T ∗ X is the identity, for τ ∼ ϕ ( τ, x, η ) = h x, η i + τ σ Q ( x, η ) + k η k O (cid:0) τ (cid:1) . (21)On the other hand the amplitude is a classical symbol a ( τ, · , · , · ) ∈ S , and a (0 , · , · , · ) = 1 / V ( y ), where V ( x ) d x is the local coordinate expression of d V X (see also the discussion in [P2]). We shall choose the cut-off function χ as follows.Given ǫ >
0, choose γ ∈ C ∞ (cid:0) ( − ǫ/ , ǫ/ (cid:1) real, simmetric and non-negative; in particular its Fourier transform b γ is real. After normalization,we may assume k γ k L = 1. Setting χ = γ ∗ γ (convolution) we obtain thefollowing well-known: Lemma 2.1.
For any ǫ > there exists χ ∈ C ∞ (cid:0) ( − ǫ, ǫ ) (cid:1) such that χ ≥ , χ (0) = 1 , b χ ≥ . With this choice, that U χ is smoothing may also be seen using (20) and(21) to integrate by parts in d τ in (19). Let γ x (cid:0) ( θ, w )) = x + ( θ, w ) be a system of Heisenberg local coordinates on X centered at x [SZ]. Then the following holds:1. The standard circle action r : S × X → X is expressed by a translationin the angular coordinate: where defined, r e iϑ (cid:0) x + ( θ, w ) (cid:1) = x + ( ϑ + θ, w ) .
2. If m = π ( x ) let us set m + w =: π (cid:0) x + (0 , w ) (cid:1) ; then w ∈ B d ( , δ ) m + w is a local coordinate chart centered at m , inducing a unitaryisomorphism C d ∼ = T m M ; in other words, ω m and J m correspond to thestandard complex and symplectic structures on R d ∼ = C d .3. γ x induces at x an isomorphism R ⊕ R d ∼ = T x X compatible with thedirect sum decomposition T x X ∼ = V x ⊕ H x , that is, V x ∼ = R ⊕ ( ) , H x ∼ = (0) ⊕ R d .
14. If we write w ∈ R d as a complex vector z ∈ C d , the local coordinateexpression of of α at x + ( θ, z ) is α = d θ + d X j =1 (cid:16) A j ( z , z ) d z j + A j ( z , z ) d z j (cid:17) , where A j = − ( i/ z j + O ( k z k ). The Szeg¨o projector of X is the orthogonal projector Π : L ( X ) → H ( X ). Itsdistributional kernel, the Szeg¨o kernel Π ∈ D ′ ( X × X ), has singular supportalong the diagonal [F]. The following analysis is based on the description ofΠ as an FIO with a complex phase of positive type in [BtSj]: up to smoothingterms, Π( x, y ) = Z + ∞ e itψ ( x,y ) s ( x, y, t ) d t. (22)Here ψ is essentially determined along the diagonal by the metric, and theamplitude is a classical symbol admitting an asymptotic expansion of theform s ( x, y, t ) ∼ X j ≥ t d − j s j ( x, y )(see also the discussion in § × Σ:WF(Π) = n(cid:0) ( x, r α x ) , ( x, − r α x ) (cid:1) : x ∈ X, r > o (23)([BtSj], [BtG]). In a system of Heisenberg local coordinates centered at x ∈ X , by the discussion in § t ψ (cid:0) x + ( θ, v ) , x + ( θ ′ , v ′ ) (cid:1) (24)= it h − e i ( θ − θ ′ ) i − it ψ ( v , v ′ ) e i ( θ − θ ′ ) + t R ( v , v ′ ) e i ( θ − θ ′ ) . Furthermore, s ( x, x ) = π − d . X The arguments in this paper are based on the local analysis of the FourierT¨oplitz operator U T ( τ ). In the S -invariant case, ς T is (the pull-back to X of) a C ∞ function on M , and U T ( τ ) may be regarded as a quantization of the15lassical dynamics of ς T . Before embarking on the actual proof, it is in orderto put things in perspective by showing that even in this more general casethere is a ‘conformally contact’ dynamics in the picture, which is ‘quantized’by U T ( τ ). Before doing so, however, let us briefly clarify the relation betweenthe flows of σ Q on T ∗ X and of σ T on Σ. σ Q and σ T Recall that among the principal symbols σ T : Σ → R of T and σ Q : ( T ∗ X ) → R of Q , where ( T ∗ X ) ⊆ T ∗ X is the complement of the zero section, and thereduced symbol ς T : X → R there are the relations σ Q (cid:0) x, r α x (cid:1) = σ T (cid:0) x, r α x (cid:1) = r ς T ( x ) . (25)Let φ T ∗ Xτ : ( T ∗ X ) → ( T ∗ X ) and φ Σ τ : Σ → Σ be the Hamiltonian flowsgenerated by σ Q on ( T ∗ X ) and by σ T on the symplectic submanifold Σ. Lemma 2.2.
Given that [Π , Q ] = 0 , for every τ ∈ R we have φ T ∗ Xτ (Σ) = Σ . Proof.
Since [Π , U ( τ )] = 0, wave fronts satisfyWF ′ (Π) ◦ WF ′ (cid:0) U ( τ ) (cid:1) = WF ′ (cid:0) Π ◦ U ( τ ) (cid:1) (26)= WF ′ (cid:0) U ( τ ) ◦ Π (cid:1) = WF ′ (cid:0) U ( τ ) (cid:1) ◦ WF ′ (Π) . Now by (23) and [DG]WF ′ (Π) = n(cid:0) ( x, r α x ) , ( x, r α x ) (cid:1) : x ∈ X, r > o , (27)WF ′ (cid:0) U ( τ ) (cid:1) = graph (cid:0) φ T ∗ X − τ (cid:1) (28)= (cid:8)(cid:0) φ T ∗ Xτ ( x, η ) , ( x, η ) (cid:1) : ( x, η ) ∈ ( T ∗ X ) (cid:9) . By (26) and (27) - (28) we obtain (cid:8)(cid:0) φ T ∗ Xτ ( x, r α x ) , ( x, r α x ) (cid:1) : x ∈ X, r > (cid:9) (29)= (cid:8)(cid:0) ( x, r α x ) , φ T ∗ X − τ ( x, r α x ) (cid:1) : x ∈ X, r > (cid:9) . Clearly (29) implies the statement.
Corollary 2.1.
Given that [Π , Q ] = 0 , φ Σ τ is the restriction of φ T ∗ Xτ to Σ . .4.2 The contact flow on X In the S -invariant case, U T ( τ ) is a quantization of a Hamiltonian flow on M . Namely, let f = ς T , naturally interpreted as a C ∞ real function on M ,and let φ Mτ : M → M be the flow of f generated by the Hamiltonian vectorfield υ f ∈ X ( M ) of f with respect to ( M, ω ). Then e υ f =: υ ♯f − f ∂∂θ is a contact vector field on X lifting υ f , and U T ( τ ) is a T¨oplitz Fourieroperator associated to the corresponding contact flow φ Xτ .By way of motivation, we shall show here that even in the general case U T ( τ ) is associated to a suitable underlying conformally contact dynamicson X .Given the direct sum decomposition T X ∼ = V ⊕ H , we also have T ∗ X ∼ = V ∗ ⊕ H ∗ , where V ∗ = H = span( α ), H ∗ = V (annihilators). We shallaccordingly writed ς T = d v ς T + d h ς T with d v ς T ∈ C ∞ ( X, V ) , d h ς T ∈ C ∞ ( X, H ) . (30)Since ∂/∂θ and α are in duality, we have d v ς T = ( ∂ θ ς T ) · α . On theother hand, the pull back ω = π ∗ ( ω ) is a symplectic structure on the vec-tor sub-bundle H ⊆ T ∗ X (here and in the following we shall more or lesssystematically omit symbols of pull-back in order to simplify notation). Let X h ( X ) be the space of horizontal vector fields on X , that is, smooth sectionsof H . Then d h ς T corresponds to a unique υ h T ∈ X h ( X ) under 2 ω Clearly, Σ q → X is a trivial R + -bundle. In terms of the diffeomorphism( x, r α x ) ∈ Σ ( x, r ) ∈ X × R + and the decomposition T X ∼ = V ⊕ H ,omitting the pull-back symbol q ∗ we have T Σ ∼ = T X ⊕ span (cid:26) ∂∂r (cid:27) (31) ∼ = V ⊕ H ⊕ span (cid:26) ∂∂r (cid:27) ∼ = H ⊕ span (cid:26) ∂∂θ , ∂∂r (cid:27) . Let X h (Σ) ⊆ X (Σ) be the space of vector fields on Σ tangent to the distribu-tion H = q ∗ ( H ). Given υ ∈ X (Σ), one can accordingly write υ = υ h + a ∂∂θ + b ∂∂r for unique υ h ∈ X h (Σ) and a, b ∈ C ∞ (Σ).17et ω Σ the symplectic structure on Σ, given by ω Σ = d( r α ) = d r ∧ α + r d α = d r ∧ α + 2 r ω, (32)where ω = q ∗ ( ω ). Let υ T ∈ X (Σ) be the Hamiltonian vector field of σ T withrespect to ω Σ .We haved σ T = d( r ς T ) = ς T d r + r d ς T (33)= ς T d r + r d v ς T + r d h ς T = ς T d r + r ( ∂ θ ς T ) · α + r d h ς T . Let υ h T ∈ X h ( X ) be the horizontal vector field on X that corresponds tod h ς T under 2 ω (pulled back to H ):d h ς T = 2 ι (cid:0) υ h T (cid:1) ω = 2 ω (cid:0) υ h T , · (cid:1) and let us define υ XT =: υ h T − ς T ∂∂θ ∈ X ( X ) , and view υ XT as a vector field on Σ in a natural manner. Inspection of (33)then shows the following: Lemma 2.3.
The Hamiltonian vector field of σ T with respect to ω Σ is givenby υ T = υ h T − ς T ∂∂θ + r ( ∂ θ ς T ) ∂∂r = υ XT + r ( ∂ θ ς T ) ∂∂r . Corollary 2.2.
The Hamiltonian flow φ Σ τ : Σ → Σ of σ T is a lifting the flow φ Xτ : X → X of υ XT . As we shall see presently, in general the flow φ Xτ generated by υ XT doesnot preserve α , and ς T is not a constant of motion; nonetheless, φ Xτ alwaysleaves Σ invariant. Lemma 2.4.
1. The Lie derivative of α with respect to υ XT is given by L υ XT ( α ) = − ( ∂ θ ς T ) · α.
2. The derivative of ς T along φ Xτ is υ XT ( ς T ) = − ς T · ( ∂ θ ς T ) .
3. The 1-form (1 /ς T ) · α is a contact form on X , and is φ Xτ -invariant. roof. Regarding the first point, L υ XT ( α ) = d (cid:0) ι (cid:0) υ Xf (cid:1) α (cid:1) + ι (cid:0) υ XT (cid:1) d α = − d ς T + ι (cid:0) υ h T (cid:1) ω = − d v ς T = − ( ∂ θ ς T ) · α. Next, υ XT ( ς T ) = υ h T ( ς T ) − ς T ∂∂θ ς T = d ς T (cid:0) υ h T (cid:1) − ς T ∂∂θ ς T = d h ς T (cid:0) υ h T (cid:1) − ς T ∂∂θ ς T = 2 ω (cid:0) υ h T , υ h T (cid:1) − ς T ∂∂θ ς T = − ς T · ( ∂ θ ς T ) . Finally, L υ XT (cid:18) ς T · α (cid:19) = − ς T υ XT ( ς T ) α + 1 ς T · L υ XT ( α )= − ς T ( − ς T · ( ∂ θ ς T )) · α + 1 ς T · (cid:0) − ( ∂ θ ς T ) (cid:1) · α = 0 Corollary 2.3.
For any t ∈ R , we have (cid:0) φ Xτ (cid:1) ∗ ( α ) = ς T ◦ φ Xτ ς T · α. In particular, the cotangent lift of φ Xτ leaves Σ invariant. On the other hand, the cotangent lift of φ Xτ is Hamiltonian, generatedby the Hamiltonian function H =: − λ can (cid:0) υ XT (cid:1) , where λ can is the tautological1-form on T ∗ X ; on Σ, H (cid:0) ( x, r α x ) (cid:1) = − r α x (cid:0) υ XT ( x ) (cid:1) = r ς T ( x ) = σ T (cid:0) ( x, r α x ) (cid:1) . Corollary 2.4.
The flow φ Σ τ coincides with the restriction to Σ ⊆ T ∗ X ofthe cotangent lift of φ Xτ . Let us collect here some facts from linear algebra that will be handy in thefollowing. 19 emma 2.5.
Let
R, S be r × r matrices, with S symmetric, and considerthe symmetric r × r symmetric matrix C = C ( R, S ) =: (cid:18) R T R S (cid:19) . Then det( C ) = ( − r det( R ) . If furthermore det( R ) > then the signatureof C is sgn( C ) = 0 .Proof. The first statement is a straightforward computation by row oper-ations. As to the second, let us remark that the signature (actually, thenumber of positive and negative eigenvalues) is locally constant on the spaceof non-degenerate symmetric matrices. If det( R ) >
0, we may find a contin-uous path R t , 0 ≤ t ≤
1, with R = R and R = I r (the identity matrix),and det( R t ) > t . Thus C t =: (cid:18) R Tt R t (1 − t ) S (cid:19) is a family of non-degenerate symmetric matrices, with C = C , and oneeasily checks that sgn( C ) = 0. For ξ ∈ g , let ξ M ∈ X ( M ) be the vector field it induces on M under µ , and fora given m ∈ M let g M ( m ) ⊆ T m M be the vector subspace of all the ξ M ( m )( ξ ∈ g ). If ∈ g ∨ is a regular value of Φ, then G acts on M ′ = Φ − ( )locally freely. If m ∈ M ′ , therefore, the evaluation map val m : g → T m M is injective, and a linear isomorphism g → g M ( m ). Let B = ( ξ j ) be afixed orthonormal basis of g , and B m = ( u j ) be an orthonormal basis of g M ( m ) ⊆ T m M (with the restricted metric), and let C m = M BB m (val m ) be the d × d matrix representing val m : g → g M ( m ) with respect to these two basis.Thus if ξ ∈ g and ν = M B ( ξ ) ∈ R e is its coordinate vector with respect to B ,then k ξ M ( m ) k m = k C m ν k = ν t C t C ν, (34)where k · k m is the norm on T m M , and k · k is the standard Euclidean normon R e .Although C m depends on the choice of the orthonormal basis B and B m ,det( C m ) is invariantly defined (up to sign), and has the following geometricsignificance. Let G · m ⊆ M be the G -orbit of m ∈ M ′ , so that the choiceof an orientation on g determines an orientation on G · m ; then V eff ( m ) is,by definition, the volume of G · m with respect to the Riemannian volume20orm for the restricted metric. We may assume without loss that B and B m have been chosen oriented, so that det( C m ) >
0. In view of (34), thepull-back to G of the Riemannian volume form on G · m under the (cid:12)(cid:12) G Mm (cid:12)(cid:12) : 1covering map µ m : G → G · m , g µ g ( m ), is det( C m ) vol G , where vol G isthe Haar volume form; det( C m ) is clearly G -invariant, since G acts on M byRiemannian isometries. Therefore, (cid:12)(cid:12) G Mm (cid:12)(cid:12) V M eff ( m ) = Z G det( C m ) vol G = det( C m ) . (35) Proof.
We shall first prove the Theorem under the assumption x = y .Let d V G be the Haar measure on G . Let ρ : G → U ( H ) be a unitary G -action on a Hilbert space H ; then the orthogonal projection P ( ̟ ) : H → H ( ̟ ) onto the ̟ -th isotype, with irreducible representation ( ρ ( ̟ ) , V ̟ ), is given by P ( ̟ ) = d ̟ · Z G χ ̟ (cid:0) g − (cid:1) ρ ( g ) d V G ( g ) , where d ̟ =: dim( V ̟ ) [Dix], χ ̟ ( g ) =: trace (cid:0) ρ ̟ ( g ) (cid:1) . In our case, U ( ̟ ) T ( τ ) = P ( ̟ ) ◦ U T ( τ ) = P ( ̟ ) ◦ U ( τ ) ◦ Π = U ( τ ) ( ̟ ) ◦ Π, where U ( ̟ ) ( τ ) =: P ( ̟ ) ◦ U ( τ ).Thus, in terms of distributional kernels, U ( ̟ ) ( τ )( x, y ) = d ̟ · Z G χ ̟ ( g ) U ( τ ) (cid:0) µ Xg − ( x ) , y (cid:1) d V G ( g )= d ̟ · Z G χ ̟ ( g ) U ( τ ) (cid:0) x, µ Xg ( y ) (cid:1) d V G ( g ) , (36)and on the other hand U ( ̟ ) T ( τ ) ( x ′ , x ′′ ) = Z X U ( ̟ ) ( τ ) ( x ′ , y ) Π ( y, x ′′ ) d V X ( y ) . (37)Using (36) and (37) we obtain for (8) along the diagonal: S ( ̟ ) χ · e − iλ ( · ) ( x, x ) (38)= d ̟ · Z G Z X Z ǫ − ǫ χ ( τ ) e − iλτ χ ̟ ( g ) U ( τ ) (cid:0) x, µ Xg ( y ) (cid:1) Π ( y, x ) · d τ d V X ( y ) d V G ( g ) . Let X ⊆ X be an arbitrarily small open neighborhood of x ∈ X , andlet ̺ x ∈ C ∞ ( X ) be identically = 1 in an open neighborhood X ⋐ X of x .Write S ( ̟ ) χ · e − iλ ( · ) ( x, x ) = S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ′ + S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ′′ , ̺ z ( y ), and in the latter by 1 − ̺ x ( y ). Lemma 3.1. As λ → ∞ , we have S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ′′ = O (cid:0) λ −∞ (cid:1) . Proof.
On the support of 1 − ̺ x , we have dist X ( y, x ) ≥ c for some fixed c >
0. Thus, F x ( y ) =: (cid:0) − ̺ x ( y ) (cid:1) Π( y, x ) is C ∞ function of y . Therefore, thefunction Γ x ( τ ) =: χ ( τ ) · Z G χ ̟ ( g ) U ( τ )( F x ) (cid:0) µ Xg − ( x ) (cid:1) d V G ( g )is C ∞ and compactly supported, so that its Fourier transform b Γ x is of rapiddecrease. On the other hand, we have S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ′′ = d ̟ · Z ǫ − ǫ e − iλτ e F x ( τ ) d τ = d ̟ · b Γ x ( λ ) . Thus we are reduced to considering the asymptotics of S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ′ .On the support of Π ′ ( y, x ) =: ̺ x ( y ) Π( y, x ), we may represent Π as an FIOusing (22).We can adopt the same principle to make one more similar reduction.In fact, if ǫ is small enough then the singular support of U ( τ ) for χ ( τ ) = 0lies within a small tubular neighborhood of the diagonal in X × X . Thus,the contribution to the asymptotics of S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ′ coming from the locuswhere dist X (cid:0) µ Xg ( y ) , x (cid:1) ≥ a > a > y itself (for ̺ x ( y ) = 0) belongs to a small neighborhood of x , in orderfor µ Xg ( y ) to belong to a small neighborhood of x we need to assume that g belongs to a small neighborhood of the stabilizer G Xm ⊆ G of x . Therefore,we only lose a negligible contribution to the asymptotics, if the integrand in(38) is further multiplied by a cut-off of the form ρ x ( g ), supported in a smallneighborhood of G Xm , and identically = 1 sufficiently close to G Xm .If G Xm = (cid:8) g = e, g , . . . , g r (cid:9) (here r = r m and m = π ( x )), we may assumefor simplicity that ρ x ( g ) = P rj =1 ρ ( g j g ), where ρ is a fixed cut-off supportedin a small neighborhood of the unit e ∈ G .Furthermore, on the support of the cut-offs introduced, up to smooth-ing terms contributing negligibly to the asymptotics U ( τ ) and Π may bedescribed as FIOs using (20) and (22).22hus we conclude that asymptotically for λ → ∞ S ( ̟ ) χ · e − iλ ( · ) ( x, x ) (39) ∼ d ̟ (2 π ) d +1 · Z + ∞ Z R d +1 Z G Z X Z ǫ − ǫ e i [ ϕ ( τ,x,η ) −h g · y,η i + tψ ( y,x ) − λτ ] · χ ( τ ) χ ̟ ( g ) ̺ x ( y ) ρ x ( g ) a ( τ, x, g · y, η ) s ( y, x, t ) · d τ d V X ( y ) d V G ( g ) d η d t = d ̟ (2 π ) d +1 · Z + ∞ Z R d +1 Z G Z X Z ǫ − ǫ e i Ψ A d τ d V X ( y ) d V G ( g ) d η d t, where we have set g · y =: µ Xg ( y ). Furthermore in view of (21) the phase Ψis given byΨ =: ϕ ( τ, x, η ) − h g · y, η i + tψ ( y, x ) − λτ (40)= h x, η i + τ σ Q ( x, η ) − h g · y, η i + tψ ( y, x ) − λτ + k η k O (cid:0) τ (cid:1) . The amplitude on the other had is given by A =: χ ( τ ) χ ̟ ( g ) ̺ x ( y ) ρ x ( g ) a ( τ, x, g · y, η ) s ( y, x, t ) . (41)For λ ≪
0, we have ∂ τ Ψ ≥ C ( k η k + | λ | ), and integrating by parts in d τ shows that the right hand side of (39) is O ( λ −∞ ) for λ → −∞ . This provesthe first statement of the Proposition.Let us then focus on the asymptotics for λ → + ∞ . To this end, letus operate the change of variables t λ t , η λ η , so that (39) may berewritten: S ( ̟ ) χ · e − iλ ( · ) ( x, x ) (42) ∼ π d ̟ (cid:18) λ π (cid:19) d +2 · Z + ∞ Z R d +1 Z G Z X Z ǫ − ǫ e i λ Ψ A · d τ d V X ( y ) d V G ( g ) d η d t, where nowΨ =: h x − g · y, η i + τ σ Q ( x, η ) + tψ ( y, x ) − τ + k η k O (cid:0) τ (cid:1) , (43) A =: χ ( τ ) χ ̟ ( g ) ̺ x ( y ) ρ x ( g ) a (cid:0) τ, x, g · y, λ η (cid:1) s (cid:0) y, x, λ t (cid:1) . (44)23f we set η = r Ω, with r > ∈ S d ⊆ R d +1 (the unit sphere), wemay further rewrite (43) as S ( ̟ ) χ · e − iλ ( · ) ( x, x ) (45) ∼ π d ̟ (cid:18) λ π (cid:19) d +2 · Z + ∞ Z + ∞ Z S d Z G Z X Z ǫ − ǫ e i λ Ψ A · r d d τ d V X ( y ) d V G ( g ) dΩ d r d t, with Ψ =: r h x − g · y, Ω i + r τ σ Q ( x, Ω) + tψ ( y, x ) − τ + r O (cid:0) τ (cid:1) , (46) A =: χ ( τ ) χ ̟ ( g ) ̺ x ( y ) ρ x ( g ) a (cid:0) τ, x, g · y, λ r Ω (cid:1) s (cid:0) y, x, λ t (cid:1) . (47)So far we have not made a specific choice of local coordinates. It is nowconvenient to assume that the computation is being carried out in a systemof Heisenberg local coordinates centered at x , and we write y = x + ( θ, v ),with the replacement Z X d V X ( y ) −→ Z R d Z π − π V M ( θ, v ) d θ d v , (48)where V M is the local coordinate expression of the volume density on M , andin particular V M ( θ, ) = 1 / (2 π ).We shall write accordingly η = ( η ′ , η ′′ ) = r Ω = r · (Ω ′ , Ω ) ∈ R × R d with (Ω ′ ) + k Ω k = 1 , (49)and make the replacement Z R d +1 d η −→ Z + ∞ r d d r Z S d dΩ (50)In Heisenberg local coordinates, η = (1 , ) corresponds the cotangentvector α x . Using this and (23), (28) one can prove the following: Lemma 3.2.
Only a negligible contribution to the asymptotics is lost in(45), if the amplitude A is multiplied by a cut off function γ (Ω) , compactlysupported in a small neighborhood S ⊆ S d of (1 , ) and identically = 1 ina smaller neighborhood of (1 , ) .Proof. See Lemma 2.2 of [P2]. 24ow small the neighborhoods in Lemma 3.2 may be chosen depends onhow small ǫ is.Now S ⊆ S + , the upper hemisphere, and on S + we have a system oflocal coordinates for S d , given by Ω ∈ B d ( , (cid:0)p − k Ω k , Ω (cid:1) ; wecan then make the replacement Z S d dΩ −→ Z B d ( , V S ( Ω ) d Ω , (51)where V S ( Ω ) is the local coordinate expression of the volume form on thesphere, and integration is compactly supported.Integration in d t d r may also assumed to be compactly supported: Lemma 3.3.
Only a negligible contribution to the asymptotics is lost in(45), if for some D ≫ the amplitude A is further multiplied by a cut offfunction γ ( t, r ) , compactly supported in (1 /D, D ) and identically = 1 in (2 /D, D/ .Proof. This follows integrating by parts in ( τ, y ), by arguments on the lineof those in Lemma 2.3 of [P2].At this point, integrating parts in d t will show that integration in y maybe restricted to a suitable shrinking neighborhood of x ; more precisely, wehave: Lemma 3.4.
For any
C, a > the locus where k ( θ, v ) k ≥ C λ a − / con-tributes negligibly to the asymptotics of (45).Proof. By Corollary 1.3 of [BtSj], for some C ′ > y, z ) ∈ X × X ℑ ψ ( y, z ) ≥ C dist( y, z ) . Therefore, on the locus where k ( θ, v ) k ≥ C λ a − / we have (cid:12)(cid:12) ∂ t Ψ (cid:12)(cid:12) = (cid:12)(cid:12) ψ (cid:0) x + ( θ, v ) , x (cid:1)(cid:12)(cid:12) ≥ ℑ ψ (cid:0) x + ( θ, v ) (cid:1) ≥ C ′′ λ a − . Iteratively integrating by parts in d t then introduces at each step a factor λ − a .To fix ideas, we shall take in the following a = 1 /
24. Summing up,multiplying the amplitude in (45) by β λ (Ω , t, r, θ, v ) =: γ (Ω) γ ( t, r ) γ (cid:0) λ / k ( θ, v ) k (cid:1) (52)does not alter the asymptotics.We can now at least verify that (45) is rapidly decreasing if x does notbelong to a fixed small tubular neighborhood of X ′ .25 emma 3.5. Given ǫ > , there exists δ ′ > , which may chosen very smallif ǫ is sufficiently small, such that S ( ̟ ) χ · e − iλ ( · ) ( x, x ) = O (cid:0) λ −∞ (cid:1) for dist X ( x, X ′ ) ≥ δ ′ .Proof. We have restricted integration in d Ω to the locus where k Ω k < δ ,say, so that 1 ≥ Ω ′ ≥ − cδ for some c > ∈ g ∨ is a regular value of Φ, we have for some C > m = π ( x )) (cid:13)(cid:13) Φ( m ) (cid:13)(cid:13) ≥ C dist M ( π ( x ) , M ′ ) = C dist X ( x, X ′ ) . On the other hand, integration in d V G ( g ) is localized in a small neigh-borhood of G Xm . Let e G : g → G denote the exponential map Thus any g in a neighborhood of g ℓ ∈ G Xm may be uniquely written g = e G ( ξ ) g ℓ , forsome ξ ∈ B g ( , δ ); the latter denotes an open ball of some small radius δ > G . Again, δ may be chosen arbi-trarily small at the price of making ǫ itself small enough. Thus in the range k ( θ, v ) k ≤ C λ − / and k Ω k ≤ δ , we have −h g · y, Ω i = r h h Φ( m ) , ξ i Ω ′ − h ξ ♯ ( m ) , Ω i + O (cid:0) k ξ k (cid:1) + O (cid:0) λ − / (cid:1) i . It follows in view of (46) that for λ ≫ b > (cid:13)(cid:13) ∇ ξ Ψ (cid:13)(cid:13) = r (cid:13)(cid:13)(cid:13) Φ( m ) Ω ′ − F m ( Ω ) + O ( k ξ k ) (cid:13)(cid:13)(cid:13) ≥ D (cid:2) C (cid:0) − δ (cid:1) dist X ( x, X ′ ) − b ( δ + δ ) (cid:3) (53)(here F m is an appropriate linear map).We see from (53) that (cid:13)(cid:13) ∇ ξ Ψ (cid:13)(cid:13) is bounded below by a fixed positiveconstant for dist X ( x, X ′ ) ≥ C b ( δ + δ ) . (54)Here C and b are fixed, while δ , δ may be taken arbitrarily small with ǫ small enough. Integrating by parts in d V G ( g ), we conclude that the locus(54) contributes negligibly to the asymptotics of (45), as claimed.Therefore, we shall assume in the following that x belongs to some smalltubular neighborhood of X ′ in X . 26et us also fix an orthonormal basis ( ξ j ) of g . On the other hand, thechoice of ( ξ j ) determines a unitary isomorphism g ∼ = R e (the latter with thestandard Euclidean structure). We shall thus identify ξ = P j ν j ξ j ∈ g withits coordinate vector ν ∈ R e , and with this understanding write g = e G ( ν ) g a .Similarly, we shall write ν M ∈ X ( M ) and ν X ∈ X ( X ) for the vector fieldsgenerated on M and X by ν ∈ R e ∼ = g .Let Φ j = h Φ , ξ j i be the components of the moment map with respect tothe ξ j ’s; then under the previous identification h Φ , ξ i = h Φ , ν i = X j Φ j ν j . (55)Since the metric on G is bi-invariant, we can make the replacement Z G ρ x ( g ) dV G ( g ) −→ X ℓ Z B e ( ,δ ) V G ( ν ) ρ (cid:0) e G ( ν ) (cid:1) d ν, (56)where now B e ( , δ ) ⊆ R e is the open ball centered at the origin and radius δ , and ρ is a fixed cut off supported in it; V G ( ν ) is the local coordinateexpression of the Haar density. Dependence on ℓ is of course in the rest ofthe amplitude (see below).Before proceeding, let us show that a cut-off similar to the one in Lemma3.4 may be applied to integration in d ν , in each summand of (56) (we take a = 1 / Lemma 3.6. If C is as in Lemma 3.4 and C ≫ C then the locus where k ν k ≥ C λ − / contributes negligibly to the asymptotics of (45).Proof. Let us go back to (43), which may be rewritten asΨ =: (cid:2) tψ ( y, x ) − τ (cid:3) + h x − g · y, η i + τ F ( x, η, τ ) , (57)where F ( x, η, τ ) =: σ Q ( x, η ) + k η k O ( τ ).In a natural manner, T ∗ x X ֒ → T ( x,β ) ( T ∗ X ) for any ( x, β ) ∈ T ∗ X . Inparticular, we can view α x as a tangent vector to Σ ⊆ T ∗ X at ( x, rα x ) forsome r >
0, and interpret ∇ η Ψ as a tangent vector to x by duality.If y = x + ( θ, v ) with k ( θ, v ) k ≤ C λ − / then the Euclidean gradient ofΨ with respect to η is ∇ η Ψ = − ξ X ( x ) + O (cid:0) λ − / (cid:1) + τ ∇ η F. (58)Now if x ∈ X ′ then h α x , ξ X ( x ) i = 0; on the other hand F ( x, r α x , τ ) = r ς T ( x, η ) + r O ( τ ) ,
27o that h α x , ∇ η F i = d η F ( α x ) = ς T ( x, η ) + O ( τ ) ≥ C > C > ǫ has been chosen small enough.It follows that for any x ∈ X ′ we have g X ( x ) ⊕ span (cid:0) ∇ η F (cid:1) = { } over X ′ , and by continuity the same holds in an open (fixed) tubular neigh-borhood of X ⊆ X of X ′ ; here g X ( x ) ⊆ T x X is the image of the evaluationmap ev x : ξ ∈ g ξ X ( x ) ∈ T x X . Since ev x is injective near X ′ , we concludefrom (58) that over X for some D > (cid:13)(cid:13) ∇ η Ψ (cid:13)(cid:13) ≥ D k ξ k + O (cid:0) λ − / (cid:1) = D k ν k + O (cid:0) λ − / (cid:1) . (59)Thus, if C ≫ k ν k ≥ C λ − / then (cid:13)(cid:13) ∇ η Ψ (cid:13)(cid:13) ≥ D λ − / for some D >
0. The claim then follows as in Lemma 3.4 integrating by parts in d η ,which is legitimate since integration in d η is now compactly supported.Arguing as for Lemma 3.2 of [P4] (with µ Xg ℓ in place of φ X − τ ) we have inHeisenberg local coordinates g ℓ · y = g ℓ · (cid:0) x + ( θ, v ) (cid:1) (60)= g ℓ · (cid:0) x + ( θ, v ) (cid:1) = x + ( θ + R ( v ) , A ℓ v + R ( v )) . Here and in the following R j will denote a generic function, allowed tovary from line to line, defined on some open neighborhood of the origin in aEuclidean space, and vanishing to j -th order at .Given this, in view of Corollary 2.2 of [P3] applied with ϑ = − ν we obtain e G ( ν ) g ℓ · y = e G ( ν ) · (cid:16) x + (cid:0) θ + R ( v ) , A ℓ v + R ( v ) (cid:1)(cid:17) (61)= x + (cid:16) Θ ℓ ( θ, v , ν ) , V ℓ ( v , ν ) (cid:17) , where Θ ℓ ( θ, v , ν ) =: θ − h Φ( m ) , ν i − ω m (cid:0) ν M ( m ) , A ℓ v ) + R ( ν, v ) , V ℓ ( v , ν ) =: A ℓ v + ν M ( m ) + R ( ν, v ) . Thus, S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ∼ r m X ℓ =1 S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ( ℓ ) , (62)28here the sum is over G Xm , and the ℓ -th summand is given by S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ( ℓ ) (63) ∼ π d ̟ (cid:18) λ π (cid:19) d +2 · Z D /D Z D /D Z B d ( , Z R e Z R d Z π − π Z ǫ − ǫ e i λ Ψ ( ℓ )4 A ( ℓ )4 ·V S ( Ω ) V G ( ν ) V ( θ, v ) r d d τ d θ d v d ν d Ω d r d t, where now, in view of (24),Ψ ( ℓ )4 =: − r (cid:10) (Θ ℓ , V ℓ ) , (Ω ′ , Ω ) (cid:11) + r τ σ Q ( x, Ω) − τ (64)+ tψ (cid:16) x + ( θ, v ) , x (cid:17) + r O (cid:0) τ (cid:1) = − r Θ ℓ Ω ′ − r h V ℓ , Ω i + r τ σ Q ( x, Ω) − τ + r O (cid:0) τ (cid:1) + it (cid:2) − e iθ (cid:3) + i t k v k e iθ + t R ( v ) e iθ = − r h θ − h Φ( m ) , ν i − ω m (cid:0) ν M ( m ) , A ℓ v ) + R ( ν, v ) i Ω ′ − r D A ℓ v + ν M ( m ) + R ( ν, v ) , Ω E + r τ σ Q ( x, Ω) − τ + it (cid:2) − e iθ (cid:3) + i t k v k e iθ + t R ( v ) e iθ + r O (cid:0) τ (cid:1) = it (cid:2) − e iθ (cid:3) − r θ Ω ′ + r τ σ Q ( x, Ω) − τ + r O (cid:0) τ (cid:1) + r h h Φ( m ) , ν i Ω ′ − h A ℓ v + ν M ( m ) , Ω i i + r ω m (cid:0) ν M ( m ) , A ℓ v ) Ω ′ + i t k v k e iθ − r (cid:10) R ( ν, v ) , Ω (cid:11) − r R ( ν, v ) Ω ′ + t R ( v ) e iθ . A ( ℓ )4 =: χ ( τ ) χ ̟ (cid:0) e G ( ξ ) g ℓ (cid:1) ̺ x (cid:0) x + ( θ, v ) (cid:1) ρ (cid:0) e G ( ν ) (cid:1) β λ (Ω , t, r, θ, v , ν ) · a (cid:16) τ, x, g · (cid:0) x + ( θ, v ) (cid:1) , λ r Ω (cid:17) s (cid:0) x + ( θ, v ) , x, λ t (cid:1) , (65)where Ω = (cid:0) Ω ′ , Ω (cid:1) = (cid:0)p − k Ω k , Ω (cid:1) , g = e G ( ξ ) g ℓ . We have set β λ (Ω , t, r, θ, v , ν ) =: β λ (Ω , t, r, θ, v ) · γ (cid:0) λ / ν (cid:1) , (66)for an appropriate cut-off γ supported near the origin in g ∨ .Here the expression Φ ν ( m ) = h Φ( m ) , ν i refers to the pairing g ∨ × g → R ,while h A ℓ v + ν M ( m ) , Ω i refers to the pairing T m M × T m M ∨ → R .Now let us introduce the rescaled variables v r √ λ v , ν r √ λ ν. (67)29hus integration in d v d ν will now be on an expanding ball in R d × R e ofradius O (cid:0) λ / (cid:1) . Then (63) may be further rewritten S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ( ℓ ) ∼ d ̟ (2 π ) d +1 λ d +2 − e/ (68) · Z D /D Z D /D Z B d ( , Z R e Z R d Z π − π Z ǫ − ǫ e iλ Γ+ i √ λ Υ e B ( ℓ ) e λ R (cid:16) ν √ λ , v √ λ (cid:17) ·A ( ℓ ) · V S ( Ω ) V G (cid:18) νr √ λ (cid:19) V (cid:18) θ, v r √ λ (cid:19) r − e d τ d θ d v d ν d Ω d r d t, where, in view of (64) and (65) we haveΓ =: it (cid:2) − e iθ (cid:3) − r θ Ω ′ + r τ σ Q ( x, Ω) − τ + r O (cid:0) τ (cid:1) (69)Υ ( ℓ ) =: h Φ( m ) , ν i Ω ′ − h A ℓ v + ν M ( m ) , Ω i B ( ℓ ) =: ir ω m (cid:0) ν M ( m ) , A ℓ v ) Ω ′ − r t k v k e iθ − ir (cid:10) R ( ν, v ) , Ω (cid:11) , while A ( ℓ ) is A ( ℓ )4 expressed in terms of the new rescaled variables.We may then rewrite (68) the following form: S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ( ℓ ) ∼ (2 π ) − (2 d +1) d ̟ λ d +2 − e/ (70) · Z R e (cid:20)Z B d ( , Z R d e i √ λ Υ ( ℓ ) I ( ℓ ) λ ( v , Ω , ν ) d v d Ω (cid:21) d ν, where I λ ( v , Ω , ν ) ( ℓ ) =: Z D /D Z D /D Z ǫ − ǫ Z π − π e iλ Γ B ( ℓ ) d θ d τ d r d t, (71)with B ( ℓ ) =: e B ( ℓ ) · e λ R (cid:16) ν √ λ , v √ λ (cid:17) A ( ℓ ) · V S ( Ω ) V G (cid:18) νr √ λ (cid:19) V (cid:18) θ, v r √ λ (cid:19) r − e . (72)For k ν k , k v k ≤ C λ / , we have λ R (cid:18) ν √ λ , v √ λ (cid:19) = O (cid:0) λ − / (cid:1) . Therefore, in the same range |B ( ℓ ) | ≤ C ′ λ d e − c k v k C ′ , c >
0. Also, we have an asymptotic expansion, coming from theasymptotic expansions of a and s as classical symbols and from the Taylorexpansions in the rescaled variables, A ( ℓ ) · V S ( Ω ) V G (cid:18) νr √ λ (cid:19) V (cid:18) θ, v r √ λ (cid:19) r − e ∼ X k ≥ λ d − k/ P ( ℓ ) k ( ν, v ) , where P ( ℓ ) k ( ν, v ) is a polynomial of joint degree ≤ k (with coefficients de-pending on all the other variables, which we omit). It follows that B ( ℓ ) hasan asymptotic expansion of the form: B ( ℓ ) ∼ e B ( ℓ ) · X k ≥ λ d − k/ Q ( ℓ ) k ( ν, v , Ω ) , (73)where Q ( ℓ ) k ( ν, v , Ω ) is a polynomial in ( ν, v ), of joint degree ≤ k (again, withcoefficients depending on all the other variables). Furthermore, the leadingcoefficient is Q ( ℓ )0 = (cid:18) tπ (cid:19) d r − e · χ ̟ ( g ℓ ) . Let us evaluate I ( ℓ ) λ ( v , Ω , ν ) asymptotically for λ → + ∞ , viewing it asan oscillatory integral in ( θ, t, τ, r ), with oscillatory parameter λ and phaseΓ with non-negative imaginary part. A computation that we leave to thereader shows that for | τ | < ǫ and ǫ small enough we have: Lemma 3.7. Γ has a unique stationary point P = ( θ , t , τ , r ) =: (cid:18) , Ω ′ σ Q ( x, Ω) , , σ Q ( x, Ω) (cid:19) . The Hessian matrix at the stationary point satisfies det (cid:18) λ π i H(Ψ)( P ) (cid:19) = (cid:18) λ π (cid:19) σ Q ( x, Ω) . In particular, the stationary point is non-degenerate.
We have Γ( P ) = 0 and B ( ℓ ) ( P , v , Ω , ν ) = σ Q ( x, Ω) Ω ′ (cid:20) i ω m (cid:0) ν M ( m ) , A ℓ v ) − k v k (cid:21) − i (cid:10) R ( ν, v ) , Ω (cid:11) I λ ( v , Ω , ν ) ( ℓ ) ∼ (2 π ) π d e B ( ℓ ) ( P , v , Ω ,ν ) λ d − σ Q ( x, Ω) e − (1+ d ) · Ω ′ d · χ ̟ ( g ℓ ) · " X j ≥ λ − j/ e Q ( l ) j ( ν, v ) . (74)Since the remaining integration is compactly supported in Ω , and overan expanding ball of radius O (cid:0) λ / (cid:1) , the expansion may be integrated termby term.Let us next view the integral in d v d Ω in (70) as an oscillatory integralwith parameter √ λ and real phase Υ ( ℓ ) . Again, a computation that we leaveto the reader and application of Lemma 2.5 with r = 2 d and R = − A (asymplectic matrix) shows the following: Lemma 3.8. Υ ( ℓ ) has a unique critical point, given by ( v , Ω ) =: (cid:0) − A − ℓ ν M ( m ) , (cid:1) . The Hessian matrix at the critical point has the form
Hess (cid:0) Υ ( ℓ ) (cid:1) ( v , Ω ) = (cid:18) d − A tℓ − A ℓ − Φ ν ( m ) I d (cid:19) , where Φ ν = h Φ , ν i . In particular, its determinant and signature are det (cid:16) Hess (cid:0) Υ ( ℓ ) (cid:1) ( v , Ω ) (cid:17) = 1 , sgn (cid:16) Hess (cid:0) Υ ( ℓ ) (cid:1) ( v , Ω ) (cid:17) = 0 . Also, we have (recalling that A ℓ is unitary (i.e., symplectic and orthogo-nal): i √ λ Υ ( ℓ ) ( P , v , Ω , ν ) + B ( ℓ ) ( P , v , Ω , ν ) (75)= i √ λ · h Φ( m ) , ν i − ς T ( x ) · k ν M ( m ) k . (76)Thus we obtain for the inner integral in (70) an asymptotic expansion Z B d ( , Z R d e i √ λ Υ ( ℓ ) I ( ℓ ) λ ( v , Ω , ν ) d v d Ω (77) ∼ π d (2 π ) d +2 λ − ς T ( x ) e − (1+ d ) χ ̟ ( g ℓ ) ·· e i √ λ ·h Φ( m ) ,ν i− ς T ( x ) ·k ν M ( m ) k X j ≥ λ − j/ R j ( ν ) (78)32here R j is a polynomial in ν (of degree ≤ j ), and R = 1.The expansion may be integrated in d ν . To perform the computation, let C m be the matrix representing the evaluation map val m : g → g M ( m ), ξ ξ M ( m ), with respect to the orthonormal basis ( ξ j ) of g and an orthonormalbasis of ξ M ( m ) ⊆ T m M . Thus k ν M ( m ) k = ν t C tm C m ν . If we set s = C m ν ,and then r =: p ς T ( x ) · s , we get for every j Z R e R j ( ν ) e i √ λ ·h Φ( m ) ,ν i− ς T ( x ) ·k ν M ( m ) k d ν = 1det( C m ) Z R e e R j ( s ) e i √ λ · (cid:10) ( C − ) t Φ( m ) , s (cid:11) − ς T ( x ) ·k s k d s = 1det( C m ) ς T ( x ) − e/ Z R e b R j ( r ) e i √ λ · (cid:10) L (cid:0) Φ( m ) (cid:1) , r (cid:11) − ·k r k d r (79)where L (cid:0) Φ( m ) (cid:1) =: ς T ( x ) − / ( C − ) t Φ( m ), while e R j and b R j are obtained from R j by substitution.For j = 0, we get1det( C m ) ς T ( x ) − e/ Z R e e i √ λ · (cid:10) L (cid:0) Φ( m ) (cid:1) , s (cid:11) − ·k r k d r (80)= 1det( C m ) (2 π ) e/ ς T ( x ) − e/ exp (cid:18) − λ (cid:13)(cid:13) L (cid:0) Φ( m ) (cid:1)(cid:13)(cid:13) (cid:19) . For general j , on the other hand we obtain Z R e b R j ( r ) e i √ λ · (cid:10) L (cid:0) Φ( m ) (cid:1) , r (cid:11) − ·k r k d r (81)= 1det( C m ) ς T ( x ) − e/ S j (cid:16) √ λ L (cid:0) Φ( m ) (cid:1)(cid:17) exp (cid:18) − λ (cid:13)(cid:13) L (cid:0) Φ( m ) (cid:1)(cid:13)(cid:13) (cid:19) , where S j is a polynomial of degree ≤ j , and (cid:13)(cid:13) L (cid:0) Φ( m ) (cid:1)(cid:13)(cid:13) ≥ c k Φ( m ) k forsome c >
0. Thus for every N ≫ S ( ̟ ) χ · e − iλ ( · ) ( x, x ) ( ℓ ) ∼ N X j =0 λ − j/ Q ( ℓ ) j (cid:16) √ λ Φ( m ) (cid:17) exp (cid:18) − λ (cid:13)(cid:13) L (cid:0) Φ( m ) (cid:1)(cid:13)(cid:13) (cid:19) + O (cid:0) λ − ( N +1) (cid:1) , (82)where Q ( ℓ ) j is a polynomial of degree ≤ j .If as assumed dist X ( x, X ′ ) = dist M ( m, M ′ ) ≥ C λ − , then k Φ( m ) k ≥ C ′ λ − for some C ′ >
0, because ∈ g ∨ is a regular value of Φ; so thatexp (cid:18) − λ (cid:13)(cid:13) L (cid:0) Φ( m ) (cid:1)(cid:13)(cid:13) (cid:19) ≤ exp (cid:18) − C ′ λ / (cid:19) . (83)33n the case x = y , Theorem 1.1 follows from (82) and (83).Given this, the general case now follows from the Cauchy-Schwartz in-equality: (cid:12)(cid:12)(cid:12) S ( ̟ ) χ · e − iλ ( · ) ( x, y ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j b χ (cid:0) λ − λ ( ̟ ) j (cid:1) / e ( ̟ ) j ( x ) · b χ (cid:0) λ − λ ( ̟ ) j (cid:1) / e ( ̟ ) j ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sX j b χ (cid:0) λ − λ ( ̟ ) j (cid:1) (cid:12)(cid:12)(cid:12) e ( ̟ ) j ( x ) (cid:12)(cid:12)(cid:12) · sX j b χ (cid:0) λ − λ ( ̟ ) j (cid:1) (cid:12)(cid:12)(cid:12) e ( ̟ ) j ( y ) (cid:12)(cid:12)(cid:12) = S ( ̟ ) χ · e − iλ ( · ) ( x, x ) / S ( ̟ ) χ · e − iλ ( · ) ( y, y ) / . Proof.
It suffices to prove the Theorem in case θ = 0. Let us set for λ > x λ =: x + (cid:18) θ √ λ , w √ λ (cid:19) , x λ =: x + w √ λ . Following the same line of argument as in the proof of Theorem 1.1, we obtainthe analogues of (62) and (63): S ( ̟ ) χ · e − iλ ( · ) ( x λ , x λ ) ∼ r m X ℓ =1 S ( ̟ ) χ · e − iλ ( · ) ( x λ , x λ ) ( ℓ ) , (84)and S ( ̟ ) χ · e − iλ ( · ) ( x λ , x λ ) ( ℓ ) (85) ∼ π d ̟ (cid:18) λ π (cid:19) d +2 · Z D /D Z D /D Z B d ( , Z R e Z R d Z π − π Z ǫ − ǫ e i λ Ψ ( ℓ )4 λ A ( ℓ )4 λ ·V S ( Ω ) V G ( ν ) V ( θ, v ) r d d τ d θ d v d ν d Ω d r d t, where nowΨ ( ℓ )4 λ =: it (cid:2) − e iθ (cid:3) − r θ Ω ′ + r τ σ Q ( x λ , Ω) − τ + r O (cid:0) τ (cid:1) (86)+ r (cid:20)(cid:18) θ √ λ + h Φ( m ) , ν i (cid:19) Ω ′ + (cid:28) w √ λ − (cid:0) A ℓ v + ν M ( m ) (cid:1) , Ω (cid:29)(cid:21) + r ω m (cid:0) ν M ( m ) , A ℓ v ) Ω ′ − i t ψ (cid:18) v , w √ λ (cid:19) e iθ − r (cid:28) R (cid:18) ν, v , w j √ λ (cid:19) , Ω (cid:29) + R (cid:18) ν, v , w j √ λ (cid:19) , ( ℓ )4 λ =: χ ( τ ) χ ̟ (cid:0) e G ( ξ ) g ℓ (cid:1) ̺ x (cid:0) x + ( θ, v ) (cid:1) ρ (cid:0) e G ( ν ) (cid:1) β λ (Ω , t, r, θ, v , ν ) · a (cid:16) τ, x λ , g · (cid:0) x + ( θ, v ) (cid:1) , λ r Ω (cid:17) s (cid:0) x + ( θ, v ) , x λ , λ t (cid:1) . (87)Here R j ( a, b, . . . ) denotes as before a function of a, b, . . . vanishing to j -th or-der at the origin a = 0 , b = 0 , · · · , and possibly depending on other variables,which are omitted.Given that we have now reduced integration to a shrinking domain where( v , θ, ν ) = O (cid:0) λ − / (cid:1) , and by assumption k ( θ , w ) k / √ λ < C λ − / , wesee from (86) that ∂ r Ψ ( ℓ )4 λ = τ [ σ Q ( x λ , Ω) + O ( τ )] + O (cid:0) λ − / (cid:1) . (88)Since σ Q ( x λ , Ω) is bounded from below by a positive constant, (88) im-plies (cid:12)(cid:12)(cid:12) ∂ r Ψ ( ℓ )4 λ (cid:12)(cid:12)(cid:12) ≥ C | τ | + O (cid:0) λ − / (cid:1) ; thus, where | τ | ≥ D ′ λ − / for some D ′ ≫ (cid:12)(cid:12)(cid:12) ∂ r Ψ ( ℓ )4 λ (cid:12)(cid:12)(cid:12) ≥ D ′ λ − / . Again, integration by parts in d r implies that the corresponding contribution to the asymptotics of (85) isnegligible. Thus we can multiply the amplitude (87) by a further cut-off ofthe form γ (cid:0) λ / τ (cid:1) , which we implicitly incorporate in β λ , without alteringthe asymptotics.Let us now operate the change of variables v v √ λ , ν ν √ λ , θ θ √ λ , τ τ √ λ , (89)and rewrite (85) in the form S ( ̟ ) χ · e − iλ ( · ) ( x λ , x λ ) ( ℓ ) ∼ π d ̟ (cid:18) λ π (cid:19) d +2 λ − d − − e/ (90) · Z D /D Z D /D Z B d ( , Z R e Z R d Z ∞−∞ Z ∞−∞ e i λ e Ψ ( ℓ )4 λ e A ( ℓ )4 λ ·V S ( Ω ) V G (cid:18) ν √ λ (cid:19) V (cid:18) θ √ λ , v √ λ (cid:19) r d d τ d θ d v d ν d Ω d r d t, where e Ψ ( ℓ )4 λ and e A ( ℓ )4 λ are Ψ ( ℓ )4 λ and A ( ℓ )4 λ , respectively, with the rescaled variablesinserted. Explicitly, keeping in mind that Φ( m ) = because m ∈ M ′ , we35ave e Ψ ( ℓ )4 λ (91)= 1 √ λ · n t θ − τ + r h τ σ Q ( x, Ω) + ( θ − θ ) Ω ′ + h w − (cid:0) A ℓ v + ν M ( m ) (cid:1) , Ω i io + 1 λ h rτ (cid:10) ∂ x σ Q ( x, Ω) , w (cid:11) + r O (cid:0) τ (cid:1) + r ω m (cid:0) ν M ( m ) , A ℓ v ) Ω ′ − i t ψ ( v , w ) e iθ/ √ λ − r (cid:10) R ( ν, v , w j ) , Ω (cid:11)i + R (cid:18) τ √ λ , v √ λ , w j √ λ , ν √ λ (cid:19) = 1 √ λ K ( ℓ ) ν + 1 λ H ( ℓ ) ν + R (cid:18) θ √ λ , τ √ λ , v √ λ , w j √ λ , ν √ λ (cid:19) , where K ( ℓ ) ν ( t, θ, r, τ, v , Ω ) (92)=: t θ − τ + r h τ σ Q ( x, Ω) + ( θ − θ ) Ω ′ + h w − (cid:0) A ℓ v + ν M ( m ) (cid:1) , Ω i i ,H ( ℓ ) ν ( t, θ, r, τ, v , Ω ) =: rτ (cid:10) ∂ x σ Q ( x, Ω) , w (cid:11) + r O (cid:0) τ (cid:1) (93)+ r ω m (cid:0) ν M ( m ) , A ℓ v ) Ω ′ − i t ψ ( v , w ) − r (cid:10) R ( ν, v , w j ) , Ω (cid:11) . Here O ( τ ) is to be interpreted as a homogeneous quadratic term, sinceterms in τ of order ≥ R , and similarly for R ( ν, v , w j ). In particular, H ( ℓ ) ν ( t, θ, r, τ, v , Ω ) is homogeneous of degree twoin the rescaled variables.Thus S ( ̟ ) χ · e − iλ ( · ) ( x λ , x λ ) ( ℓ ) = (2 π ) − d − d ̟ λ d +1 − e/ · Z R e I ( ν, λ ) d ν, (94)where I ( ν, λ ) = Z D /D Z D /D Z B d ( , Z R d Z ∞−∞ Z ∞−∞ e i √ λ K ( ℓ ) ν · e iH ( ℓ ) ν + λ R (95) · e A ( ℓ )4 λ · V S ( Ω ) V G (cid:18) ν √ λ (cid:19) V (cid:18) θ √ λ , v √ λ (cid:19) r d d τ d θ d v d Ω d r d t Integration in the rescaled variables is over an expanding ball of radius O (cid:0) λ / (cid:1) . Let us view (95) as an oscillatory integral in √ λ , with real phase K ( ℓ ) ν given by (92), and amplitude B ( ℓ ) ν =: e iH ( ℓ ) ν + λ R · e A ( ℓ )4 λ · V S ( Ω ) V G (cid:18) ν √ λ (cid:19) V (cid:18) θ √ λ , v √ λ (cid:19) r d . (96)36n HLC centered at x ∈ X , the amplitude s ( t, x, y ) ∼ P l ≥ t d − l s l ( x, y ) ofΠ in (22) satisfies s ( x, x ) = π − d . The amplitude a of V ( τ ) in (20), on theother hand, satisfies a (0 , x, x, η ) = 1 / V ( x ) = 2 π (see the discussion following(7) in [P2]). Therefore, recalling the rescaling t λ t and η λ η , we obtainfor B ( ℓ ) ν an asymptotic expansion B ( ℓ ) ν ∼ e iH ( ℓ ) ν r d · χ ̟ ( g ℓ ) · β λ ( ν, θ, τ, v , w j ) (97) · (cid:18) λπ (cid:19) d X k ≥ λ − k/ P k ( ν, θ, τ, v , w j ) , where P k ∈ C ∞ ( t, r, Ω )[ ν, θ, τ, v , w j ] is a polynomial in the rescaled variablesof joint degree ≤ k , P = t d , and β λ is a compactly supported bump function,whose support in the rescaled variables is a ball of radius O (cid:0) λ / (cid:1) . Theexpansion may be integrated term by term, so that I ( ν, λ ) ∼ X k ≥ I ( ν, λ ) k , (98)where I ( ν, λ ) k ∼ λ d − k/ π d χ ̟ ( g ℓ ) · Z D /D Z D /D Z B d ( , Z R d Z ∞−∞ Z ∞−∞ e i √ λ K ( ℓ ) ν · e iH ( ℓ ) ν r d · β λ ( ν, θ, τ, v , w j ) P k ( ν, θ, τ, v , w j ) · d τ d θ d v d Ω d r d t. (99)Let us note the following: Lemma 4.1.
For any k ≥ , P k has the same parity as k .Proof. The asymptotic expansions in t of the amplitude s of Π in (22) andin η of the amplitude a of V ( τ ) in (20) go down by integer steps of degreeof homogeneity. Now consider A ( ℓ )4 λ ( x ′ , x ′′ ) given by A ( ℓ )4 λ in (87) with x λ and x λ replaced by some fixed x ′ , x ′′ ∈ X . In view of substitutions t λ t and η λ η , A ( ℓ )4 λ is given by an asymptotic expansion in descending integerpowers of λ . Therefore the appearance of fractional powers of λ in (97) isdue solely to Taylor expansion in the rescaled variables, and this implies theclaim.The proof of the following is left to the reader: Lemma 4.2.
The following holds: . K ( ℓ ) ν has a unique stationary point P ℓ ( ν ) = ( t , θ , r , τ , v ( ℓ )0 ( ν ) , Ω )=: (cid:18) ς T ( x ) , , ς T ( x ) , , A − ℓ (cid:0) w − ν M ( m ) (cid:1) , (cid:19) .
2. Let us define the column vector D =: ∂ Ω ln σ Q | ( x,α x ) . Then the Hessian matrix at the critical point is
Hess( K ( ℓ ) ν ) P ℓ ( ν ) = t t − t t − ς T ( x ) t t ς T ( x ) 0 t D t [0] − A tℓ /ς T ( x ) − A ℓ /ς T ( x ) − θ I d .
3. The determinant of
Hess( K ( ℓ ) ν ) P ℓ ( ν ) is det (cid:16) Hess( K ( ℓ ) ν ) P ( ℓ )0 ( ν ) (cid:17) = ς T ( x ) − d .
4. The signature of
Hess( K ( ℓ ) ν ) P ( ℓ )0 ( ν ) is zero.5. At the critical point, we have K ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) = θ /ς T ( x ) and i H ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) (100)= 1 ς T ( x ) h i ω m (cid:0) ν M ( m ) , w ) + ψ (cid:0) w − ν M ( m ) , A ℓ w (cid:1)i .
6. The inverse of the Hessian matrix at the critical point is:
Hess( K ( ℓ ) ν ) − P ℓ ( ν ) = /ς T ( x ) D t A ℓ t t t /ς T ( x ) D t A ℓ t /ς T ( x ) 0 1 /ς T ( x ) 0 t t A tℓ D 0 A tℓ D 0 θ ς T ( x ) I d − ς T ( x ) A tℓ − ς T ( x ) A ℓ [0] (recall that A ℓ A tℓ = I d ). . The third order remainder of the phase at the critical point is R (3) K = r h τ · σ Q ( x, Ω) (1) − θ · (cid:0) Ω ′ (cid:1) (1) i , where σ Q ( x, Ω) (1) and (cid:0) Ω ′ (cid:1) (1) are the first order remainders at the origin ∈ R d as functions of Ω . In particular, as far as the rescaled variablesare concerned, it only depends on τ and θ , and is linear in them. Therefore, the gradient of K ( ℓ ) ν is bounded below in norm, uniformly in ν and ℓ , by a fixed positive constant when P = ( t, θ, r, τ, v , Ω ) ∈ R × R d × B d ( ,
1) remains at distance ≥ a from P , where a > κ ∈ C ∞ (cid:0) R × R d × B d ( , (cid:1) be identically ≡ ≤ a fromthe origin, and set κ ν ( P ) =: κ (cid:0) P − P ( ℓ )0 ( ν ) (cid:1) . Then we can write I ( ν, λ ) = I ( ν, λ ) ′ + I ( ν, λ ) ′′ , where I ( ν, λ ) ′ and I ( ν, λ ) ′′ are given by (95) with theamplitude multiplied by κ ν ( P ) and 1 − κ ν ( P ), respectively. Lemma 4.3.
Uniformly in ν , we have I ( ν, λ ) ′′ = O ( λ −∞ ) as λ → + ∞ .Proof. Where 1 − κ ν = 0, we can ‘integrate by parts’ in d t d θ d r d τ d v d Ω using the previous remark, and noting that the integrand is compactly sup-ported (with an expanding support). At each step, as in the usual proofof the stationary phase Lemma, we get a factor λ − ; furthermore, differ-entiation of the amplitude, and of the coefficients of the first order ope-rator involved, introduces in view of the factor e iH ( ℓ ) ν and (93) a factor O (cid:0) λ / λ / (cid:1) = O (cid:0) λ / (cid:1) . Thus we obtain at each step a factor O (cid:0) λ − / (cid:1) ,whence after N step a factor O (cid:0) λ − N/ (cid:1) . Since integration is over a ball ofradius O (cid:0) λ / (cid:1) , the statement follows.Hence as far at the asymptotics are concerned we need only consider I ( ν, λ ) ′ , which can be estimated applying the stationary phase Lemma. Letus first note the following, whose proof is also left to the reader: Lemma 4.4.
Let the components w j h , w j v , w j t be as in (11). Then (100)may be rewritten i H ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) = T ℓ ( w , w )+ 1 ς T ( x ) h i ω (cid:0) ν M ( m ) , w + A ℓ w (cid:1) − (cid:13)(cid:13)(cid:13) ν M ( m ) (cid:13)(cid:13)(cid:13) i , where T ℓ ( w , w ) =: 1 ς T ( x ) h ψ (cid:0) w , A ℓ w (cid:1) − (cid:13)(cid:13) w − A ℓ w (cid:13)(cid:13) i + iς T ( x ) h ω (cid:0) w , w (cid:1) − ω (cid:0) w , w (cid:1)i . emark . Here ω and k · k are the standard symplectic structure andnorm, respectively, on R d , and they correspond to the symplectic structure ω m and norm k · k m on T m M in the given HLC system centered at x .Let us set D =: (cid:0) ∂ t , ∂ θ , ∂ r , ∂ τ , ∂ v , ∂ Ω (cid:1) T and L K =: D D , Hess( K ( ℓ ) ν ) − P ℓ ( ν ) D E (101)= 2 (cid:20) ∂ ∂t∂θ + 1 ς T ( x ) ∂ ∂t∂τ + 1 ς T ( x ) ∂ ∂r∂τ + D t A ℓ (cid:18) ∂ ∂t∂ v + ∂ ∂r∂ v (cid:19) − ς T ( x ) (cid:28) ∂∂ Ω , A ℓ ∂∂ v (cid:29)(cid:21) + θ ς T ( x ) (cid:28) ∂∂ v , ∂∂ v (cid:29) . By the stationary phase Lemma [H], each I ( ν, λ ) k is given by an asymp-totic expansion of the following form I ( ν, λ ) k ∼ (2 π ) d π − d λ − − k/ ς T ( x ) d − χ ̟ ( g ℓ ) · e i √ λ θ /ς T ( x ) · X j ≥ λ − j/ i j L j (cid:16) e iH ( ℓ ) ν r d · P k (cid:17)(cid:12)(cid:12)(cid:12) P (102)where L j ( ϕ ) =: X a − b = j X a ≥ b a a ! b ! L aK (cid:18) ϕ · (cid:16) R (3) K (cid:17) b (cid:19) ; (103)notice that β λ is identically equal to 1 on the present restricted domain ofintegration. Furthermore a ≤ j in the range of summation.The leading term in (102) is therefore(2 π ) d ) π d λ − − k/ ς T ( x ) − − d χ ̟ ( g ℓ ) · e i √ λ θ /ς T ( x )+ i H ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) · R k ( P ) , where R k ( P ) is a polynomial of degree ≤ k in the rescaled variables.Since furthermore L K is homogeneous of degree − H ( ℓ ) ν is homogeneous of degree 2 and R (3) K is linear in them,applying (103) with ϕ = e iH ( ℓ ) ν r d · P k , we conclude that the general term in(102) is a linear combination of terms of the form λ − − ( k + j ) / e i √ λ θ /ς T ( x )+ i H ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) P a,b ( θ , w , w , ν M ( m )) , where P a,b has parity − a + b + k ≡ a − b + k = j + k .40n view of the last summand in (101) and of (93), where a − b = j ,2 a ≥ b , and P a,b ( P ) is now a polynomial in ( ν, θ , w , w ) of joint degree ≤ k + 3 a + b ≤
11 ( k + j ). Thus (cid:12)(cid:12)(cid:12)(cid:12) λ − − ( k + j ) / e i √ λ θ /ς T ( x )+ i H ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) P a,b ( θ , w , w , ν M ( m )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k,j λ − − ( k + j ) λ ( k + j ) = C k,j λ − − ( k + j ) . Since integration in d ν is on a domain of radius O (cid:0) λ / (cid:1) the expansionmay be integrated term by term. Therefore, we obtain for (94) an asymptoticexpansion S ( ̟ ) χ · e − iλ ( · ) ( x λ , x λ ) ( ℓ ) (104) ∼ (2 π ) π d λ d − e/ ς T ( x ) − (1+ d ) d ̟ χ ̟ ( g ℓ ) e i √ λ θ /ς T ( x ) · X k ≥ λ − k/ Z R e ̺ λ ( ν ) e i H ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) Q k (cid:0) θ , w , w , ν (cid:1) d ν for certain polynomials Q k , of degree ≤ k and parity k ; we have Q = 1.Furthermore, ̺ λ ( ν ) is again a bump function supported in an expanding ballof radius O (cid:0) λ / (cid:1) .Let us consider the Gaussian integrals in (104). First of all, for k = 0 wehave Z R e ̺ λ ( ν ) e i H ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) d ν ∼ Z R e e i H ( ℓ ) ν (cid:0) P ( ℓ )0 ( ν ) (cid:1) d ν = e T ℓ ( w , w ) (105) · Z R e exp (cid:18) ς T ( x ) (cid:20) i ω (cid:0) ν M ( m ) , w + A ℓ w (cid:1) − (cid:13)(cid:13)(cid:13) ν M ( m ) (cid:13)(cid:13)(cid:13) (cid:21)(cid:19) d ν. Recall from the discussion preceding (55) that ν ∈ R e represents thecoordinates of ξ ∈ g with respect to a chosen orthonormal basis ( ξ j ) of g . If C m is as in § ς T ( x ) (cid:20) i ω (cid:0) ν M ( m ) , w + A ℓ w (cid:1) − (cid:13)(cid:13)(cid:13) ν M ( m ) (cid:13)(cid:13)(cid:13) (cid:21) (106)= 1 ς T ( x ) (cid:20) − i ν t C t J (cid:0) w + A ℓ w (cid:1) − h C ν, C ν i (cid:21) = − i a t J (cid:0) w + A ℓ w (cid:1)p ς T ( x ) − h a , a i (107)where J is the matrix of the standard complex structure, and in the latterline we have set a =: C ν/ p ς T ( X ). 41e then obtain, in view of (35), Z R e exp (cid:18) ς T ( x ) (cid:20) i ω (cid:0) ν M ( m ) , w + A ℓ w (cid:1) − (cid:13)(cid:13)(cid:13) ν M ( m ) (cid:13)(cid:13)(cid:13) (cid:21)(cid:19) d ν (108)= ς T ( x ) e/ C m ) Z R e exp − i a t J (cid:0) w + A ℓ w (cid:1)p ς T ( x ) − k a k ! d a = (2 π ) e/ ς T ( x ) e/ | G Mm | V M eff ( m ) exp (cid:18) − ς T ( x ) k w + A ℓ w k (cid:19) . Now T ℓ ( w , w ) − ς T ( x ) k w + A ℓ w k (109)= 1 ς T ( x ) n i h ω (cid:0) w , w (cid:1) − ω (cid:0) w , w (cid:1)i + ψ (cid:0) w , A ℓ w (cid:1) − (cid:13)(cid:13) w (cid:13)(cid:13) − k A ℓ w k o = Q T vt ( w , w ) + Q T h ( w , A ℓ w ) , since k A ℓ w k = k w k by the unitarity of A ℓ .Thus the leading term in (104) is(2 π ) e/ π d λ d − e/ | G Mm | V M eff ( m ) d ̟ χ ̟ ( g ℓ ) (110) · ς T ( x ) e/ − (1+ d ) exp (cid:18) i √ λ θ ς T ( x ) + Q T vt ( w , w ) + Q T h ( w , A ℓ w ) (cid:19) . More generally, let us write every Q k (cid:0) θ , w , w , ν (cid:1) in (104) as a polyno-mial in ν with coefficients in C [ θ , w , w ]. If a I is a mononomial of degree | I | ≥
0, then Z R e a I exp (cid:18) − i a t x − k a k (cid:19) d a = P I ( x ) e − k x k , where P I is a polynomial of degree | I | , in general non-homogeneous but ofthe same parity as | I | . Therefore, the k -th term in (104) has the form2 π · e/ | G Mm | V M eff ( m ) d ̟ χ ̟ ( g ℓ ) ς T ( x ) − (1+ d − e/ (cid:18) λπ (cid:19) d − e/ λ − k/ (111) · R k ( θ , w , w ) exp (cid:18) i √ λ θ ς T ( x ) + Q T vt ( w , w ) + Q T h ( w , A ℓ w ) (cid:19) , where R k is a polynomial of degree k and parity k . Theorem 1.2 now followsby Remark 1.2. 42 Proof of Corollary 1.2
Proof.
Locally near any given x ∈ X , a HLC system centered at x may bedeformed smoothly with x . More precisely, there exists an open neighborhood Y ⊆ X of x and a C ∞ mapΓ : ( y, θ, v ) ∈ Y × ( − π, π ) × B d ( , δ ) γ y ( θ, v ) = y + ( θ, v ) , where γ y is a HLC system on X centered at y . It may be assumed that Y = π − ( N ), where N ⊆ M is an open neighborhood of m =: π ( x ), and thatfor any y ∈ X we have (cid:0) y + ( θ, (cid:1) + ( ϑ, v ) = y + ( θ + ϑ, v ), when both sidesare defined.We may then find a finite open cover { M ′ j } of M ′ such that, setting X ′ j =: π − (cid:0) M ′ j (cid:1) , we have maps Γ j : X ′ j × ( − π, π ) × B d ( , δ ) → X as above. For any x ∈ X ′ j , recall the unitary isomorphism of R e with the summand R e ver in (14),which associated to any w ∈ R e a transverse vector w t ∈ R e ver ∼ = T m M trasv .We obtain S -equivariant maps Λ j : X ′ j × B e ( , δ ) → X , given by Λ j ( x, w ) =: x + w t , (112)and it is easily seen that these maps are local diffeomorphisms, and actuallydiffeomorphisms onto their images X j =: Λ j (cid:0) X ′ j × B e ( , δ ) (cid:1) if δ > X ′ j , meaning that the pull back of the volume form is Λ ∗ j (d V X ) = E j ( x, w ) d w d V X ′ , (113)where d V X ′ is the volume form on X ′ for the induced Riemannian structure(and orientation), and E j ( x, ) = 1 identically.Thus e X =: S j X j is an open neighborhood of X ′ , ( X j ) is an open coverof e X , and each X j is S -invariant. Furthermore, ( X ′ j ) is an open cover of X ′ ,where X ′ j =: X j ∩ X ′ .Let ( ̺ j ) be a partition of unity on e X subordinate to ( X j ); we may assumewithout loss that ̺ j is S -invariant, hence the pull-back of a C ∞ -functionon M , that we shall still denote by ̺ j . We may assume without loss that ̺ j ( x + w t ) = ̺ j ( x − w t ).Also, let ̺ ∈ C ∞ (cid:0) e X (cid:1) be identically equal to 1 on a small tubular neigh-borhood of X ′ . 43hen we havetrace (cid:16) S ( ̟ ) χ · e − iλ ( · ) (cid:17) = Z X S ( ̟ ) χ · e − iλ ( · ) ( y, y ) d V X ( Y ) (114) ∼ Z e X ̺ ( y ) S ( ̟ ) χ · e − iλ ( · ) ( y, y ) d V X ( y ) = X j Z X j ̺ ( y ) ̺ j ( y ) S ( ̟ ) χ · e − iλ ( · ) ( y, y ) d V X ( y )= X j Z X ′ j Z B e ( ,δ ) ̺ ( x + w t ) ̺ j ( x + w t ) S ( ̟ ) χ · e − iλ ( · ) ( x + w t , x + w t ) ·E j ( x, w ) d w d V X ′ ( x )= λ − e/ X j Z X ′ j Z B e ( ,δ ) ̺ (cid:18) x + w t √ λ (cid:19) ̺ j (cid:18) x + w t √ λ (cid:19) · S ( ̟ ) χ · e − iλ ( · ) (cid:18) x + w t √ λ , x + w t √ λ (cid:19) E j (cid:18) x, w √ λ (cid:19) d w d V X ′ ( x ) , where we have performed the change of coordinates w w / √ λ .We can now make use of the local asymptotic expansion from Corollary1.1. Using that Z R e exp (cid:18) − ς T ( x ) k w k (cid:19) d w = (cid:16) π (cid:17) e/ ς T ( x ) e/ , and in view of the parity statement on the F j ’s, we gettrace (cid:16) S ( ̟ ) χ · e − iλ ( · ) (cid:17) ∼ π · dim( V ̟ ) (cid:18) λπ (cid:19) d − e · Z X ′ F ( λ, x ) d V X ′ ( x ) , where F ( λ, · ) : X ′ → R has an asymptotic expansion F ( λ, x ) ∼ a Φ ,̟ ( m ) V X eff ( m ) · ς T ( x ) − ( d +1 − e ) · " X j ≥ λ − j β j ( x ) ;here a Φ ,̟ is as in Definition 1.8. We concludetrace (cid:16) S ( ̟ ) χ · e − iλ ( · ) (cid:17) ∼ π · dim( V ̟ ) (cid:18) λπ (cid:19) d − e a gen (Φ , ̟ ) Γ(Φ , ̟ ) · " X j ≥ λ − j E j , where Γ(Φ , ̟ ) =: Z X ′ V X eff ( m ) ς T ( x ) − ( d +1 − e ) d V X ′ ( x ) . Proof of Corollary 1.3
Proof.
As in the proof of Theorem 1.1 of [P2], we shall adapt the classicalTauberian argument in §
12 of [GrSj]. Given Corollary 1.2, we have that for λ ≫ N ( ̟ ) T ( λ + 1) − N ( ̟ ) T ( λ ) ≤ C λ d − e , and therefore for any τ and λ > (cid:12)(cid:12) N ( ̟ ) T ( λ − τ ) − N ( ̟ ) T ( λ ) (cid:12)(cid:12) ≤ C ′ (cid:0) | τ | (cid:1) d − e +1 λ d − e . Hence Z + ∞−∞ h N ( ̟ ) T ( λ − τ ) − N ( ̟ ) T ( λ ) i χ ( τ ) d τ = O (cid:0) λ d − e (cid:1) . (115)Let us consider the spectral measure d T ( ̟ ) = P j δ λ ( ̟ ) j on R , where δ a isDirac’s delta at a ∈ R . Thus N ( ̟ ) T ( λ ) = Z λ −∞ d T ( ̟ ) ( η ) , X j b χ (cid:0) λ − λ ( ̟ ) j (cid:1) = Z + ∞−∞ b χ ( λ − η ) d T ( ̟ ) ( η ) . Now we set G ( λ ) =: R λ −∞ b χ ( τ ) d τ , and compute R + ∞−∞ G ( λ − η ) d T ( ̟ ) ( η )in two different manners. On the one hand, by change of variable and theTonelli-Fubini Theorem, Z + ∞−∞ G ( λ − η ) d T ( ̟ ) ( η ) (116)= Z + ∞−∞ (cid:20)Z λ − η −∞ b χ ( τ ) d τ (cid:21) d T ( ̟ ) ( η ) = Z + ∞−∞ (cid:20)Z λ −∞ b χ ( τ − η ) d τ (cid:21) d T ( ̟ ) ( η )= Z λ −∞ (cid:20)Z + ∞−∞ b χ ( τ − η ) d T ( ̟ ) ( η ) (cid:21) d τ = Z λ −∞ X j b χ (cid:0) τ − λ ( ̟ ) j (cid:1) d τ = Z λ −∞ trace (cid:16) S ( ̟ ) χ · e − iτ ( · ) (cid:17) d τ = 2 π d − e + 1 · dim( V ̟ ) a gen (Φ , ̟ ) Γ(Φ , ̟ ) (cid:18) λπ (cid:19) d − e +1 + O (cid:0) λ d − e (cid:1) . On the other hand, letting H denote the Heavyside function, we also45ave: Z + ∞−∞ G ( λ − η ) d T ( ̟ ) ( η ) = Z + ∞−∞ (cid:20)Z λ − η −∞ b χ ( τ ) d τ (cid:21) d T ( ̟ ) ( η )= Z + ∞−∞ (cid:20)Z + ∞−∞ H ( λ − η − τ ) b χ ( τ ) d τ (cid:21) d T ( ̟ ) ( η )= Z + ∞−∞ (cid:20)Z + ∞−∞ H ( λ − η − τ ) d T ( ̟ ) ( η ) (cid:21) b χ ( τ ) d τ = Z + ∞−∞ N ( ̟ ) T ( λ − τ ) b χ ( τ ) d τ = N ( ̟ ) T ( λ ) Z + ∞−∞ b χ ( τ ) d τ + Z + ∞−∞ h N ( ̟ ) T ( λ − τ ) − N ( ̟ ) T ( λ ) i b χ ( τ ) d τ = 2 π N ( ̟ ) T ( λ ) + O (cid:0) λ d − e (cid:1) , (117)where in the last line we have used (115) and that1 = χ (0) = 12 π Z + ∞−∞ b χ ( τ ) d τ by our choice of χ and the Fourier inversion formula. Comparing (116) and(117), we conclude that N ( ̟ ) T ( λ ) = πd − e + 1 · dim( V ̟ ) a gen (Φ , ̟ ) Γ(Φ , ̟ ) (cid:18) λπ (cid:19) d − e +1 + O (cid:0) λ d − e (cid:1) , as claimed. Proof.
By first integrating along the orbits of e µ on X ′ (that is, the fibers ofthe projection X ′ → b X ) and then on the base b X , we have in view of (16):Γ(Φ , ̟ ) = Z b X b ς T (cid:0)b x (cid:1) − ( d − e +1) d V b X (cid:0)b x (cid:1) . (118)Furthermore, we have b Σ =: n ( b x, r b α b x ) : b x ∈ b X, r > o , b Σ is given by the analogue of (32): ω b Σ = d ( r b α ) = d r ∧ b α + r d b α = d r ∧ b α + 2 r b ω, where we omit the pull-back symbols in front of b α and b ω .Since b Σ has dimension 2 ( d − e + 1), the symplectic volume form isd V b Σ = 1( d − e + 1)! ω ∧ ( d − e +1) b Σ = (2 r ) d − e d − e )! ω ∧ ( d − e ) c M ∧ d r ∧ b α = (2 r ) d − e d V c M ∧ d r ∧ b α = − d − e +1 π r d − e d V b X ∧ d r. Since b σ T = r b ς T , b Σ = n ( b x, r b α b x ) ∈ b Σ : b x ∈ b X, r < / b ς T (cid:0)b x (cid:1)o and so in view of (118) its symplectic volume isvol (cid:16)b Σ (cid:17) = 2 d − e +1 π Z b X Z / c ς T (cid:0) b x (cid:1) r d − e d r d V b X = 2 d − e +1 πd − e + 1 Z b X b ς T ( b x ) − ( d − e +1) d V b X = 2 d − e +1 πd − e + 1 Γ(Φ , ̟ ) . Inserting this in the estimate of Corollary 1.3 we obtain N ( ̟ ) T ( λ ) = dim( V ̟ ) a gen (Φ , ̟ ) vol (cid:16)b Σ (cid:17) (cid:18) λ π (cid:19) d − e +1 + O (cid:0) λ d − e (cid:1) , as claimed, since a gen (Φ , ̟ ) = dim( V ̟ ) in this case. References [BSZ] P. Bleher, B. Shiffman, S. Zelditch,
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