Charles H Conley
email: conley@unt.edu

Valentin Ovsienko
email: ovsienko@math.univ-lyon1.fr

LINEAR DIFFERENTIAL OPERATORS ON CONTACT MANIFOLDS

Keywords: Contact geometry, differential operators, Heisenberg calculus, infinitesimal characters, differential invariants

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Linear differential operators on contact manifolds

Introduction

The space D(M ) of linear differential operators on a smooth manifold M has a rich geometric structure. By the geometry of D(M ), we understand its structure as a module over the group of all diffeomorphisms of M , and thereby also over its Lie algebra, the space Vect(M ) of smooth vector fields on M . The most interesting geometric properties of D(M ) are described by its invariants under the group of diffeomorphisms.

Additional structure on M leads to a smaller group of diffeomorphisms, and therefore a richer set of invariants of D(M ). Contact manifolds provide an important class of examples of geometric structures. In this paper we study the geometric properties of D(M ) viewed as a module over the Lie algebra K(M ) of all contact vector fields on M . This viewpoint fits into the general framework of Heisenberg calculus, see [START_REF] Beals | Calculus on Heisenberg manifolds[END_REF][START_REF] Epstein | Contact degree and the index of Fourier integral operators[END_REF][START_REF] Van Erp | The Atiyah-Singer index formula for subelliptic operators on contact manifolds, Part I[END_REF], where the geometric structure is a codimension-1 distribution in T (M ).

Our first main result is the association of a contact vector field to an arbitrary second order linear differential operator in a contact-invariant manner. Although we do not carry out the investigation here, this could provide a means to associate topological invariants to second order operators. We generalize the result to differential operators of arbitrary order, associating to each a certain tensor density on M . This tensor density is independent of the symbol of the operator and may be thought of as a partial "subsymbol".

By a tensor density, we mean a section of a power of the determinant line bundle. In fact, we state our results in the more general context of the spaces D λ,µ (M ) of differential operators between such line bundles, rather than simply for differential operators on functions. An interesting feature is appearance of contact resonances, that is, of special powers of the determinant line bundle for which the geometric properties of differential operators are more complicated. These resonances were already observed in [START_REF] Frégier | Decomposition of symmetric tensor fields in the presence of a flat contact projective structure[END_REF]. Let us mention that the usual case where λ = µ is non-resonant.

Our second main result is the existence of a filtration refining the usual filtration given by the Heisenberg calculus. Recall that differential operators on a contact manifold have a contact order, in which vector fields tangent to the distribution are of order 1, and contact vector fields are of order 2. We introduce a contact-invariant filtration on D λ,µ (M ) for which, roughly speaking, tangential vector fields have order 1 and contact vector fields have order 3. However, this filtration is not compatible with composition. We prove its existence in the non-resonant case.

We remark that there is also an invariant double filtration on the space of differential forms on M [START_REF] Rumin | Formes différentielles sur les variétés de contact[END_REF]. In some ways, the situation for contact manifolds appears to be analogous to that for foliated manifolds. This may at first be surprising, as contact distributions are completely non-integrable, but such analogies have been observed before [START_REF] Eliashberg | Confoliations[END_REF].

Although our results and their applications are essentially geometric, the proofs are algebraic. The considerations are local, so we may work in the Euclidean case, replacing M by R m , where m = 2ℓ + 1. Moreover, we need only consider the Lie algebra of polynomial contact vector fields on R m , which is the classical infinite dimensional Cartan algebra K m ; see [START_REF] Fuchs | Cohomology of Infinite Dimensional Lie Algebras[END_REF].

Our main theorems are proven using certain underlying structural results concerning the cohomology of K m with coefficients in spaces of differential operators between refined symbol modules. These results in turn are obtained using a quantization map which is equivariant with respect to the projective subalgebra s m of K m , a maximal subalgebra isomorphic to sp 2(ℓ+1) . We prove the existence and uniqueness of this quantization map using the description of the infinitesimal characters of s m given by the Harish-Chandra homomorphism. We also calculate the map explicitly.

An s m -module is said to have an infinitesimal character if the center Z(s m ) of the universal enveloping algebra of s m acts on it by scalars. The infinitesimal character is then the resulting homomorphism from Z(s m ) to C. If an s m -module has a finite Jordan-Hölder composition series of modules with distinct infinitesimal characters, then the module splits as the direct sum of its composition series modules.

Let us discuss at this point the role of infinitesimal characters in other forms of quantization. The Casimir element is the best known and simplest element of Z(s m ). It turns out that for the contact projective quantization studied in this paper, it is not sufficient to consider the eigenvalues of the Casimir element alone, because there are fine symbol modules with distinct infinitesimal characters but identical Casimir eigenvalues. This is in contrast with the situation for projective quantization with respect to the full vector field Lie algebra Vect(R m ), whose projective subalgebra is sl m+1 . The full principal symbol modules have infinitesimal characters under the action of sl m+1 , and these infinitesimal characters are distinct if and only if their Casimir eigenvalues are different [START_REF] Lecomte | On the cohomology of sl(m+1, R) acting on differential operators and sl(m+1, R)-equivariant symbol[END_REF]. Therefore in this setting there is no need to consider infinitesimal characters.

For conformal quantization, one replaces the projective subalgebra with the conformal subalgebra o p+1,q+1 , a maximal subalgebra of Vect(R p+q ). As was first observed in [START_REF] Duval | Conformally equivariant quantization: existence and uniqueness[END_REF], in this setting the Casimir element of o p+1,q+1 is not sufficient to detect distinct infinitesimal characters among these submodules. Complete results concerning the existence and uniqueness of conformal quantization for differential operators between tensor density modules have recently been obtained in [START_REF] Silhan | Conformally equivariant quantization -towards complete classification[END_REF] and [START_REF] Michel | Conformally equivariant quantization -a complete classification[END_REF]. It would be interesting to determine to what extent infinitesimal characters can be used to replicate them.

The crucial property that allows us to apply algebraic results in the geometric situation of an arbitrary contact manifold M is the uniqueness of the sp 2(ℓ+1) -equivariant quantization map. For example, the subsymbol is first defined locally in Darboux coordinates. Its uniqueness then implies that it is defined globally on M . This paper is organized as follows. In Section 2 we define the modules of tensor densities, differential operators, and symbols, and formulate our main results. In Section 3 we fix local Darboux coordinates and review local properties of the contact Lie algebra. In Section 4 we study the modules of Section 2 under the action of the projective subalgebra, using infinitesimal characters to compute contact resonances. In Section 5 we prove the existence and uniqueness of the projective quantization, the natural projective equivalence from symbols to differential operators. Section 6 contains the proofs of two of our main results: the existence and uniqueness of the subsymbol and of the fine filtration. Section 7 gives the explicit formula for the projective quantization and the subsymbol and proves our third main result.

Main results

Fix m = 2ℓ + 1 odd, and let M be a smooth m-dimensional manifold equipped with a contact distribution Ξ: a completely non-integrable distribution of codimension 1. As usual, locally we define the contact structure in terms of a contact form θ whose kernel is Ξ. The non-integrability of Ξ is equivalent to the fact that θ ∧ (dθ) ℓ is a local volume form. We define the subspace Tan(M ) of Vect(M ) to consist of the sections of Ξ, that is, the vector fields annihilated by θ. We will refer to such vector fields as tangential vector fields.

2.1. Definitions. We will use the following notation throughout this paper. For X ∈ Vect(M ), we write L(X) for the associated Lie derivative. The non-negative integers will be denoted by N, and the positive integers by Z + . For x ∈ R, we use the floor notation for the greatest integer ≤ x and the ceiling notation for the least integer ≥ x:

⌊x⌋ := sup{n ∈ Z : n ≤ x}, ⌈x⌉ := inf{n ∈ Z : n ≥ x}.
Within Vect(M ) we have the Lie subalgebra K(M ) of contact vector fields, those which preserve Ξ. Contact vector fields are characterized locally as those whose Lie derivatives preserve the conformal class of θ. More precisely, a vector field

X on M is contact if (1)
L(X)θ = 1 ℓ+1 Div(X)θ, where Div is the divergence with respect to the volume form θ ∧ (dθ) ℓ . The complete nonintegrability of Ξ translates to

Vect(M ) = K(M ) ⊕ Tan(M ).
This decomposition is invariant under the Lie action of K(M ). Observe that K(M ) is not invariant under multiplication by functions, and Tan(M ) is not a Lie algebra. Let π : Vect(M ) → K(M ) be the projection along Tan(M ).

We now make several definitions valid for arbitrary (not necessarily contact) manifolds.

Definition.

(i) For λ ∈ C, let |Λ m T * (M )| λ be the line bundle of homogeneous functions of degree λ on the determinant bundle. The space F λ (M ) of tensor densities of degree λ consists of the smooth sections of |Λ m T * (M )| λ with complex coefficients. It is a module for Vect(M ), and we write L λ (X) for the action of a vector field X on it. (ii) Let D λ,µ (M ) be the space of differential operators from F λ (M ) to F µ (M ), and let L λ,µ be the natural action of Vect(M ) on it. For k ∈ N, let D k λ,µ (M ) be the subspace of operators of order ≤ k. The spaces D k λ,µ (M ) comprise the order filtration of D λ,µ (M ) and are invariant under Vect(M ). (iii) We write δ for the difference between µ and λ:

δ := µ -λ.
(iv) The space of principal symbols of degree k is the quotient

S k δ (M ) := D k λ,µ (M )/D k-1 λ,µ (M ).
It is well-known that its Vect(M )-module structure depends only on δ.

(v) The principal symbol is the natural projection

σ k λ,µ : D k λ,µ (M ) → S k δ (M ).
Let us give some natural examples of tensor density modules. The simplest is C ∞ (M ), which is F 0 (M ). In the contact setting, the following facts are well-known.

• The adjoint action of the Lie algebra K(M ) of contact vector fields on itself is equivalent to F -1 ℓ+1 (M ). In other words, there is a K(M )-equivalence

X : F -1 ℓ+1 (M ) → K(M ), associating a contact vector field X ϕ to each element ϕ of F -1 ℓ+1 (M ).
The tensor density ϕ is called the contact Hamiltonian of X ϕ . The notion of the contact Hamiltonian is independent of the choice of a contact form θ. However, fixing θ one can (locally) identify tensor densities and functions and think of a contact Hamiltonian as of a function.

• The conformal class C ∞ (M )θ of the contact form θ is equivalent to F 1 ℓ+1 (M ) as a K(M )- module.
In fact, the second statement follows from (1), and the first follows from Lemma 3.1 below. Thus the K(M )-modules of contact Hamiltonians and contact forms are dual over C ∞ (M ). We remark that the algebraic direct sum λ F λ (M ) of all tensor density modules is a Poisson algebra under the Lagrange bracket.

The space D λ,µ (M ) generalizes D 0,0 (M ), which is the usual space of differential operators acting on functions. Geometric properties of D λ,µ (M ) vary with the parameters, and the structure of D λ,µ (M ) viewed as a K(M )-module can be special for certain values of λ and µ. Let us stress the fact that differential operators between tensor densities appear naturally in many geometric situations. We mention for example the classical notion of the conformally invariant Laplace operator, also known as the Yamabe Laplacian, which is an element of D 1 2 -1 m , 1 2 + 1 m (M ). The case λ + µ = 1 is particularly special. This is the case where the notions of symmetric and skewsymmetric operators are well-defined. More generally, if A ∈ D λ,µ (M ), then the adjoint operator A * belongs to D 1-µ,1-λ (M ).

We now recall the classical notion of the Heisenberg order of a differential operator on a contact manifold; see for example [vE10] and references therein.

Definition.

(i) The space of differential operators of Heisenberg order ≤ d is

P d λ,µ (M ) := Span T c • L λ (Y 1 ) • • • • • L λ (Y t ) : T c ∈ D c λ,µ (M ), Y i ∈ Tan(M ), 2c + t ≤ d . The spaces P d λ,µ (M ), comprise the Heisenberg filtration of D λ,µ (M ). They are invariant under K(M ). (ii) The bifiltration D k,d λ,µ (M ) := D k λ,µ (M ) ∩ P d λ,µ (
M ) gives rise to the fine symbol modules:

Σ k,d δ (M ) := D k,d λ,µ (M )/ D k-1,d λ,µ (M ) + D k,d-1 λ,µ (M ) . (iii) The fine symbol is the corresponding projection fσ k,d λ,µ : D k,d λ,µ (M ) → Σ k,d δ (M ).
In the simplest case k = 1 and δ = 0, Σ 1,1 0 (M ) is Tan(M ) and Σ 1,2 0 (M ) is K(M ). Here fσ 1,2 λ,λ is nothing but the projection π defined above. More generally, it follows from Proposition 3.

3 below that Σ k,2k δ (M ) ∼ = F δ-k ℓ+1 (M ).
Therefore fσ k,2k λ,µ may be regarded as a K(M )-equivariant linear projection from D k λ,µ (M ) to F δ-k ℓ+1 (M ). This type of equivariant assignment of a tensor density to a differential operator is known as a differential invariant.

Definition. We say that δ is contact-resonant if it lies in the set

1 ℓ + 1 + n 2(ℓ + 1) n ∈ N .
We will see in Section 4.5 that contact resonances arise from the representation theory of sp 2(ℓ+1) .

2.2. The subsymbol. Our first main theorem gives a new contact differential invariant.

Theorem A. If δ is not contact-resonant, then there exists a unique K(M )-equivariant linear map sσ k λ,µ :

D k λ,µ (M ) → Σ k-1, 2(k-1) δ (M ) whose restriction to D k-1 λ,µ (M ) is fσ k-1, 2(k-1) λ,µ
.

We refer to sσ k λ,µ as the contact subsymbol. We will give an explicit formula for it in Proposition 7.4. It may be regarded as a K(M )-equivariant projection from D k λ,µ (M ) to F δ-k-1 ℓ+1 (M ). We remark that in the general self-adjoint case, where λ + µ = 1 and k is arbitrary, the existence of such a differential invariant is obvious. Indeed, for T in

D k λ,µ (M ), the operator T -(-1) k T * is in D k-1
λ,µ (M ), and so can be projected to Σ

k-1, 2(k-1) δ (M ). Since F -1 ℓ+1 (M ) is equivalent to K(M )
, the case that k = 2 and µ = λ is of particular interest, as there the differential invariant given by the contact subsymbol may be viewed as a contact vector field. In other words, for all λ ∈ C, the subsymbol sσ 2 λ,λ defines a K(M )-equivariant projection from D 2 λ,λ (M ) to K(M ). In order to give an intrinsically defined and manifestly contact-invariant formula for sσ 2 λ,λ , observe that any second order differential operator can be represented as a linear combination of compositions of vector fields. On contact manifolds, contact vector fields and tangential vector fields are intrinsically distinguished. Thus we are led to express an arbitrary second order operator on F λ (M ) as a linear combination of operators of the form

(2) T = L λ (X ϕ1 ) • L λ (X ϕ2 ) + L λ (X ϕ3 ) • L λ (Y 1 ) + L λ (Y 2 ) • L λ (Y 3 ) +L λ (X ϕ4 ) + L λ (Y 4 ) + f,
where the ϕ i are arbitrary contact Hamiltonians, the Y i are tangential vector fields, and f is a function.

Theorem B. The subsymbol sσ 2 λ,λ (T ) is the contact vector field

1 2 X ϕ1 , X ϕ2 -ℓ+1 ℓ+2 λ -1 2 X L(Y1)ϕ3 + 1 2 π Y 2 , Y 3 + X ϕ4
, where L(Y 1 )ϕ 3 denotes the natural action of Y 1 on the -1 ℓ+1 -density ϕ 3 . Let us comment on this formula. It only contains natural operations, so it is clearly contactinvariant. Conversely, equivariance with respect to K(M ) (in fact the affine subalgebra suffices) implies that sσ 2 λ,λ (T ) has to be of the form

c 12 X ϕ1 , X ϕ2 + c 13 X L(Y1)ϕ3 + c 23 π Y 2 , Y 3 + c 4 X ϕ4 ,
where the c's are constants. The normalization condition on D 1 λ,λ gives c 4 = 1. Skew-symmetrizing the expression then yields c 12 = c 23 = 1 2 . Symmetrizing the expression implies that c 13 vanishes in the self-adjoint case λ = 1 2 , but its exact form must be deduced by computation.

The main content of the theorem is that the formula is actually well-defined. Indeed, the choice of the ϕ i and Y i in (2) is not unique: one can write an operator as a linear combination of such expressions in many different ways. However, the formula is independent of the choice. Moreover, the uniqueness statement of Theorem A implies that, up to a scalar, this is not true for any other choice of the c's.

2.3. The fine filtration. In order to explain the significance of our next theorem, consider the following arrangement of the fine symbol modules (we have omitted M and δ for clarity):

Σ 6,12 Σ 5,10 Σ 6,11 Σ 4,8 Σ 5,9 Σ 6,10 Σ 3,6 Σ 4,7 Σ 5,8 Σ 6,9 • • • Σ 2,4 Σ 3,5 Σ 4,6 Σ 5,7 Σ 6,8 Σ 1,2 Σ 2,3 Σ 3,4 Σ 4,5 Σ 5,6 Σ 6,7 Σ 0,0 Σ 1,1 Σ 2,2 Σ 3,3 Σ 4,4 Σ 5,5 Σ 6,6
Observe that the graded module of S k δ (M ) defined by the bifiltration

D k,d λ,µ (M ) is the "vertical" sum gr S k δ (M ) = k≤d≤2k Σ k,d δ (M ).
The graded module of

P d λ,µ (M )/P d-1 λ,µ (M ) is the "slope -1" sum gr P d λ,µ (M )/P d-1 λ,µ (M ) = ⌈ d 2 ⌉≤k≤d Σ k,d δ (M ).
The content of our next theorem is that there exists a K(M )-invariant filtration that strengthens the filtration P d λ,µ (M ). The graded modules of its subquotients are the "slope -1 2 " sums.

Theorem C. Assume that δ is not contact-resonant. Then there is a unique

K(M )-invariant filtration of D λ,µ (M ), D (0) λ,µ (M ) ⊂ • • • ⊂ D (b) λ,µ (M ) ⊂ D (b+1) λ,µ (M ) ⊂ • • • , such that the graded module of D (b) λ,µ (M ) is given by gr D (b) λ,µ (M ) = 2d-k≤b Σ k,d δ (M ).
For example, gr

D (6) λ,µ (M )/D (5) λ,µ (M ) = Σ 6,6 δ (M ) ⊕ Σ 4,5 δ (M ) ⊕ Σ 2,4
δ (M ), as indicated by the boundaries in the diagram above. We will define D (b) λ,µ (M ) via the projective quantization: see Section 6.3.

Comments on non-existence and conjectures.

In this paper we are concerned with existence results rather than non-existence results, but we remark that in the contact-resonant case our theorems are false for most values of λ. For example, if k = δ(ℓ + 1) -ℓ in Theorem A, then there exists no subsymbol except in the self-adjoint case λ + µ = 1, where there exists a 1parameter family of such maps. Non-existence results can be interpreted in terms of cohomological obstructions. We conjecture that the filtration of Theorem C does not exist in the contact-resonant case. These questions will be addressed elsewhere.

A module is called uniserial, or completely indecomposable, if it has a unique maximal invariant filtration. Such modules may be thought of as opposite to completely reducible modules. We conjecture that for generic values of λ and µ, D

(b) λ,µ (M )/D (b-1) λ,µ (M ) is uniserial, because we expect that in the Euclidean case, there is a non-trivial projectively relative 1-cohomology class of K m linking Σ k,d δ to Σ k-2,d+1
δ : an analog of the Schwarzian derivative. This conjecture says essentially that the bifiltration

D k λ,µ (M ) ∩ D (b)
λ,µ (M ) is the best possible. In other words, if the conjecture is true then there is no K(M )-invariant filtration of D λ,µ (M ) whose elements are composed of all those fine symbol modules on or below the lines of any fixed slope shallower than -1 2 passing through the above diagram.

The Euclidean contact Lie algebra

Since all of the theorems in Section 2 are local, their proofs essentially reduce to the case M = R m . Therefore in this section we establish notation and state some well-known results for Euclidean contact manifolds: the proofs are straightforward and are usually omitted. All C ∞ (R m )modules of finite rank are equipped with their usual topologies as Frechet spaces, and by definition all Hom spaces between such modules include only continuous linear maps.

3.1. Darboux coordinates. Fix coordinates x i , y i , and z on R m , where m = 2ℓ+1 and 1 ≤ i ≤ ℓ. Henceforth we use Einstein's summation convention: unless stated otherwise, repeated indices are summed over from 1 to ℓ. Let θ and ω be the standard contact and volume forms on R m :

θ := dz + 1 2 (x i dy i -y i dx i ), ω := 1 ℓ! θ ∧ (dθ) ℓ = dz ∧ ℓ 1 (dx i ∧ dy i ). The standard divergence operator Div : Vect(R m ) → C ∞ (R m ) is defined by L(X) ω = Div(X)ω.
Recall from Section 2 the definitions of the Lie algebra of contact vector fields and the space of tangent vector fields:

K(R m ) := X ∈ Vect(R m ) : L(X) θ = 1 ℓ+1 Div(X)θ , Tan(R m ) := X ∈ Vect(R m ) : θ, X = 0 .
The space Tan(R m ) is a module over K(R m ), and so one has the

K(R m )-invariant decomposition Vect(R m ) = K(R m ) ⊕ Tan(R m ).
It is important to keep in mind that while Tan(R m ) is closed under multiplication by smooth functions, K(R m ) is not. We now give explicit descriptions of both spaces. Define the following Euler operators:

E z := z∂ z , E xy := x i ∂ xi + y i ∂ yi .
For 1 ≤ i ≤ ℓ, define the vector fields

A i := ∂ xi + 1 2 y i ∂ z , B i := -∂ yi + 1 2 x i ∂ z .

The following statements are classical.

• There is a linear bijection X : C ∞ (R m ) → K(R m ) mapping f to X f , the unique contact vector field such that θ, X f = f . It has the following explicit formulas:

(3)

X f = f ∂ z + B i (f )A i -A i (f )B i = 1 -1 2 E xy (f )∂ z + 1 2 ∂ z (f )E xy + ∂ xi (f )∂ yi -∂ yi (f )∂ xi . • {A i , B i : 1 ≤ i ≤ ℓ} is a basis of Tan(R m ) over C ∞ (R m ), and [A i , B j ] = δ ij ∂ z , [∂ z , A i ] = 0, [∂ z , B i ] = 0.
We remark that one can verify (1) directly in this setting by checking that (4)

L(X f ) θ = ∂ z (f ) θ, Div(X f ) = (ℓ + 1)∂ z (f ).
Definition. The Lagrange bracket {f, g} on C ∞ (R m ) is defined by

X {f,g} := [X f , X g ].
The following formulas for {f, g} may be deduced from (3).

Lemma 3.1. The Lagrange bracket is given by

{f, g} = X f (g) -g∂ z (f ) = f ∂ z (g) -X g (f ) = f ∂ z (g) -∂ z (f )g + B i (f )A i (g) -A i (f )B i (g) = 1 -1 2 E xy (f )∂ z (g) -1 -1 2 E xy (g)∂ z (f ) + ∂ xi (f )∂ yi (g) -∂ yi (f )∂ xi (g) . 3.2. Tensor density modules. Recall from Section 2 the tensor density module F λ (R m ) of Vect(R m ). As a vector space, F λ (R m ) is nothing but C ∞ (R m ).
However, the action of Vect(R m ) depends on λ and is given by L λ (X)(g) := X(g) + λ Div(X) g, where g ∈ C ∞ (R m ) and λ ∈ R. Since the volume form ω is global, we may regard F λ (R m ) as ω λ C ∞ (R m ), so that the action of Vect(R m ) is identified with the usual Lie derivative:

L(X)(ω λ g) = ω λ L λ (X)(g).
We remark that the full family F λ (R m ) : λ ∈ C of Vect(R m )-modules can be understood algebraically as a non-trivial deformation of the module C ∞ (R m ).

We will consider F λ (R m ) as a module over the subalgebra of contact vector fields K(R m ). In light of (1) and the global contact form θ, we may regard

F λ (R m ) in this context as either ω λ C ∞ (R m ) or θ λ(ℓ+1) C ∞ (R m ). In particular, (4) gives L(X f )(θ λ(ℓ+1) g) = θ λ(ℓ+1) L λ (X f )(g) = θ λ(ℓ+1) X f (g) + λ(ℓ + 1)∂ z (f )g .
As mentioned in Section 2, the adjoint action of K(R m ) on itself is equivalent to the module of -1 ℓ+1 -densities. The following definition and lemma state this formally. Definition. Henceforth we regard X as the map

X : F -1 ℓ+1 (R m ) → K(R m ), X(θ -1 f ) := X f .
Lemma 3.2. The map X is a linear bijection and a K(R m )-equivalence.

Proof. By Lemma 3.1, {f, g} = L -1 ℓ+1 (X f )(g). Therefore X intertwines the K(R m )-action on F -1 ℓ+1 (R m
) and the adjoint action.

3.3. Differential operator modules. We now turn to the focus of the paper, the modules

D λ,µ (R m ) of differential operators from F λ (R m ) to F µ (R m ).
We can write the action L λ,µ of Vect(R m ) on these modules concretely as follows:

L λ,µ (X)(T ) := L µ (X) • T -T • L λ (X).
The structure of the spaces D λ,µ (R m ) viewed as Vect(R m )-modules has been thoroughly studied; see for example [LMT96, DO97, GMO05, Co09] and the references therein. We will be interested in these spaces viewed as K(R m )-modules, as which they are less rigid, because K(R m ) is smaller than Vect(R m ). In particular, the D λ,µ (R m ) admit more K(R m )-invariant operations than they do Vect(R m )-invariant operations, such as projections to tensor fields. We will understand such operations as differential invariants.

3.4. Symbol modules. In Section 2 we defined the principal symbol modules S k δ (R m ) and the fine symbol modules Σ k,d δ (R m ), k ≤ d ≤ 2k. We will write S δ (R m ) and Σ δ (R m ) for the total symbol module and the total fine symbol module, respectively; the graded modules associated to the order filtration and the bifiltration of D λ,µ (R m ):

S δ (R m ) := ∞ k=0 S k δ (R m ), Σ δ (R m ) := ∞ k=0 2k d=k Σ k,d δ (R m ).
Let L S δ be the natural action of Vect(R m ) on S δ (R m ), and let L Σ δ be the natural action of K(R m ) on Σ δ (R m ). Our next proposition gives formulas for L S δ (X f ) and L Σ δ (X f ). In order to state it we must develop a variation of the usual symbol calculus which is adapted to the fine filtration. Let α i , β i , and ζ be the symbols associated to the vector fields A i , B i , and ∂ z . More explicitly, (5)

α i = ξ xi + 1 2 y i ξ z , β i = -ξ yi + 1 2 x i ξ z , ζ = ξ z ,
where ξ x1 , ξ y1 , . . . , ξ x ℓ , ξ y ℓ , ξ z are the coordinates on T * R m dual to x 1 , y 1 , . . . , x ℓ , y ℓ , z.

We shall abuse notation and use α i and β i also to denote fine symbols. Thus (6) d , where I and J are multi-indices: I = (I 1 , . . . , I ℓ ) and J = (J 1 , . . . , J ℓ ). Note that in local coordinates we do not write the shift ω δ in the tensor density degree explicitly.

S k δ (R m ) = Span C ∞ (R m ) ζ c α I β J : |I| + |J| + c = k , Σ k,d δ (R m ) = Span C ∞ (R m ) ζ d-k α I β J : |I| + |J| = 2k -
Any differential operator from one symbol module to another, or from one fine symbol module to another, may be written as a linear combination over C ∞ (R m ) of products of the operators

A i , B i , ∂ z , α i , β i , ζ, ∂ αi , ∂ βi , ∂ ζ .
Such a combination is to be interpreted as follows. The operators A i , B i , and ∂ z act solely on the C ∞ (R m ) coefficients of the basis elements in (6), while the remaining operators act solely on the basis elements themselves.

Imitating the definitions of E z and E xy , we set

E ζ := ζ∂ ζ , E αβ := α i ∂ αi + β i ∂ βi . Proposition 3.3. (i) The action L Σ δ of K(R m ) on Σ δ (R m ) is L Σ δ (X f ) = f ∂ z + B i (f )A i -A i (f )B i + ∂ z (f ) δ(ℓ + 1) -E ζ -1 2 E αβ + 1 2 (A i B i + B i A i )(f )(β i ∂ βj -α i ∂ αj ) + A i A j (f )β i ∂ αj -B i B j (f )α i ∂ βj . (ii) The action L Σ δ of K(R m ) on S δ (R m ) is L S δ (X f ) = L Σ δ (X f ) + ∂ z A i (f )β i -∂ z B i (f )α i ∂ ζ .
Proof. Taking in to account the shift ω δ in tensor density degree, L S and L Σ are derivations in an obvious sense. Therefore it is only necessary to check the formulas on the generators α i , β i , and ζ and on functions g. Keep in mind that A i and B i do not commute, although α i and β i , being symbols, do.

The difference between L S δ and L Σ δ is due to the fact that Σ δ (R m ) is the graded module of

S δ (R m ). Observe that L S δ -L Σ δ maps Σ k,d δ (R m ) to Σ k,d-1 δ (R m
). Henceforth we will frequently drop the argument R m of the various tensor density, differential operator, and symbol modules, writing simply F λ , D λ,µ , S k δ , Σ k,d δ , and so on.

The projective subalgebra

Here we recall the projective subalgebra s m of K(R m ), which is isomorphic to sp 2(ℓ+1) . Restriction of K(R m )-modules to s m will be central to our strategy throughout this paper, for two reasons. First, s m is a maximal polynomial subalgebra of K(R m ), and it turns out that for most values of their parameters, the tensor density modules and the fine symbol modules are not only algebraically irreducible under K(R m ), they remain so under restriction to s m . (By "algebraically irreducible", we mean irreducible in the polynomial category.) Second, s m is a finite dimensional semisimple Lie algebra, and so we can bring the representation theory of such algebras to bear on the restricted modules.

In fact, the restrictions to s m of the tensor density modules and fine symbol modules are duals of sp 2ℓ -relative Verma modules. Our approach to the construction of the projective quantization referred to in the introduction will be to observe that for generic δ, the fine symbol modules composing D λ,µ have distinct infinitesimal characters under the action of s m . This implies that there is a unique s m -equivariant splitting of D λ,µ into the sum of its fine symbol modules. This splitting may be regarded as a projectively invariant total symbol. The projective quantization is, by definition, its inverse.

In addition to the projective subalgebra s m , two other subalgebras of K(R m ) will be important to us: the affine subalgebra t m and its nilradical u m . 4.1. The projective subalgebra of the full vector field Lie algebra. We first recall the definitions of the analogous subalgebras of the full vector field Lie algebra. Let u 1 , . . . , u m be any coordinates on R m . The full Euler operator is

E u := m i=1 u i ∂ ui .
Within Vect(R m ) we have the projective subalgebra a m , the affine subalgebra b m , and the constant coefficient subalgebra c m :

a m := Span C ∂ ui , u j ∂ ui , u j E u : 1 ≤ i, j ≤ m , b m := Span C ∂ ui , u j ∂ ui : 1 ≤ i, j ≤ m , c m := Span C ∂ ui : 1 ≤ i ≤ m .
Clearly a m ⊃ b m ⊃ c m . In fact, a m is isomorphic to sl m+1 , and b m is a maximal parabolic subalgebra of a m with Levi factor gl m and nilradical c m . The center of the Levi factor is CE u .

There is a standard conceptual proof of a m ∼ = sl m+1 which we briefly sketch. Let u 0 , . . . , u m be coordinates on R m+1 , and note that Span C {u j ∂ ui : 0 ≤ i, j ≤ m} is a subalgebra of Vect(R m+1 ) isomorphic to gl m+1 . Restricting the action of Vect(R m+1 ) to functions on P m defines the canonical projection from this copy of gl m+1 to sl m+1 . Regarding P m locally as the hyperplane defined by u 0 = 1 and identifying this hyperplane with R m yields the isomorphism. 4.2. The projective subalgebra of the contact Lie algebra. We now define the projective and affine subalgebras of K(R m ): they are simply the intersections of the projective and affine subalgebras of Vect(R m ) with K(R m ).

Definition. The projective subalgebra s m and the affine subalgebra t m of K(R m ) are

s m := a m ∩ K(R m ), t m := b m ∩ K(R m ).
We define also the following subspaces of K(R m ):

u m := Span C X 1 , X xi , X yi : 1 ≤ i ≤ ℓ , l s m := Span C X xiyj , X xixj , X yiyj : 1 ≤ i, j ≤ ℓ , l m := l s m ⊕ CX z . Observe that X z = E z + 1 2 E xy . This is the natural Euler operator in K(R m ).
The space l m is the 0-weight space of its adjoint action, and u m is the sum of its -1 2 -eigenspace Span C {X xi , X yi : 1 ≤ i ≤ ℓ} and its -1-eigenspace CX 1 .

Next we give explicit descriptions of s m and t m and prove that s m is symplectic. We also show that u m , l s m , and l m are subalgebras of t m , and that u m is the contact-analog of the constant coefficient algebra c m .

Lemma 4.1.

(i) The space l s m is a subalgebra of s m isomorphic to sp 2ℓ . (ii) The space l m is a Levi subalgebra of s m , with semisimple part l s m and center CX z . (iii) The space u m is a Heisenberg Lie algebra with center CX 1 . (iv) The affine subalgebra t m is the semidirect sum l m ⊕ s u m . It is a maximal parabolic subalgebra of s m with nilradical u m . (v) The projective subalgebra s m is isomorphic to sp 2(ℓ+1) , and

s m = X θ -1 Polynomials of degree ≤ 2 on R m .
Proof. Most of this can be left to the reader. To prove (i), use (3) to obtain

X xiyj = y j ∂ yi -x i ∂ xj , X xixj = x i ∂ yj + x j ∂ yi , X yiyj = -y i ∂ xj -y j ∂ xi .
Then note that the natural action of l s m on Span C {x i , y i : 1 ≤ i ≤ ℓ} preserves the skew-symmetric form defined by x i , x j = 0, y i , y j = 0, and x i , y j = δ ij . (Alternately, observe that the Lagrange bracket defines a non-degenerate l m -invariant skew-symmetric form on the -1 2 -eigenspace of X z .) Parts (ii), (iii), and (iv) now follow by computation.

For a direct proof of the displayed equation in (v), first check that E xy = x i A i -y i B i . Then verify that the vector fields

A i , B i , x j A i -y i B j , x j B i -x i B j , y j A i -y i A j , zA i + 1 2 y i E xy , zB i + 1 2 x i E xy , 1 ≤ i, j ≤ ℓ, span a (2ℓ + 3)ℓ-dimensional subspace of a m ∩ Tan(R m ).
Next, use Lemma 3.2 to verify that the right side of the display in (v) is a (2ℓ + 3)(ℓ + 1)dimensional subspace of a m ∩ K(R m ). Since a m is (2ℓ + 3)(2ℓ + 1)-dimensional, (v) is proven.

The fact that s m is a copy of sp 2(ℓ+1) can now be proven using (ii) and an adaptation of the above argument proving that a m is a copy of sl m+1 .

Let us establish some notation for later use: we will write C 2ℓ for the basic module of l s m ∼ = sp 2ℓ , and Sym r C 2ℓ for its r th symmetric power. It is well-known that Sym r C 2ℓ is self-dual and irreducible for all r.

The following lemma defines a Cartan subalgebra h m of s m and gives the associated root system and Weyl group. Its proof may be found in any text on Lie theory; see e.g., [START_REF] Varadarajan | Lie groups, Lie algebras, and their representations[END_REF].

Lemma 4.2.

(i) The set {2X z , X x1y1 , . . . , X x ℓ y ℓ } is a basis of a Cartan subalgebra h m of s m . Under the Cartan-Killing form this basis is orthogonal and all of its elements are of length 2 √ ℓ + 2. Let {e 0 , e 1 , . . . , e ℓ } be the dual basis of h * m . (ii) The adjoint action of h m on the polynomial contact vector fields is semisimple:

X(x I y J z c ) is of h m -weight (2c + |I| + |J| -2)e 0 -(I i -J i )e i . (
iii) The roots of s m are all of the non-zero ±e i ± e j . The Weyl group W (s m ) is S ℓ+1 ⋉ Z ℓ+1 2 , acting in the usual way on the basis {e 0 , . . . , e ℓ }. (iv) The order 0 < e ℓ < • • • < e 0 gives the following simple root system Π + (s m ) and positive root system ∆ + (s m ) of s m :

Π + (s m ) := e i-1 -e i , 2e ℓ ℓ i=1 , ∆ + (s m ) := e i -e j } i<j ∪ {e i + e j i,j .
The half-sum of the positive roots is ρ(s m ) = ℓ 0 (ℓ + 1 -i)e i . The dominant weights are those ℓ 0 γ i e i with 0 ≤ γ ℓ ≤ • • • ≤ γ 0 . The negative root vectors are X 1 , X xi , X yi , X xixj , and those X xiyj with i < j.

(v)

The algebra h m is also a Cartan subalgebra of the Levi subalgebra l m of s m . The roots of l m are the non-zero ±e i ± e j with i, j > 0. It inherits the simple root system

Π + (l m ) = e i-1 -e i , 2e ℓ ℓ i=2 . (vi) The subalgebra h s m := Span{X x1y1 , . . . , X x ℓ y ℓ } of h m is a Cartan subalgebra of l s m .
The roots of l s m are the same as those of l m .

4.3.

The s m -structure of the symbol modules. In this subsection we analyze the action of s m on the principal and fine symbol modules. Our first lemma gives the restriction to s m of the actions L S δ and L Σ δ on S δ and Σ δ . It is a corollary of Proposition 3.3. In order to state it concisely, we define the total weight operator W to act on both S δ and Σ δ by

W | S δ = W | Σ δ := (E z + 1 2 E xy ) -(E ζ + 1 2 E αβ ) + δ(ℓ + 1).
Lemma 4.3.

(i) The restrictions of L S δ and L Σ δ to the affine subalgebra t m coincide. (ii) Their restriction to the nilradical u m of t m is the identity map X f → X f :

X 1 → ∂ z , X xi → x i ∂ z -B i , X yi → y i ∂ z -A i .
(iii) Their restriction to the Levi factor l m of t m is given by

X z → W, X xiyj → x i y j ∂ z -(x i A j + y j B i ) + (α j ∂ αi -β i ∂ βj ), X xixj → x i x j ∂ z -(x i B j + x j B i ) + (β i ∂ αj + β j ∂ αi ), X yiyj → y i y j ∂ z -(y i A j + y j A i ) -(α i ∂ βj + α j ∂ βi ).
(iv) The action of the rest of s m under L Σ δ is given by

L Σ δ (X xiz ) = x i W -zB i -1 2 (x r α r -y r β r )∂ αi + 1 2 β i (x s ∂ βs + y s ∂ αs ), L Σ δ (X yiz ) = y i W -zA i + 1 2 (x r α r -y r β r )∂ βi + 1 2 α i (x s ∂ βs + y s ∂ αs ), L Σ δ (X z 2 ) = 2zW -z 2 ∂ z -1 2 (x r α r -y r β r )(x s ∂ βs + y s ∂ αs ).
(v) The action of the rest of s m under L S δ is given by

L S δ (X xiz ) = L Σ δ (X xiz ) + β i ∂ ζ , L S δ (X yiz ) = L Σ δ (X yiz ) + α i ∂ ζ , L S δ (X z 2 ) = L Σ δ (X z 2 ) -(x r α r -y r β r )∂ ζ .
Note that the operators x r α r -y r β r and x s ∂ βs + y s ∂ αs occurring in (iv) are l s m -invariants of total weights 0 and 1, respectively.

We now define a space which will regularly play an important role in our arguments:

(Σ k,d δ ) um := P ∈ Σ k,d δ : L Σ δ (u m )P = 0 , the subspace of Σ k,d δ invariant under the Heisenberg algebra u m . Since t m normalizes u m , (Σ k,d δ ) um is a t m -module on which u m acts trivially. Lemma 4.4. (i) (Σ k,d δ ) um = Span C {ζ d-k α I β J : |I| + |J| = 2k -d}. (ii) Under W , (Σ k,d δ ) um has total weight δ(ℓ + 1) -1 2 d. (iii) Under l s m ∼ = sp 2ℓ , (Σ k,d δ ) um is equivalent to Sym 2k-d C 2ℓ . Proof.
Part (i) is clear from the action of u m given in Lemma 4.3. Part (ii) is straightforward, and Part (iii) follows from the fact that as a module of l s m , Span C {α i , β i } i is equivalent to the irreducible module C 2ℓ of sp 2ℓ .

Thus the u m -invariant fine symbols are precisely the constant fine symbols. In addition to the spaces of constant symbols, we will encounter the spaces of polynomial fine symbols, which are modules of the Lie algebra of the polynomial contact vector fields. Under the action of s m , these modules turn out to be the restricted duals of l m -relative Verma modules.

To make this precise, let us write Poly(R m ) for the polynomials in C ∞ (R m ). We will denote the polynomial subspaces of K(R m ), F λ (R m ), D λ,µ (R m ), S δ (R m ), and Σ δ (R m ) by writing R m poly in place of R m . Thus for example K(R m poly ) is the classical Cartan algebra K m discussed in the introduction, and

Σ k,d δ (R m poly ) := Span Poly(R m ) ζ d-k α I β J : |I| + |J| = 2k -d .
Note that K m is a dense subalgebra of K(R m ) containing s m , and all of the above polynomial subspaces are dense K m -submodules of their smooth counterparts.

We will need the fact that smooth globally defined eigenfunctions of Euler operators are polynomials.

Lemma 4.5.

(i) Let f ∈ C ∞ (R m ) be an eigenfunction of the full Euler operator

E u = m i=1 u i ∂ i with eigenvalue λ. Then λ ∈ N and f is a homogeneous polynomial in u of degree λ. (ii) Let f ∈ C ∞ (R m
) be an eigenfunction of the contact Euler operator E z + 1 2 E xy with eigenvalue λ. Then λ ∈ 1 2 N and f is a homogeneous polynomial in (x, y, z) of degree λ, in the sense that x i and y i have degree 1 2 and z has degree 1. Proof. The first statement is classical, and the second follows from the first by the change of coordinates (x, y, z) → (x, y, z 2 ).

Recall from Lemma 4.2 the Cartan subalgebra h m of s m . Given any h m -module V and any ν ∈ h * m , we use the standard notation V ν for the ν-weight space of V . By the restricted dual of V , we mean the direct sum of the duals of its weight spaces. A lowest weight vector in an s m -module is a weight vector annihilated by the negative root vectors; see Lemma 4.2(iv). Recall also that total weights are eigenvalues of the operator W . Proof. Lemma 4.5 implies (i), and (ii) is due to the fact that polynomials have non-negative total weights. For (iii), recall from Lemma 4.2 the basis {e 0 , . . . , e ℓ } of the weight space h * m and its order 0 < e ℓ < • • • < e 0 . The lowest h m -weight space is contained in the lowest total weight space (Σ k,d δ ) um . Since the h m -weights of ζ, α i , and β i are, respectively, -2e 0 , -e 0 + e i , and -e 0 -e i , (iii) follows from Lemma 4.4.

We now recall the basic notions of Verma modules. Let us denote by u + m and t + m the subalgebras of s m opposite to u m and t m , respectively:

u + m = Span C X xiz , X yiz , X z 2 : 1 ≤ i ≤ ℓ , t + m = l m ⊕ s u + m .
Given an irreducible module V of l m , the associated relative Verma module of s m is U(s m ) ⊗ t + m V , where u + m acts trivially on V . 4.4. The s m -structure of K m . As a particular case of the s m -module structures of spaces of fine symbols, we investigate the s m -structure of the algebra K m itself, which is by definition Σ 1,2 0 .

Proposition 4.8. The quotient K m /s m is irreducible under s m . Its lowest weight vector is X x 3 1 , which has weight e 0 -3e 1 .

Proof. The reader may use Lemma 4.3 to check that the lowest weight vectors of K m /s m under l m are precisely all the elements of the form X x i 1 z c with i + c ≥ 3. The same lemma shows that it is possible to move from any one of these lowest weight vectors to any other using the elements X 1 , X x1 , X x1z , and X z 2 of s m . The weight of X x 3 1 is given by Lemma 4.2(ii). It will be important to understand the space of u m -invariants in K m /s m . The following result is immediate from Proposition 4.8. Corollary 4.9.

(i

) (K m /s m ) um = Span{X x I y J : |I| + |J| = 3}. (ii) (K m /s m ) um has total weight 1 2 . (iii) (K m /s m ) um is equivalent to Sym 3 C 2ℓ under the action of l s m .
4.5. Infinitesimal characters. We now turn to the infinitesimal characters of the fine symbol modules. A module of a complex semisimple Lie algebra g is said to have an infinitesimal character if the center Z(g) of the universal enveloping algebra U(g) acts on it by scalars. In this case, the infinitesimal character is the resulting homomorphism from Z(g) to C. By Schur's lemma, all irreducible modules have infinitesimal characters. Suppose that a Cartan subalgebra and a positive root system of g are fixed. Let ρ be the halfsum of the positive roots. If V and V ′ are two lowest weight modules of g with lowest weights ν and ν ′ , respectively, and both modules have infinitesimal characters, then it is a consequence of the Harish-Chandra homomorphism that their infinitesimal characters are the same if and only if ν -ρ and ν ′ -ρ lie in the same orbit of the Weyl group of g. (i)

k ′ = k, d ′ = 2δ(ℓ + 1) -1 + 2k -d. (ii) k ′ = 2(δ -1)(ℓ + 1) + 1 -k, d ′ = 2(δ -1)(ℓ + 1) + 1 -2k + d. (iii) k ′ = (2δ -1)(ℓ + 1) + k -d, d ′ = 2(2δ -1)(ℓ + 1) -d. (iv) k ′ = d -k -ℓ, d ′ = 2(δ -1)(ℓ + 1) + 1 -2k + d.
Cases (i), (ii), (iii), and (iv) cannot occur unless 2δ(ℓ + 1) is in

2 + N, 2(ℓ + 1) + N, ℓ + 2 + N, 2ℓ + 1 + N,
respectively. Moreover, if ℓ = 0, then Case (iv) cannot occur unless 2δ ∈ 2 + N. Therefore if δ is not contact-resonant (see Section 2.1), then all of the fine symbol modules of D λ,µ have distinct infinitesimal characters.

Proof. It is well-known that Verma modules have infinitesimal characters, and so their restricted duals do also. Therefore by Lemma 4.7, Z(s m ) acts by scalars on Σ k,d δ (R m poly ). By a density argument, it acts by the same scalars on Σ k,d δ (R m ). Thus the fine symbol modules have infinitesimal characters.

Recall from Lemma 4.6 the lowest weight ν k,d δ of Σ k,d δ (R m poly ), and from Lemma 4.2 the half-sum ρ(s m ) of the positive roots:

ρ(s m ) = ℓ i=0 (ℓ + 1 -i)e i .
Recall also that the Weyl group W (s m ) = S ℓ+1 ⋉ Z ℓ+1 2 acts by permutations and sign changes on the e i .

As stated above, the Harish-Chandra homomorphism shows that the infinitesimal characters of Σ k,d δ and Σ k ′ ,d ′ δ are the same if and only if there is an element w of W (s m ) such that

w ν k,d δ -ρ(s m ) = ν k ′ ,d ′ δ -ρ(s m ).
By Lemma 4.6, such a w exists if and only if the two sets

|(2δ -1)(ℓ + 1) -d|, ℓ + 2k -d, ℓ -1, ℓ -2, . . . , 2, 1 , |(2δ -1)(ℓ + 1) -d ′ |, ℓ + 2k ′ -d ′ , ℓ -1, ℓ -2, . . . , 2, 1 are equal. Since ℓ + 2k -d > ℓ -1 > • • • > 1 > 0,
this can occur only in the following ways.

First, |(2δ -1)(ℓ + 1) -d| and |(2δ -1)(ℓ + 1) -d ′ )| can be equal. In this case ℓ + 2k -d and ℓ + 2k ′ -d ′ must also be equal. Since we are assuming that (k, d) and (k ′ , d ′ ) are distinct, this leads to Case (iii).

Second, we could have

|(2δ -1)(ℓ + 1) -d| = ℓ + 2k ′ -d ′ , |(2δ -1)(ℓ + 1) -d ′ | = ℓ + 2k -d.
This can occur in three ways, depending on the signs of the arguments of the absolute values. If both are negative we arrive at Case (i), if both are positive we are in Case (ii), and if they are different we obtain Case (iv). Finally, if |(2δ -1)(ℓ + 1) -d| is equal to one of ℓ -1, . . . , 1, say i, then again ℓ + 2k -d and ℓ + 2k ′ -d ′ must be equal and |(2δ -1)(ℓ + 1) -d| must also be i. Therefore here we are still in Case (iii), albeit with a different w.

Projective quantization

Recall from Section 3.4 the total symbol modules S δ and the total fine symbol modules Σ δ . In this section we study quantizations of Σ δ invariant under the projective subalgebra s m of K(R m ). We begin in Section 5.1 with a review of quantizations of S δ invariant under the projective subalgebra a m of Vect(R m ), as these quantizations are a component of the fine projective quantizations.

Projective quantization of symbols.

A quantization of S δ is defined to be a linear bijection Q from S δ to D λ,µ which preserves degree and is the identity on symbols. By this we mean that it carries S k δ into D k λ,µ , and its restriction to S k δ is a right-inverse of the principal symbol map σ k λ,µ :

σ k λ,µ • Q| S k δ : S k δ → S k
δ is the identity map. Suppose that g is any Lie subalgebra of Vect(R m ). A quantization is said to be a g-equivariant quantization (or simply a g-quantization) if it intertwines the two g-actions L S δ | g and L λ,µ | g . The more vector fields a quantization is invariant with respect to, the more useful it is. Recall from Section 4.1 the projective subalgebra a m and the affine subalgebra b m of Vect(R m ). At one extreme, one might ask for a Vect(R m )-quantization of S δ . However, there is no such map: S δ and D λ,µ are not Vect(R m )-equivalent for any (λ, µ). At the opposite extreme, b m -quantizations are easy to find, not unique, and not very useful. The critical intermediate case is afforded by a m , because it is a simple finite dimensional maximal subalgebra of Vect(R m ).

Definition. We say that δ is projectively resonant if it lies in the set

1 + n m + 1 n ∈ N .
The following theorem was proven in [START_REF] Cohen | Automorphic pseudodifferential operators[END_REF] for m = 1, in [START_REF] Lecomte | Projectively invariant symbol calculus[END_REF] for arbitrary m at p = 0, and in general in [START_REF] Lecomte | On the cohomology of sl(m+1, R) acting on differential operators and sl(m+1, R)-equivariant symbol[END_REF].

Theorem 5.1. For δ not projectively resonant, there exists a unique a m -quantization Q am λ,µ : S δ → D λ,µ . Theorem 5.1 may be proven using only the eigenvalues of the Casimir operator; the full infinitesimal characters are not needed. This is the approach taken in [START_REF] Lecomte | On the cohomology of sl(m+1, R) acting on differential operators and sl(m+1, R)-equivariant symbol[END_REF]. The explicit formula for the projective quantization Q am λ,µ was given in [START_REF] Cohen | Automorphic pseudodifferential operators[END_REF] for m = 1, in [START_REF] Lecomte | Projectively invariant symbol calculus[END_REF] for arbitrary m at p = 0, and in general in [START_REF] Duval | Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions[END_REF]; see Section 7.2 below. 5.2. Projective quantization of fine symbols. By analogy with quantizations of S δ , we define a quantization of Σ δ , sometimes called a fine quantization, to be a linear bijection

Q from Σ δ to D λ,µ , carrying Σ k,d δ into D k,d
λ,µ , which is the identity on fine symbols in the sense that its restriction to Σ k,d δ is a right-inverse for the fine symbol map fσ k,d λ,µ :

fσ k,d λ,µ • Q| Σ k,d δ : Σ k,d δ → Σ k,d δ
is the identity map.

For any Lie subalgebra g of K(R m ), we say that a fine quantization is a fine g-equivariant quantization if it intertwines L Σ δ | g and L λ,µ | g . The picture for quantizations of Σ δ is similar to that for quantizations of S δ : there is no fine K(R m )-quantization, there are many fine t m -quantizations, and for most δ there is a unique fine s m -quantization. The following theorem makes this precise. It may be proven by combining the results of [START_REF] Duval | Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions[END_REF] and [START_REF] Frégier | Decomposition of symmetric tensor fields in the presence of a flat contact projective structure[END_REF]. We understand it as a corollary of Proposition 4.10. As remarked before that proposition, it cannot be proven using the Casimir operator of s m alone; the full infinitesimal characters of the fine symbol modules are required. The explicit formula for Q sm λ,µ will be given in Section 7.3.

Lowest weight calculations

In this section we prove Theorem A and Theorem C. The proofs rely on lowest weight calculations in modules of homomorphisms between the fine symbol spaces.

The structure of Hom

um (Σ k,d δ , Σ k ′ ,d ′ δ
). We begin with a description of the total weight spaces of Hom um (Σ k,d δ , Σ k ′ ,d ′ δ ) which will be needed in both proofs. Recall that for ν ∈ h * m , V ν denotes the ν-weight space of any h m -module V . We will also use the following notation: for w ∈ C, V (w) denotes the w-total weight space of V . Note that if ν = ℓ 0 ν i e i , then Since ∂ z , A, and B commute with u m , the u m -equivariant maps in this space are precisely those independent of x, y, and z. This proves (i).

V ν ⊆ V ( ν 0 2 ) . We will abbreviate C[∂ z , A 1 , B 1 . . . , A ℓ , B ℓ ] by C[∂ z , A, B]. Lemma 6.1. (i) The total weight space End um C ∞ (R m ) (w) is zero unless w ∈ -1 2 N, when it is C[∂ z , A, B] (w) . (ii) The total weight space Hom um (Σ k,d δ , Σ k ′ ,d ′ δ ) (w) is zero unless w ∈ 1 2 (d -d ′ ) -1 2 N, when it is Span C ζ d ′ -k ′ α I ′ β J ′ ∂ d-k ζ ∂ I α ∂ J β : |I| + |J| = 2k -d, |I ′ | + |J ′ | = 2k ′ -d ′ ⊗ C[∂ z , A, B] (w-1 2 (d-d ′ )) . Proof. Let T : C ∞ (R m ) → C ∞ (R m )
Under

u m , Σ k,d δ is equivalent to (Σ k,d δ ) um ⊗ C ∞ (R m ), so Hom um Σ k,d δ , Σ k ′ ,d ′ δ is Hom C (Σ k,d δ ) um , (Σ k ′ ,d ′ δ ) um ⊗ End um C ∞ (R m ) .
Thus (ii) follows from (i) and the fact that Hom

C (Σ k,d δ ) um , (Σ k ′ ,d ′ δ ) um is Span C ζ d ′ -k ′ α I ′ β J ′ ∂ d-k ζ ∂ I α ∂ J β : |I| + |J| = 2k -d, |I ′ | + |J ′ | = 2k ′ -d ′ .
We will also need the structure of Hom um Σ k,d δ , Σ k ′ ,d ′ δ (w) as a module of the subalgebra l s m , which recall is isomorphic to sp 2ℓ . Lemma 6.2. Suppose that w is in 1 2 (d -d ′ ) -1 2 N. Then under the action of l s m ,

Hom um Σ k,d δ , Σ k ′ ,d ′ δ (w) ∼ = Sym 2k-d C 2ℓ ⊗ Sym 2k ′ -d ′ C 2ℓ ⊗ r≥0 Sym d-d ′ -2r-2w C 2ℓ ,
where the direct sum is only over non-negative exponents.

Proof. Consider the explicit basis of Hom

um Σ k,d δ , Σ k ′ ,d ′ δ (w)
given in Lemma 6.1(ii). We have the following l s m -equivalences: 6.2. Proof of Theorem A. Assume temporarily that M is R m , equipped with the standard contact structure. Since δ is not contact-resonant, Theorem 5.2 shows that there is a unique

Span C α I ′ β J ′ : |I ′ | + |J ′ | = 2k ′ -d ′ ∼ = Sym 2k ′ -d ′ C 2ℓ , Span C ∂ I α ∂ J β : |I| + |J| = 2k -d ∼ = Sym
s m -equivariant map from D k λ,µ to Σ k-1, 2(k-1) δ , namely (7) sσ k λ,µ := π k-1, 2(k-1) • (Q sm λ,µ ) -1
, where π j,d denotes the canonical projection from Σ δ to Σ j,d δ . Our task is to prove that this map is in fact K(R m )-equivariant.

We use the projective quantization to pull the action

L λ,µ of K(R m ) on D λ,µ back to an action L λ,µ of K(R m ) on Σ δ : L λ,µ (X f ) := (Q sm λ,µ ) -1 • L λ,µ (X f ) • Q sm λ,µ . The statement that sσ k λ,µ is K(R m )-equivariant is equivalent to the statement that π k-1, 2(k-1) : 0≤j≤k j≤d≤2j Σ j,d δ → Σ k-1, 2(k-1) δ
intertwines the K(R m )-actions L λ,µ and L Σ δ . We may regard L λ,µ as a block matrix with entries

L (j,d),(j ′ ,d ′ ) λ,µ (X f ) : Σ j,d δ → Σ j ′ ,d ′ δ .
This matrix is triangular with respect to the dictionary order on (j, d), and the diagonal entries are simply the usual actions on fine symbols: L

(j,d),(j,d) λ,µ is L Σ δ restricted to Σ j,d δ . As a result of the s m -equivariance of Q sm λ,µ , the off-diagonal entries L (j,d),(j ′ ,d ′ ) λ,µ
are s m -relative 1-cochains of K(R m ): they vanish on s m and are s m -equivariant maps from K(R m )/s m to Hom(Σ j,d δ ,

Σ j ′ ,d ′ δ ). One finds that π k-1, 2(k-1) • L λ,µ restricted to 0≤j≤k j≤d≤2j Σ j,d δ is L Σ δ • π k-1, 2(k-1) + 2k d=k L (k,d),(k-1, 2(k-1)) λ,µ • π k,d .
Therefore it suffices to prove that entries

L (k,d),(k-1, 2(k-1)) λ,µ
are zero for all d. The following lemma is the key to the situation. Lemma 6.3. Let V be a module of K(R m ) such that the space V um ( 1 2 ) of u m -invariants in V of total weight 1 2 contains no copies of Sym 3 C 2ℓ under the action of l s m . Then the space

C 1 K(R m ), s m ; V of s m -relative 1-cochains of K(R m ) with values in V is zero.
It is well-known (see, e.g., [START_REF] Varadarajan | Lie groups, Lie algebras, and their representations[END_REF]) that the largest irreducible component of Sym r C 2ℓ ⊗ Sym s C 2ℓ is Sym r+s C 2ℓ , and for r ≥ s, its smallest irreducible component is its Parthasarathy -Ranga Rao -Varadarajan submodule, a copy of Sym r-s C 2ℓ . Therefore the maximal component of

Sym 2k ′ -d ′ C 2ℓ ⊗ r≥0 Sym d-d ′ -2r-1 C 2ℓ is Sym d-2d ′ +2k ′ -1 C 2ℓ . In Case (ii) we have k ′ -d ′ < k -d, so the smallest irreducible component of (8) is Sym 2(d ′ -d+k-k ′ )+1 C 2ℓ , which is larger than Sym 3 C 2ℓ .
This lemma implies that L

C := ∂ z ∂ ζ : Σ k,d δ → Σ k-1,d-2 δ .
The tangential divergence is the map

Div T := A r ∂ αr + B r ∂ βr : Σ k,d δ → Σ k-1,d-1 δ . Finally, define ∆ := (α r B r -β r A r )∂ ζ : Σ k,d δ → Σ k,d-1 δ . Lemma 7.1.
(i) Div C , Div T , and ∆ are all t m -equivariant. (ii) The contact divergence Div C commutes with both Div T and ∆, and

[Div T , ∆] = (ℓ + E αβ ) Div C . (iii) Regarded as a map from S k δ to S k-1 δ , Div T + Div C is the full divergence Div.
Proof. The statement follows from Lemma 4.3 and short computations.

Remark. It follows from the first fundamental theorem of invariant theory for sp 2ℓ that the associative algebra generated by the operators

Div C , Div T , ∆, E αβ , E ζ
coincides with the algebra End tm (Σ δ ) of all affine invariants: see [START_REF] Frégier | Dequantized differential operators between tensor densities as modules over the Lie algebra of contact vector fields[END_REF].

7.2. The sl m+1 -equivariant quantization of S δ . Continuing the discussion of Section 5.1, we now give an explicit formula for the quantization map Q am λ,µ of Theorem 5.1. This formula is part of the explicit formula for the fine projective quantization in the contact setting.

We begin with the standard symbol calculus on S δ . As in Section 4.1, fix any coordinates u 1 , . . . , u m on R m . Write ξ i for the symbol of the vector field ∂ ui . Then

S δ = Span C ∞ (R m ) ξ I : I ∈ N m , D λ,µ = Span C ∞ (R m ) ∂ I u : I ∈ N m .
As noted in Section 3.4, these spaces carry the Vect(R m )-actions L S δ and L λ,µ , respectively. One of the simplest quantizations is the normal order quantization:

N : S δ → D λ,µ , N f I (u)ξ I := f I (u)∂ I u .
In the literature, S δ and D λ,µ are frequently identified via N, which then does not appear explicitly in the formulas. Although N is not an a m -equivariant quantization, it does turn out to be a b mquantization. The first step in computing Q am λ,µ is to find explicit formulas for L S δ and L λ,µ . In fact, one computes the pull-back of L λ,µ to S δ via N, that is,

L N -1 λ,µ (X) := N -1 • L λ,µ (
X) • N. Towards this end, note that any differential operator on S δ may be written as a C ∞ (R m )-linear combination of monomials

∂ I u ξ J ∂ K ξ . Given any vector field X = m 1 X i ∂ ui , it is straightforward to obtain L N -1 λ,µ (X) = X + δ Div(X) - |I|>0 1 I! ∂ I u λ Div(X) + j X j ξ j ∂ I ξ .
The action L S δ is simply the part of L N -1 λ,µ which preserves ξ-degree:

L S δ (X) = X + δ Div(X) - Bear in mind that E ξ and Div do not commute: [E ξ , Div] = -Div. The theorem is as follows.

Theorem 7.2. [LO99, DO01] For δ non-resonant,

Q am λ,µ = N • ∞ s=0 1 s! Div s • E ξ + λ(m + 1) -1 s 2E ξ -δ(m + 1) + m -1 s -1
.

It is worth mentioning that this formula can be understood as a (non-commutative) hypergeometric function: see [START_REF] Duval | Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions[END_REF]. 7.3. The sp 2(ℓ+1) -equivariant quantization of Σ δ . We now proceed to derive an explicit formula for Q sm λ,µ . We begin by defining a map SQ sm δ from Σ δ to S δ :

(9)

SQ sm δ := ∞ s=0 2 s (s!) 2 ∆ s • 2E ζ -2δ(ℓ + 1) s -1
.

This formula is well-defined provided that δ is not contact-resonant. Recall from (6) that we are abusing notation and using the same bases for Σ δ and S δ (this is analogous to regarding the normal order quantization N as the identity). Therefore we may and do regard SQ sm δ as a map from Σ δ to S δ .

Theorem 7.3. The fine s m -equivariant quantization of Theorem 5.2 is To calculate the S k-1 δ -component of Q am λ,µ

Q sm λ,µ = Q am λ,µ
-1 , we use the formula of Theorem 7.2.

Note that the formula for sσ k λ,µ is well-defined for all but one contact-resonant value of δ, the value δ = ℓ+k ℓ+1 . By continuity, it retains K m -equivariance whenever it is well defined.

7.5. Proof of Theorem B. Let T be a second order operator from F λ to F λ . Since Σ 1,2 0 is equivalent to F -1 ℓ+1 , the subsymbol sσ 2 λ,λ (T ) may be written as a contact Hamiltonian. If the operator T is of the form

T = T 2,0,0 ∂ 2 z + T 1,i,0 ∂ z ∂ xi + T 1,0,i ∂ z ∂ yi + T 0,ij,0 ∂ xi ∂ xj + T 0,i,j ∂ xi ∂ yj + T 0,0,ij ∂ yi ∂ yj +T 1,
0,0 ∂ z + T 0,i,0 ∂ xi + T 0,0,i ∂ yi + T 0,0,0 , then the formula of Proposition 7.4 reads sσ 2 λ,λ (T ) = -1+2λ(ℓ+1) ℓ+2 ∂ z (T 2,0,0 -1 2 y i T 1,i,0 + 1 2 x i T 1,0,i ) +∂ xi (T 1,i,0 -1 2 y j T 0,ij,0 + 1 2 x j T 0,i,j )

+∂ yi (T 1,0,i + 1 2 x j T 0,0,ij -1 2 y j T 0,j,i ) +T 1,0,0 -1 2 y i T 0,i,0 + 1 2 x i T 0,0,i . One can verify directly that this formula coincides with that of Theorem B. The most efficient approach is to prove first the comments following the statement of the theorem, after which it suffices to carry out the verification for the single operator T = L λ (X z ) • L λ (A 1 ), because that determines c 13 .

  Lemma 4.6.(i) Σ k,d δ (R m poly ) is the span of those elements of Σ k,d δ with well-defined total weight.(ii) (Σ k,d δ ) um is the lowest total weight space of Σ k,d δ (R m poly ). (iii) Σ k,d δ (R m poly ) has lowest h m -weight ν k,d δ = 2δ(ℓ + 1) -d e 0 -2k -d e 1 .Its lowest h m -weight space is the line spanned by ζ d-k β 2k-d 1 .

Lemma 4. 7 .

 7 As an s m -module, Σ k,d δ (R m poly ) is equivalent to the restricted dual of the relative Verma module U(s m ) ⊗ t + m Sym 2k-d C 2ℓ . Proof. By Lemmas 4.3 and 4.4, any non-zero t m -invariant subspace of Σ k,d δ (R m poly ) contains (Σ k,d δ ) um . Taking duals, we see that the dual of (Σ k,d δ ) um generates the restricted dual Σ k,d δ (R m poly ) * under the action of t m . It follows that Σ k,d δ(R m poly ) * is a quotient of U(s m ) ⊗ t + m (Σ k,d δ ) um * .The reader may easily check that the two have the same total weight space dimensions, so they are equivalent. The result now follows from Lemma 4.4(iii) and the fact that C 2ℓ is self-dual.

Proposition 4. 10 .

 10 Under the action of s m , the fine symbol modules have infinitesimal characters. Fix two fine symbol modules Σ k,d δ and Σ k ′ ,d ′ δ such that either k ′ < k, or k ′ = k and d ′ < d. They have the same infinitesimal characters if and only if at least one of the following four conditions holds:

.

  Theorem 5.2. For δ not contact-resonant (see Section 2.1), there exists a unique fine s mquantizationQ sm λ,µ : Σ δ → D λ,µ .Proof. Write χ k,d δ for the s m -infinitesimal character of Σ k,d δ , and (D λ,µ ) χ k,d δ for the subspace of D λ,µ on which the center Z(s m ) of U(s m ) acts by χ k,d δ . The χ k,d δ are distinct by Proposition 4.10, so we have the s m -decomposition D λ,µ = k,d (D λ,µ ) χ k,d δ The fine symbol map fσ k,d λ,µ restricts to the unique fine symbol-preserving s m -equivalence from (D λ,µ ) χ k,d δ to Σ k,d δ . The fine s m -quantization Q sm λ,µ is the direct sum of the inverses of these restrictions.

  be of total weight w. Use Lemma 4.5 to see that T maps polynomials to polynomials, and verify that End Poly(R m ) is C[x, y, z] [[∂ z , A, B]]. Hence End Poly(R m ) (w) = C[x, y, z] [∂ z , A, B] (w) .

  2k-d C 2ℓ , and for v ∈ -1 2 N, C[∂ z , A, B] (v) ∼ = r≥0 Sym -2v-2r C 2ℓ . Since ζ, ∂ ζ , and ∂ z are l s m -invariant, the result follows.

(

  k,d),(k ′ ,d ′ ) λ,µ = 0 under the conditions of Cases (i) and (ii). Therefore the space D (b)λ,µ (R m ) defined by D (b) λ,µ (R m ) := Q sm λ,µ 2d-k≤b Σ k,d δ is invariant under the action L λ,µ of K(R m ). Since under s m there is unique copy of Σ k,d δ in D λ,µ for all (k, d), D (b)λ,µ is the unique subspace of D λ,µ whose graded module is as in Theorem C. This completes the proof for M = R m . For M arbitrary, local existence and uniqueness allows us to conclude global existence and uniqueness of D (b) λ,µ (M ) as in the proof of Theorem A. 7. Explicit formulas 7.1. The affine invariants. We begin by proving that the following maps between fine symbol modules are equivariant with respect to the affine subalgebra t m : Definition. The contact divergence is the map Div

(

  ∂ ui X j )ξ j ∂ ξi .Observe that these two formulas are the same if and only if X ∈ b m . Thus as claimed, N is a b m -quantization.As usual, let E ξ denote the ξ-Euler operator. The full divergence operator isDiv = 1≤i≤m ∂ ui ∂ ξi : S k δ → S k-1 δ .

.I.-

  [L Σ δ (X xiz ), ∆ s ] = -s∆ s-1 β i ∂ ζ E ζ -δ(ℓ + 1) -1 2 (s -1) , This gives the recursion relationC s-1 = 1 2 s 2c -2δ(ℓ + 1) -(s -1) C s . Since C 0 = 1, the theorem follows.7.4. The subsymbol. In order to give an explicit local formula for sσ k λ,µ , let us fix a system of Darboux coordinates as in Section 3.1. An arbitrary differential operator T of order ≤ k may be expressed asT = c+|I|+|J|≤k T c,I,J ∂ c z ∂ I x ∂ J y ,where the T c,I,J are smooth functions. As usual, we replace ∂ z , ∂ x , ∂ y by their symbols ξ z , ξ x , ξ y , respectively. This amounts to replacing T byN -1 (T ) = c+|I|+|J|≤k T c,I,J ξ c z ξ I x ξ J y .The formula has two ingredients: the full divergence Div, and the projection π k-1,2(k-1) : The full divergence Div(N -1 (T )) isc+|I|+|J|≤k c∂ z (T c,I,J ) ξ c-s ∂ xs (T c,I,J ) ξ c z ξ I-es x ξ J y + J s ∂ ys (T c,I,J ) ξ c z ξ I x ξ J-es y In the (α, β, ζ)-coordinates on S k-1 δgiven in Section 3.4, the projection π k-1,2(k-1) simply gives the ζ k-1 term. In Darboux coordinates,π k-1,2(k-1) T c,I,J ξ k-1-|I|-|J| z ξ I x ξ J y = (-1) |I| 1 2 |I|+|J| y I x J T c,I,J ζ k-1 , because by (5), ξ xi = α i -1 2 y i ζ, ξ yi = -β i + 1 2 x i ζ, and ξ z = ζ. Proposition 7.4. In Darboux coordinates, the subsymbol sσ k λ,µ (T ) is given bysσ k λ,µ (T ) = π k-1,2(k-1) • 1 -(k -1) + 2λ(ℓ + 1) 2(k -1) -2(δ -1)(ℓ + 1) Div • N -1 (T ).Proof. Using (7) and Theorem 7.3, we obtain sσ k λ,µ (T ) = π k-1,2(k-1) • SQ sm δ symbol module in S k-1 δ of the highest contact order, (9) gives π k-1,2(k-1) • SQ sm δ -1 = π k-1,2(k-1) .

  • SQ sm δ . Proof. Observe that it suffices to prove that SQ sm δ intertwines the restrictions of the actions L Σ δ and L S δ to s m . By Lemma 7.1, SQ sm δ intertwines the t m -actions. Restricted to Σ k,d δ , it is of the form ∞ 0 ∆ s C s for some constants C s . One obtains (9) by deriving a recursion relation for these constants. Since X xiz generates s m under t m for any i, we need only impose the conditionL S δ (X xiz ) • SQ sm δ = SQ sm δ • L Σ δ (X xiz ). By Lemma 4.3, L S δ (X xiz ) -L Σ δ (X xiz ) is β i ∂ ζ , which commutes with ∆. Therefore we find that SQ sm δ • β i ∂ ζ and SQ sm δ , L Σ δ (X xiz ) must be equal, i.e., 0 = s ∆ s • β i ∂ ζ -∆ s , L Σ δ (X xiz ) C s .Using the same lemma, deduce the following commutator:

Acknowledgments. The first author was partially supported by Simons Foundation Collaboration Grant 207736. The second author was partially supported by Grant PICS05974, "PENTAFRIZ", of the CNRS. The article was begun while both authors were supported by the Research in Pairs program of the Mathematisches Forschungsinstitut Oberwolfach, for whose support they are very grateful. They are also pleased to thank Dmitry Fuchs, Jérôme Germoni, Peter Littelmann, Sophie Morier-Genoud, Sergei Tabachnikov, and André Unterberger for enlightening discussions.

Proof. Apply Corollary 4.9.

In order to apply this lemma, we must prove that Hom um Σ k,d δ , Σ

2 ) contains no copies of Sym 3 C 2ℓ under l s m for k ≤ d ≤ 2k. When d = 2k, by Lemma 6.1 we obtain

which is again a copy of C 2ℓ . Finally, when d < 2k -1, we obtain zero. This completes the proof of Theorem A when M = R m . If M is an arbitrary contact manifold, Darboux's theorem implies the existence of an atlas of local charts on M that are diffeomorphic to the standard contact structure on R m . We have just proved that for every chart U , there is a unique (locally defined) map sσ k λ,µ (U ) equivariant with respect to K(U ). If U ′ is another chart, uniqueness implies that the maps sσ k λ,µ (U ) and sσ k λ,µ (U ′ ) coincide on U ∩ U ′ . Therefore sσ k λ,µ (M ) is well-defined on all of M and obviously commutes with K(M ). 6.3. Proof of Theorem C. As in the proof of Theorem A, take M = R m and consider the action L λ,µ of K(R m ) on Σ δ . We first prove that certain of its matrix elements

Lemma 6.4. In both of the following cases, the space of s m -relative 1-cochains of

Proof. By Lemma 6.3, we must prove that under l s m the space Hom

) contains no copies of Sym 3 C 2ℓ in either Case (i) or Case (ii). By Lemma 6.1, this space is zero unless d -d ′ ∈ Z + . In particular, it is zero in Case (i). In Case (ii) with d -d ′ ∈ Z + , Lemma 6.2 shows that under l s m it is equivalent to


Romuald Lepers
email: romuald.lepers@u-bourgogne.fr

Paul J Stapley

Age-related changes in conventional road versus off-road triathlon performance

Keywords: Master athlete, Swimming, Cycling, Mountain biking, Running, Xterra 

Age-related changes in conventional road versus off-road triathlon performance

Introduction

The age-related decline in endurance running performance up to the marathon distance has been well documented (e.g. [START_REF] Trappe | Marathon runners. How do they age?[END_REF][START_REF] Tanaka | Endurance exercise performance in Masters athletes: ageassociated changes and underlying physiological mechanisms[END_REF]. In general, peak endurance performance is maintained until 35 years of age, followed by modest decreases until 50-60 years of age, and a progressively steeper decline thereafter [START_REF] Tanaka | Endurance exercise performance in Masters athletes: ageassociated changes and underlying physiological mechanisms[END_REF]. In contrast, less is known about the age-related decline in performance of other modes of locomotion such as swimming [START_REF] Donato | Declines in physiological functional capacity with age: a longitudinal study in peak swimming performance[END_REF] and cycling (Balmer et al 2008a,b). The nature and the type of muscular contractions differ between the three modes of locomotion (swimming, cycling, and running). For example, cycling is a non-weight bearing activity that employs predominantly concentric muscle activation compared with the stretch-shortening activity and eccentric activations during running. Swimming involves mostly the upper body musculature with no mechanical constraints. Therefore, questions about whether age-related declines in performance differ between the modes of locomotion may be raised.

Triathlon, involving successive swimming, cycling and running bouts of exercise, represents an interesting model to examine the age-related changes in endurance performance in these different locomotion modes [START_REF] Bentley | Specific aspects of contempory triahlon: implications for physiological analysis and performance[END_REF]. It has been shown that the agerelated declines in performance times were specific for each locomotion mode. For example, there is a lesser age-related decline in cycling performance compared with running and swimming performances for both short (1.5 km swim, 40 km cycling, and 10 km running) and long (3.8 km swim, 180 km cycling, and 42.2 km running) distance triathlons [START_REF] Bernard | Age-related decline in Olympic triathlon performance: effect of locomotion mode[END_REF], Lepers et al. 2010). One explanation for the different age-related declines in cycling and running may involve the mechanical power required by these disciplines. Mechanical power output in running depends on the velocity (P = k.V) while it depends on third power of velocity in cycling (P = k.V 3 ). If we assume that the changes in aerobic capacity (e.g. VO 2 max) with age are directly related to the decline in mechanical power, a similar reduction in power output for running and cycling with advancing age would induce a lower reduction in cycling velocity than running velocity (Lepers et al. 2010). This may explain, in part, why the magnitude of the decrease in cycling performance with age was less than that for running. In addition, the lesser decline in cycling performance with advancing age could also be due to a lesser reduction in lactate threshold and/or economy compared with running.

In addition, it has been shown that in contrast to swimming, the magnitude of the declines in cycling and running performances with age in short distance triathlon were less pronounced than for long distance races (Lepers et al. 2010). Greater neuromuscular fatigue, greater muscle damage, and changes in substrate availability and utilisation with age could explain the task duration effect on triathlon performances at least in cycling and running.

In the field of endurance multisport, off-road triathlon (also named Xterra) combining swimming, mountain biking (MTB) and trail running represents another interesting model to examine the age-related changes in endurance performance in different locomotion modes. Since 1996, the world's premier off-road triathlon, also named the Xterra World championship, has comprised a 1.5 km swim, a 30 km MTB ride, and a 11 km trail run, and has taken place on the island of Maui (Hawaii, USA), [START_REF] Lepers | Differences in gender and performance in off-road triathlon[END_REF].

In contrast to conventional (road-based) triathlon, cycling and running parts of off-road triathlon take place on trails, with frequently significant ascents in the cycling leg of the race.

The major difference between off-road and conventional triathlon is found in the cycling discipline, which is performed as MTB during off-road triathlon. Physiological and technical requirements of mountain biking differ in many points from road cycling. Factors other than aerobic power and capacity such as off-road cycling economy, anaerobic power and capacity, and technical ability might influence off-road cycling performance [START_REF] Impellizzeri | The physiology of mountain biking[END_REF]. Similarly, running on trails is specific and also requires more technical ability than road running [START_REF] Anderson Ls | The impact of footwear and packweight on injury and illness among long-distance hikers[END_REF][START_REF] Hoffman | The Western states 100-mile endurance run: participation and performance trends[END_REF]. During off-road triathlon, the age-related declines in cycling and running performances maybe greater because the physiological and mechanical stresses that occur during off-road triathlon are much greater compared to those encountered during road-based triathlon.

Therefore, the purposes of this study were: 1) to determine if there was a reduced rate of decline in cycling performance with advancing age compared to running and swimming for the off-road triathlon, as has been previously shown for road-based triathlons (Bernard at al. 2010, Lepers et al. 2010), and 2) to compare age-related changes in swimming, cycling, and running performances between the road-based and off-road triathlons. Due to the nature of the disciplines with stochastic efforts (i.e. with a greater anaerobic component), mountain biking and trail running require different physiological and technical aptitudes compared to conventional triathlon cycling and running. We therefore hypothesized that age-related declines in cycling and running performance would be greater for off-road than for conventional road-based triathlon.

Methods

Approval for the project was obtained from the Burgundy University Committee on Human Research. This study involved the analysis of publicly available data so content was waived.

Averaged swimming, cycling, and running and overall time performances of the top 5 amateurs of ten male age groups were analyzed for road-based (short distance) and off-road triathlon World Championships in 2009. The age groups distinguish the categories for each period of 5 years as follows: 20-24 yrs, 25-29 yrs, 30-34 yrs, 35-39 yrs, 40-44 yrs, 45-49 yrs, 50-54 yrs, 55-59 yrs, 60-64, and 65-69 yrs. For the road-based short distance triathlon, the data set was obtained from the 2009 World Championships web site (http://www.worldtriathlongoldcoast.com/). This race took place at the Gold Coast, Australia ; the road-based World Championships was a point-to-point race, combining a 1.5 km swim in the ocean with wetsuit (Water temperature: 21°C), a 40 km [START_REF] Lepers | Differences in gender and performance in off-road triathlon[END_REF]. The Xterra Maui triathlon is the last in a series of nearly 20 offroad triathlons held in the world that serve as qualifying races. The race took place on October 25 th 2009; during the race ambient temperature and relative humidity were about 28°C and 60%, respectively.

For both based-road short distance and off-road triathlons, the swimming, cycling, running and overall time performances of each triathlete finishing in the top 5 of each age group was normalized to the mean time performance of the top 5 of the best performing age group. Thus, the age-related declines in performance were expressed using a ratio calculated between the individual and the mean time performances of the best performing age group for each mode of locomotion and total event time.

Statistical analysis

Data are reported as means  SD in the text and table and displayed as means  SE in the figures. A two-way ANOVA with repeated-measures on locomotion mode was used to compare performance ratio between modes of locomotion across ages for road-based and offroad triathlons. For each mode of locomotion and total time performance, a two way ANOVA (age x type of triathlon) was used to compare performance ratios between road-based versus off-road triathlon across age. Post hoc analysis (Tukey) was used to test for differences within the ANOVAs when appropriate. A significance level of P < 0.05 was used to identify statistical significance.

Results

At the road-based triathlon World Championship, there were 1537 finishers: 925 males (60%) and 612 females (40%), and the finishing rate was 97.4%. At the off-road triathlon World Championships, there were 486 finishers: 336 males (69%) and 150 females (31%), and the finishing rate was 94.5%. Swimming, cycling, running and total time performances for each age group for both roadbased and off-road triathlons are presented in Table 1. The relative contribution of each part of the overall performance remained constant across the ages but differed slightly between the two triathlon types. For off-road triathlon, the relative contribution of the swim part was lower while the cycle part was greater compared to road-based triathlon. For road-based triathlon, the 1.5-km swimming, 40-km cycling, and 10-km running represented on average 17.3  0.8, 49.1  1.2, 31.2  1.2 % of the total time, respectively. For off-road triathlon, the 1.5-km swimming, 30-km mountain biking, and 11-km running represented on average 12.7  1.2, 57.8  1.9, 29.4  2.0 % of the total time, respectively.

Standard deviations of mean values for each locomotion mode and for total time performances were greater for off road triathlon compared to road-based triathlon, suggesting that the density of the top 5 was lower for off-road triathlon.

(Insert table 1)

For both road-based and off-road triathlons, the performance ratios for each mode of locomotion decreased in a curvilinear manner with advancing age (Figures 1A and1B). For road-based triathlon (Figure 1A), there was a locomotion mode effect (F = 12.9, P < 0.0001) for performance ratio. Independently of age, both swimming and running performance ratios were lower (P < 0.01) than cycling performance ratios. In addition, independently of locomotion mode, performance ratio differed between the age groups (F = 117.2, P < 0.0001).

Performance ratio was significantly lower (P < 0.01) for age groups 45-49 yrs and above compared to all age groups between 20-24 yrs and 35-39 yrs. Finally, there was a significant interaction between age and mode of locomotion for performance ratio (F = 2.5, P < 0.002).

For off-road triathlon however, there was no interaction between age and mode of locomotion (F = 1.45, P = 0.13) and there was no locomotion mode main effect (F= 0.29, P = 0.74) for performance ratio. However, independently of locomotion mode, performance ratio differed between the age groups (F = 31.9, P < 0.0001). Performance ratio was significantly lower (P < 0.01) for age groups 55-59 yrs and above compared to all previous age groups comprised between 20-24 yrs and 40-44 yrs.

(Insert Figures 1A & 1B)

Comparisons of age-related declines in swimming, cycling, running and total performance for road-based and off-road triathlons are shown in Figure 2. There was a significant interaction between age and type of triathlon for swimming (F = 2.85, P = 0.01), cycling (F = 10.03, P < 0.0001), running (F = 5.36, P < 0.0001), and total event (F = 12.04, P < 0.0001) performance ratio. Age-related declines in performance were less pronounced for road-based compared to off-road triathlon for swimming for age group 65-69 yrs, for cycling for age groups 50-54 yrs and above, for running for age groups 60-64 yrs and 65-69 yrs and for total time for age groups 55-59 yrs and above, respectively.

(Insert Figure 2)

Discussion

The main findings of the present study were i) a reduced age-related decline in cycling performance compared to running and swimming performances for road-based triathlon but not for off-road triathlon, and ii) a smaller effect of age on swimming, cycling and running performances for road-based compared to off-road triathlon.

Methodological considerations

With endurance events, the environment (temperature, humidity, precipitation) affects the outcome of the results. For road-based triathlon taking place on the Gold Coast (Australia), the temperature was around 22°C, whereas during the off-road triathlon taking place in Hawaii (USA), the temperature was around 28°C. Thus, the hotter conditions for the off-road triathlon may have more greatly affected performance than for the road-based triathlon. However, by normalizing the individual time performance by the mean time performance of the top 5 of the best performing age groups for both type of triathlons, the influence of environment was minimized when comparing road-based and off road triathlon performances.

The present paper focuses on the performances of male triathletes only. Potential differences in age-related decline in triathlon performance between male and female may also occur (e.g. [START_REF] Vleck | Pacing during an elite Olympic distance triathlon: Comparison between male and female competitors[END_REF][START_REF] Lepers | Age and Gender Interactions in Ultra-Endurance Performance: Insight from Triathlon[END_REF], so it could be interesting to verify if similar results are obtained for female triathletes. However, for off-road triathlon, data for female triathletes older than 55 years of age are limited. Moreover, unpublished data from our laboratory suggest that similar findings are observed for female triathletes i.e. a reduced agerelated declines in cycling performance compared with running and swimming performances for road-based triathlon but not for off-road triathlon and a greater magnitude of age-related decline in performance for the off-road triathlon than for the road-based triathlon.

The present data do not include performance of elite/professional triathletes. This is because differences existed between the race formats for off-road and road events. At the World off-road triathlon Championships, elite and age grouper triathletes raced the same course at the same time; at the World road-based short distance triathlon Championships, elite triathletes raced a different course with different rules (drafting allowed for the elite athletes and forbidden for the age groupers).

Age-related differences in triathlon performance differ between road-based versus off-road triathlon

Age-related declines in swimming, cycling and running performances were lesser for road-based triathlon than for off-road triathlon. Lepers et al. (2010) found that age-related changes in swimming performance were not influenced by task duration; indeed the magnitudes of decreases in swimming performance were similar for short distance versus long distance triathlon. The swimming distance was similar between road-based and off-road triathlon (i.e. 1.5 km) but for road-based triathlon wetsuits were permitted while for off-road triathlon they were forbidden as the water temperature was close to 28°C. The use of wetsuits may explain firstly the fastest swimming times at the road-based triathlon than at the off-road triathlon World Championships (see table 1), and secondly the lower declines in performance at older ages for road-based triathlon. Indeed, wearing a wetsuit improves swimming performance and propulsion efficiency, but also reduces gross energy consumption in the swimming portion of triathlon races [START_REF] Tomikawa | Factors related to the advantageous effects of wearing a wetsuit during swimming at different submaximal velocity in triathletes[END_REF]. We unfortunately had no possibility of testing the swimming abilities of road-based and off-road triathletes. However, if one assumes that the advantage of wearing a wetsuit may be greater for slower swimmers (and therefore oldest swimmers) compared to the faster (younger) ones, this could explain the lower declines in swimming performance at older ages for road-based triathlon where wetsuits were used. On average, for off-road triathlon swimming performance (without wetsuits), declined by ~16% at 55 years old, and reached ~37% by 65 years of age. The present decline is somewhat greater than those found by [START_REF] Tanaka | Age and gender interactions in physiological functional capacity: insight from swimming performance[END_REF] (~32% at 65 yrs age) at the same distance swum in a swimming pool. The lower performance declines observed by these authors for the same age compared with the triathlon data might be due the differences between the water conditions: calm in swimming pool versus stochastic in open water for triathlon.

For road-based short distance triathlon, there was a lesser age-related decline in cycling performance compared to running and swimming performances. Similar findings have been already documented by [START_REF] Bernard | Age-related decline in Olympic triathlon performance: effect of locomotion mode[END_REF] for short distance triathlon and by Lepers et al. (2010) for long distance triathlon. In contrast, for off-road triathlon, cycling (mountain biking) performance declined with advancing age in the same manner to that seen in swimming and running. The absence of relative preservation of cycling performance with advancing age for off-road triathlon suggests that the effect of age on cycling performance is amplified for mountain biking compared to road cycling.

Indeed, the magnitude of the declines in cycling performance with advancing age for road-based triathlon was less pronounced than for off-road triathlon. It has been shown that task duration in triathlon exerts an influence on age-related declines in triathlon performance. Lepers et al. (2010) showed that age-related decline in cycling performance was greater for long distance triathlon (180 km cycle) compared to short distance triathlon (40 km cycle). For example, these authors found that cycling performance declined by ~20% for short distance triathlon and by 26% for long distance triathlon at 65 years, respectively. In the present study, cycling performance declined by ~17% at 65 years for short distance triathlon and by 39% for off-road triathlon, respectively. The longer duration of mountain biking part of off-road triathlon ( 2h) compared to road cycling part at short distance triathlon ( 1h) may not totally explain the greater decline in cycling performance in off-road triathlon with advancing age.

The nature of the discipline may better explain the greatest part of that difference. Indeed, mountain biking differs in many respects from road cycling. The duration of the MTB leg (2h) of the off-road triathlon is very similar to that of a cross-country MTB race for which it has been shown that more than 80% of the time is spent above lactate threshold [START_REF] Impellizzeri | Exercise intensity during off-road cycling competitions[END_REF]. This very high intensity (i.e. above lactate threshold) is related in particular to climbs, forcing off-road cyclists to expend most of their effort going against gravity, greater rolling resistance associated with the difficult terrain conditions, and the isometric contractions of arm and leg muscles necessary for bike handling and stabilisation. Therefore, a known decrease in power-to-weight ratios with advancing age [START_REF] Bonnefoy | Peak anaerobic power in elderly men[END_REF][START_REF] Kostka | Quadriceps maximal power and optimal shortening velocity in 335 men aged 23-88 years[END_REF] inevitably leaves older off-road triathletes with a disadvantage in the climbs compared to flat sections. Secondly, lower arm and leg muscle strength in older athletes [START_REF] Faulkner | The aging of elite male athletes: agerelated changes in performance and skeletal muscle structure and function[END_REF]) may also mean that they possess a reduced bike stabilisation capacity during the descents, even if decreases in muscle strength may be reduced in older athletes compared to sedentary persons of the same age [START_REF] Mccrory | Thigh muscle strength in senior athletes and healthy controls[END_REF]. Older triathletes even if they compete in off-road triathlon may probably be less experienced off-road cyclists compared to older mountain bikers. Finally, age-related changes in technical ability to control and stabilize the bicycle, if it exists [START_REF] Sturnieks | Balance disorders in the elderly[END_REF], could be another factor explaining the greater decline in MTB performance compared to road cycle performance.

The age-related declines in running performance were greater for off-road triathlon than for road-based triathlon. It has been suggested that in older athletes, skeletal muscle continues to have high aerobic potential, while declines in muscle size and contractile performance are apparent [START_REF] Trappe | Marathon runners. How do they age?[END_REF][START_REF] Tanaka | Endurance exercise performance in Masters athletes: ageassociated changes and underlying physiological mechanisms[END_REF]. These changes in the skeletal muscle profile may contribute partly to the alterations in running performance seen with increasing with age, but the difference in age-related changes in performances between trail versus road running remains unclear. Lepers et al. (2010) showed that the magnitude of the decline in running performance with advancing age was greater for long distance triathlon (42 km run) compared to short distance triathlon (10 km run). Indeed, running performance declined by ~27% for short distance triathlon and by 33 % for long distance triathlon at 65 years, respectively. In the present study, running performance declined by ~22% at 65 years for short distance triathlon and by 35% for off-road triathlon. As for cycling, the longer duration of the trail running part during off-road triathlon ( 1h) compared to the road running part during short distance triathlon ( 40 min) may explain only in part, the greater decline in running performance in off-road triathlon with advancing age.

Age-related changes in trail running performance have been few investigated and for distances much greater than 11 km. For example, [START_REF] Easthope | Effects of a trail running competition on muscular performance and efficiency in well-trained young and master athletes[END_REF] showed that performance of masters athletes (45 yrs of age) during a 55 km trail running race did not differ from those of young athletes (30 yrs of age), and that both group of athletes showed similar fatigue and muscle damage after the race even if recuperation seemed slower in masters athletes. [START_REF] Hoffman | The Western states 100-mile endurance run: participation and performance trends[END_REF] found that the finish times of a 161Km trail running race were about 40% longer for athletes of age 60-69 yrs compared to fastest athletes of age 30-40 yrs.

The specificity of the 11 km trail run for the off-road triathlon with running part along the beaches and lava formations with a total altitude change of 346 m affected more the running performance compared to a 10 km flat road run but the reasons are unclear. Lower capacity to assimilate the variation in running intensity and lower ability to change the biomechanics of running (foot support, stride frequency) due to the changes in surface (sand, rocks, etc…) may explain the lower running performances of older triathletes during off-road triathlon compared to road-based triathlon.

Conclusion

This cross sectional study found that the magnitudes of age-related declines in triathlon performance for off-road triathlon are greater than for road-based triathlon, suggesting that the type of discipline (swimming with or without wetsuit, road versus MTB cycling, and road versus trail running) exerts an important influence on the age-associated changes in triathlon performance. The specific aspects of mountain biking and trail running may explain why agerelated decline in off-road triathlon is more pronounced for off-road compared to conventional on road triathlon. Further studies examining the changes in training volume and physiological characteristics of older road-based and off-road triathletes are required to better understand the age-associated changes in triathlon performance and will provide valuable information for understanding the way to maintain physical capacity and performance with advanced age. In addition, future analysis of the age-related changes in performance for other endurance or ultra-endurance single discipline events, such as mountain bike races and trail or road running races, may also provide valuable insights into the maintenance of performance in older athletes.

Figure 1

Age-related declines in swimming, cycling and running performances for road-based (Panel A) and off-road triathlons (Panel B), (mean  SE). For road-based triathlon, the swimming performance ratio for age groups 45-49 yrs and above was significantly different from the best age group (i.e. 20-24 yrs). The cycling performance ratio for age groups 55-59 yrs and above was significantly different from the best age group (i.e. 25-29 yrs). The running performance ratio for age groups 50-54 yrs and above was significantly different from the best age group (i.e. 20-24 yrs). For off-road triathlon, performance ratio differed between the age groups, independently of locomotion mode. Performance ratio was significantly lower for age groups 55-59 yrs and above compared to all previous age groups comprised between 20-24 yrs and 40-44 yrs. * : P< 0.05, ** : P < 0.001: Cycling significantly different from running.

Figure 2

Age-related declines in swimming, cycling, running and total performances for road-based and off-road triathlons (mean  SE). Age-related decline in swimming, cycling, running, and total performance was less pronounced for road-based triathlon. *: P < 0.05 ; ** : P < 0.01: Significantly different from off-road triathlon for the same age group.

( 2

 2 laps of 20 km) road bike ride almost flat (altitude change < 5 m) and a 10 km flat run on the road along the ocean. The race took place on September 12 th 2009; during the race ambient temperature and relative humidity were about 22°C and 65%, respectively. For the off-road (Xterra) triathlon, the data set was obtained from the Xterra triathlon World Championships web site: http://www.xterraplanet.com/maui/past.html. Since 2005 the Xterra World Championships has taken place in Makena, Maui (Hawaii, USA) and held each year in October with only minor changes to the course. The championships is a point-to-point race, combining a 1.5 km swim in the Pacific ocean without wetsuit (Water temperature: 28°C), a 30km MTB ride that involves a total altitude change of 915 m on rocky, dirt-strewn trails on the lower slopes of Haleakala Volcano, and a 11 km trail run along the beaches and lava formations along south Maui's scenic south shore (total altitude change of 346 m)

  Figure1

Conflict of interest

The authors have non conflict of interests to report.