# Conformally equivariant quantization for spinning particles

J.-P. Michel  University of Luxembourg, Campus Kirchberg, Mathematics Research
Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Grand Duchy
of Luxembourg<EMAIL_ADDRESS>
###### Abstract.

This work takes place over a conformally flat spin manifold $(M,\text{})$. We
prove existence and uniqueness of the conformally equivariant quantization
valued in spinor differential operators, and provide an explicit formula for
it when restricted to first order operators. The Poisson algebra of symbols is
realized as a space of functions on the supercotangent bundle
${\mathcal{M}}=T^{*}M\oplus\Pi TM$, endowed with a symplectic form depending
on . It admits two different actions of the conformal Lie algebra: one
tensorial and one Hamiltonian. They are intertwined by the uniquely defined
conformally equivariant superization, for which an explicit formula is given.
This map allows us to classify all the conformal supercharges of the spinning
particle in terms of conformal Killing tensors, which are symmetric, skew-
symmetric or with mixed symmetry. Higher symmetries of the Dirac operator are
obtained by quantization of the conformal supercharges.

###### Key words and phrases:

Quantization, Conformal geometry, graded geometry, spinor differential
operators, Dirac operator, higher symmetries.

###### 2010 Mathematics Subject Classification:

17B66, 17B81, 53A30, 53D55, 58J70, 70S10, 81S10

I thank the Luxembourgian NRF for support via the AFR grant PDR-09-063.

##  1\. Introduction

Whereas there exists a pseudo-classical model for the spinning particle, due
to Berezin and Marinov [4], quantization is scarcely developed in that
setting. We propose in this paper a natural extension of the conformally
equivariant quantization for spinning particles, so that it is valued in
spinor differential operators. We start this introduction with the concept of
quantization and especially of conformally equivariant quantization of
cotangent bundles as introduced in [12]. Then, we review the formalism of
pseudo-classical spinning particle and the known quantizations in that
context. Finally we present our main results and detail the content of the
paper.

### 1.1.

The prototypical example of quantization is the one of cotangent bundles
$T^{*}M$, endowed with their canonical symplectic structure. The quantum phase
space $\mathcal{H}$ is then given by the Hilbert space of square integrable
functions on the smooth manifold $M$. The functions on $T^{*}M$ are expected
to have a linear operator on $\mathcal{H}$ as quantum counterpart. In
particular, we call quantization of $T^{*}M$ a linear map
${\mathcal{Q}}:\mathrm{Pol}(T^{*}M)\rightarrow\mathcal{D}(M)$, valued in the
space of differential operators $\mathcal{D}(M)$ and which is the inverse of a
full symbol map. The algebra of polynomial fiberwise functions
$\mathrm{Pol}(T^{*}M)$, or equivalently the algebra of symmetric tensors
$\Gamma({\mathcal{S}}TM)$, identifies to the graded algebra of symbols
associated to $\mathcal{D}(M)$. Via the principal symbol map, it inherits of a
Poisson bracket and an Hamilonian action of vector fields. The first one
coincide with the canonical Poisson bracket on functions of $T^{*}M$, or with
the Schouten bracket on $\Gamma({\mathcal{S}}TM)$, and the second one with the
natural $\mathrm{Vect}(M)$-action on
$\mathrm{Pol}(T^{*}M)\cong\Gamma({\mathcal{S}}TM)$.

The celebrated Weyl quantization of $T^{*}M$ is characterized, if
$M=\mathbb{R}^{n}$, by its equivariance property under the action of the
symplectic affine Lie algebra
$\mathfrak{sp}(2n,\mathbb{R})\ltimes\mathbb{R}^{n}$ on $T^{*}M$. Focusing
rather on equivariance under a Lie algebra ${\mathfrak{g}}$ acting by vector
fields on $M$, Duval, Lecomte and Ovsienko have introduced the concept of
${\mathfrak{g}}$-equivariant quantization. In particular, they prove its
existence and uniqueness for ${\mathfrak{g}}$ the projective [27] or the
conformal Lie algebra [12], acting locally on manifolds $M$ which are
projectively or conformally flat respectively. It has been intensely developed
since then, see e.g. [8].

A deep motivation to build quantizations is to obtain a correspondence between
classical and quantum symmetries whenever there exists one, and an efficient
way to measure the quantum anomalies else. For a free massive scalar particle
on $(M,\text{})$, this amounts to determine whether the first integrals of the
geodesic flow of , given by symmetric Killing tensors, can be quantized into
differential operators commuting with the Laplacian. This has been
investigated up to second order by Carter, via his ad hoc minimal quantization
procedure [10]. There is no quantum anomalies in a number of examples, see
e.g. [14], in particular if $(M,\text{})$ is Ricci-flat. Over a conformally
flat manifold, the situation is now well-understood thanks to the conformally
equivariant quantization introduced above. Indeed, the latter establishes a
bijection between the classical and quantum symmetries of a free massless
particle [32], i.e. between the constants of motion along the null geodesic
flow, given by conformal Killing symmetric tensors, and the higher symmetries
of Laplacian, studied by Eastwood [16]. Moreover, this correspondence extends
to the massive case, for the symmetries given by differential operators of
second order, and leads ultimately to new quantum integrable systems [1, 15].

### 1.2.

To develop the spin counterpart of the preceding picture, we suppose that
$(M,\text{})$ is a spin manifold, with spinor bundle $S$. A quantum particle
is now described by a square integrable section of $S$ and we focus on the
algebra of differential operators $\mathcal{D}(M,S)$ acting on such sections.
Weyl quantization [42] and conformally equivariant quantization [8] have been
generalized to that setting, but with the usual algebra of symbols
$\mathrm{Pol}(T^{*}M)\otimes_{{\mathcal{C}^{\infty}}(M)}\Gamma(\mathrm{\mathbb{C}l}(M,\text{}))$
as source space. We rather follow Getzler [18], which uses in addition the
filtration of the Clifford bundle, to derive a supercommutative bigraded
algebra of symbols from the usual one. This superalgebra identifies to the one
of tensors $\Gamma({\mathcal{S}}TM\otimes\Lambda T^{*}M)$ or equivalently to
the algebra $\mathcal{O}({\mathcal{M}})$ of fiberwise polynomial functions on
the supercotangent bundle ${\mathcal{M}}=T^{*}M\oplus\Pi TM$, where $\Pi$
denotes the reverse parity functor. This supermanifold is precisely the phase
space introduced by Berezin and Marinov [4] to deal with pseudo-classical
spinning particles on $M=\mathbb{R}^{n}$. Besides, the author proved in [31]
the existence of a Hamiltonian filtration on $\mathcal{D}(M,S)$, assigning
order $2$ to spinor covariant derivatives $\nabla_{X}$, with
$X\in\mathrm{Vect}(M)$, and order $1$ to the Clifford elements $\gamma(\xi)$,
with $\xi\in\Omega^{1}(M)$. This filtration is compatible with the commutator
of $\mathcal{D}(M,S)$ and induces a Poisson bracket and a new gradation on
${\mathcal{O}}({\mathcal{M}})$. This grading stems from the identification of
${\mathcal{M}}$ with the graded manifold $T^{*}[2]M\oplus T[1]M$, whereas the
Poisson bracket comes from the symplectic structure on ${\mathcal{M}}$ induced
by the metric . The latter follows from a general construction, due to
Rothstein in the super setting [36] and to Roytenberg in the graded setting
[37]. In addition, the obtained symplectic structure on ${\mathcal{M}}$
corresponds to the one coming from the Lagrangian of a free pseudo-classical
spinning particle on $(M,\text{})$ [35, 26], so that the graded Poisson
algebra ${\mathcal{O}}({\mathcal{M}})$ is a classical counterpart to
$\mathcal{D}(M,S)$. This is confirmed by the geometric quantization scheme,
which associates the Hilbert space of square integrable spinors to the
supercotangent bundle ${\mathcal{M}}$ [31].

We name quantizations of ${\mathcal{M}}$ the linear maps
${\mathcal{Q}}:{\mathcal{O}}({\mathcal{M}})\rightarrow\mathcal{D}(M,S)$ which
are the inverse of a full symbol map for the Hamiltonian filtration of
$\mathcal{D}(M,S)$. The Weyl quantization developed by Getzler [18] and
Voronov [40] is precisely of this type. In particular, it enables to quantize
supercharges linear in momenta, built from Killing forms, into symmetries of
the Dirac operator $\not{D}$ [19, 39]. However, Weyl quantization fails to
quantize conformal supercharges, i.e. constants of motion of free massless
spinning particles, into higher symmetries of the Dirac operator, even the
simplest ones built from conformal Killing forms [2, 3].

### 1.3.

In this paper, we prove existence and uniqueness of a family
$(\mathcal{Q}^{\lambda,\mu})_{\lambda,\mu\in\mathbb{R}}$ of quantizations of
the supercotangent bundle which are conformally equivariant, assuming that the
base manifold $(M,\text{})$ is conformally flat. Such a map
$\mathcal{Q}^{\lambda,\mu}$ allows, for right parameters $(\lambda,\mu)$, to
quantize all the conformal supercharges into higher symmetries of the Dirac
operator, i.e. into operators $D_{1}\in\mathcal{D}(M,S)$ such that
$\not{D}D_{1}=D_{2}\not{D}$ for some $D_{2}\in\mathcal{D}(M,S)$. In
particular, we recover the symmetries of $\not{D}$ given by first order
differential operators, classified previously by Benn and Kreiss [3]. By the
way, we also build a family $(\mathfrak{S}^{\delta})_{\delta\in\mathbb{R}}$ of
so-called conformally equivariant superizations and provide explicit formulæ
for them. For $\delta=0$, the map $\mathfrak{S}^{\delta}$ establishes a
correspondence between conformal Killing tensors with mixed symmetry, lying in
$\Gamma({\mathcal{S}}TM\otimes\Lambda T^{*}M)$, and conformal supercharges.
Hence we obtain a classification of the conformal supercharges, which extends
previous results by physicists [19, 39] and may prove useful for integrability
of spinning particle motion, like in [26]. Future applications to separability
of the Dirac equation are also expected. This work is a first step toward the
study of the algebra structure of the higher symmetries of the Dirac operator,
which should be of interest for higher spin field theories.

### 1.4.

We detail now the content of the present paper.

In Section $2$, we review the needed results of [31], for conformally
equivariant quantization of ${\mathcal{M}}$ to make sense, i.e. we define the
gradation and the filtration on the source and target spaces of such
quantization and provide the actions of the Lie algebra ${\mathfrak{g}}$ of
conformal Killing vector fields on them. All through this section, we assume
that $(M,\text{})$ is conformally flat and of signature $(p,q)$ so that
${\mathfrak{g}}\cong\mathfrak{o}(p+1,q+1)$. Its action on the spinor bundle
$S$ is given by the Kosmann's Lie derivative of spinors [25], which admits a
one parameter deformation obtained by tensoring $S$ with a real power of the
determinant bundle, i.e. with a density bundle. We denote by
$(\mathsf{D^{\lambda,\mu}})_{\lambda,\mu\in\mathbb{R}}$ the family of
${\mathfrak{g}}$-modules defined by the induced deformed adjoint actions of
${\mathfrak{g}}$ on $\mathcal{D}(M,S)$. According to [31], for given weights
$\lambda,\mu$, the action of ${\mathfrak{g}}$ on $\mathsf{D^{\lambda,\mu}}$
induces two different actions on the symbol algebra
${\mathcal{O}}({\mathcal{M}})$. The first one is Hamiltonian and induced by
the principal symbol map associated to the Hamiltonian filtration of
$\mathcal{D}(M,S)$, it depends only on the shift $\delta=\mu-\lambda$ and
leads to the ${\mathfrak{g}}$-module denoted $\mathsf{S}^{\delta}$. It should
be understood as the module of classical observables. The latter
${\mathfrak{g}}$-action does not preserve the bigradation of
${\mathcal{O}}({\mathcal{M}})\cong\Gamma({\mathcal{S}}TM\otimes\Lambda
T^{*}M)$, but via the principal tensorial symbol map, it induces a second
${\mathfrak{g}}$-action on ${\mathcal{O}}({\mathcal{M}})$ which does. We
denote by $\mathsf{T}^{\delta}$ this extra ${\mathfrak{g}}$-module, which
identifies to the tensor module $\Gamma({\mathcal{S}}TM\otimes\Lambda T^{*}M)$
up to twist by appropriate density bundles. Then, we define the conformally
equivariant superization
$\mathfrak{S}^{\delta}:\mathsf{T}^{\delta}\rightarrow\mathsf{S}^{\delta}$,
inverse to a full tensorial symbol map, and the conformally equivariant
quantization
$\mathcal{Q}^{\lambda,\mu}:\mathsf{S}^{\delta}\rightarrow\mathsf{D^{\lambda,\mu}}$,
inverse to a full Hamiltonian symbol map. The name superization comes from the
inclusion $\mathrm{Pol}(T^{*}M)\subset\mathsf{T}^{0}$ as
$\mathrm{Vect}(M)$-modules, hence to each classical observable in
$\mathrm{Pol}(T^{*}M)$ corresponds via $\mathfrak{S}^{0}$ a spinning analog in
$\mathsf{S}^{0}$.

In Section $3$, we introduce the building blocks of the maps
$\mathfrak{S}^{\delta}$ and $\mathcal{Q}^{\lambda,\mu}$, which are the
isometric invariant differential operators acting on $\mathsf{T}^{\delta}$.
They form an algebra which is easily determined thanks to the Weyl's theory of
invariants [41], it is generated by $13$ operators, most of which are well-
known like divergence, gradient, de Rham differential… Some variants of this
algebra naturally appear in the context of Howe dual pairs [24, 28] and were
investigated over curved backgrounds [20, 21, 22]. Except the computation of
the Casimir operators of the three ${\mathfrak{g}}$-modules
$\mathsf{T}^{\delta}$, $\mathsf{S}^{\delta}$ and $\mathsf{D^{\lambda,\mu}}$,
the material in this section is probably not new. However, the presentation,
which uses the identification of differential operators on the tensor space
$\mathsf{T}^{\delta}$ with scalar differential operators on ${\mathcal{M}}$,
is original and proves to be convenient.

In Section $4$, we prove our main theorem: existence and uniqueness of the
conformally equivariant superization
$\mathfrak{S}^{\delta}:\mathsf{T}^{\delta}\rightarrow\mathsf{S}^{\delta}$ and
quantization
$\mathcal{Q}^{\lambda,\mu}:\mathsf{S}^{\delta}\rightarrow\mathsf{D^{\lambda,\mu}}$,
with $\delta=\mu-\lambda$, except for critical weights $\delta$. So far, only
the existence and uniqueness of the composition of these two maps was known,
as a consequence of the general work [8] on equivariant quantizations. Our
proof follows the one of existence and uniqueness of conformally equivariant
quantization in the scalar setting [12]. The main point is the diagonalization
of the Casimir operator of $\mathsf{T}^{\delta}$, which proves to be
equivalent to the harmonic decomposition of the polynomial space
${\mathcal{S}}\mathbb{R}^{n}\otimes\Lambda\mathbb{R}^{n}$. After deriving it,
we discover that it was the purpose of [23]. We nevertheless include it for
completeness. Moreover, we determine the critical weights $\delta$ for
$\mathfrak{S}^{\delta}$ (resp. $\mathcal{Q}^{\lambda,\mu}$). According to
[33], they coincide with the weights $\delta$ for which there exists
conformally invariant operators on $\mathsf{T}^{\delta}$ (resp.
$\mathsf{S}^{\delta}$), which strictly lowers the degree. The classification
of such operators can be deduced from [5, 6], we get it here from the more
basic Weyl's theory of invariants. It allows us to show that critical weights
are positive, so that existence and uniqueness of $\mathfrak{S}^{\delta}$ and
$\mathcal{Q}^{\lambda,\mu}$ holds in the most usual case
$\delta=\mu-\lambda=0$.

In Section $5$, we compute explicit formulæ for $\mathfrak{S}^{\delta}$ and
$\mathcal{Q}^{\lambda,\mu}$. Following the method in the reference [13], we
get local formulae for the conformally equivariant superization. Then, we
obtain such formulae for the conformally equivariant quantization also, when
it is valued in first order operators. After that, we derive global
expressions in terms of the Levi-Civita connection of the metric  for both
maps. Contrary to the scalar case, none of them is conformally invariant, i.e.
depend only on the conformal class of . This is due to the fact that the
symplectic form on ${\mathcal{M}}$ depends on  and therefore the Hamiltonian
${\mathfrak{g}}$-action on $\mathsf{S}^{\delta}$ also. Only their composition
$\mathcal{Q}^{\lambda,\mu}\circ\mathfrak{S}^{\delta}$, with
$\delta=\mu-\lambda$, is a conformally invariant map.

In Section $6$, we classify the symmetries of a free massless spinning
particle over a conformally flat manifold. In the pseudo-classical case, they
are the conformal supercharges, which generalize the symmetries of the massive
particle, given e.g. by Killing-Yano tensors [19, 39] or symmetric Killing
$2$-tensors [26]. They arise as the conformally equivariant superization of
conformal Killing hook tensors, the latter including the purely symmetric and
skew-symmetric ones. After quantization, the conformal supercharges correspond
to the higher symmetries of the Dirac operator, and we recover in particular
the ones given by first order operators classified in [3], and used for
integrability purpose in [9]. Both classifications, of conformal supercharges
and of higher symmetries of the Dirac operator, are new.

We include an appendix, where we collect the needed basic computations in the
algebra of isometric invariant differential operators acting on the tensor
space $\mathsf{T}^{\delta}$.

In all that paper, we work over a pseudo-Riemannian manifold $(M,\text{})$,
with  a metric of signature $(p,q)$. That manifold is supposed to be spin and
of even dimension $n=p+q$. We freely use the Einstein's summation convention.
The tensor products of ${\mathcal{C}^{\infty}}(M)$-modules are understood over
the algebra ${\mathcal{C}^{\infty}}(M)$.

##  2\. Spinor differential operators and their symbols

In order to keep a self-contained expository, this section presents a
recollection of our previous work [31]. Namely, we introduce the
$\mathfrak{o}(p+1,q+1)$-modules structures of the space of differential
operators acting on weighted spinors as well as the ones of its two spaces of
symbols, and we compare these three modules.

###  2.1. The supercotangent bundle

Let $E$ be a vector bundle over the smooth manifold $M$ and $d$ an integer. We
denote by $E[d]$ the $\mathbb{N}$-graded manifold with base $M$ and structural
sheaf ${\mathcal{O}}(E[d])$, which identifies to the graded sheaf of
complexified sections of ${\mathcal{S}}E^{*}$ if $d$ is even and of $\Lambda
E^{*}$ if $d$ is odd. The sections of ${\mathcal{S}}^{k}E^{*}$, or
$\Lambda^{k}E^{*}$, receive a degree $kd$. Such a formalism allows to encode
symmetric contravariant tensors, i.e. sections of the bundle
${\mathcal{S}}TM$, as functions on $T^{*}[2]M$, the order of tensors being
twice the degree of corresponding functions. Analogously, the algebra of
complexified differential forms $\Omega^{\mathbb{C}}(M)$ is viewed as the
algebra of functions on $T[1]M$, the degree of forms equating the one of
functions.

Both types of tensors are encompassed into the algebra of functions of the
supercotangent bundle of $M$, defined as the fiber product
${\mathcal{M}}=T^{*}[2]M\times_{M}T[1]M$. The gradation of its structural
sheaf is given by the following subspaces, for $\ell\in\mathbb{N}$,

(2.1)
${\mathcal{O}}_{[\ell]}({\mathcal{M}})=\bigoplus_{2k+\kappa=\ell}{\mathcal{O}}_{k,\kappa}({\mathcal{M}})\quad\text{and}\quad{\mathcal{O}}_{k,\kappa}({\mathcal{M}})=\mathrm{Pol}_{k}(T^{*}M)\otimes_{{\mathcal{C}^{\infty}}(M)}\Omega^{\mathbb{C}}_{\kappa}(M),$

where $\mathrm{Pol}_{k}(T^{*}M)\cong\Gamma({\mathcal{S}}^{k}TM)$ is the space
of functions on $T^{*}M$ of degree $k$ in the fiber variables and
$\Omega^{\mathbb{C}}_{\kappa}(M)$ is the space of complex differential forms
of degree $\kappa$. Starting from a local coordinate system $(x^{i})$ on $M$,
we can build a natural one $(x^{i},p_{i},\xi^{i})$ on ${\mathcal{M}}$, with
$p_{i}$ identifying to the partial derivative $\partial_{i}:=\partial/\partial
x^{i}$ and $\xi^{i}$ to the differential $1$-form $dx^{i}$.

Rothstein has classified the even symplectic structures on supermanifolds in
[36]. Accordingly, a symplectic structure on ${\mathcal{M}}$ is equivalent to
the three following piece of data: a symplectic form on $T^{*}M$, a metric on
the vector bundle $TM$, i.e. a metric on $M$, and a compatible connection. In
particular, the Rothstein results lead to the following proposition.

######  Proposition 2.1.

[31] Let $(M,\text{})$ be a pseudo-Riemannian manifold and
$\hbar\in\mathbb{R}$. Its supercotangent bundle
${\mathcal{M}}=T^{*}[2]M\times_{M}T[1]M$ admits an exact symplectic structure
$\omega=\boldsymbol{d}\alpha$, which reads in local natural coordinates as

(2.2)
$\alpha=p_{i}\boldsymbol{d}x^{i}+\frac{\hbar}{2\mathsf{i}}\text{}_{ij}\xi^{i}\boldsymbol{d}^{\nabla}\xi^{j},$

with $\boldsymbol{d}^{\nabla}$ the covariant differential associated to the
Levi-Civita connection.

The introduction of the factor $\hbar/\mathsf{i}$ into the symplectic form
$\omega$ deserves some explanations. First, from a physical point of view, a
symplectic form should have the dimension of an action. Assigning no dimension
to the odd variables $\xi$ forces us to insert a constant with the dimension
of an action in front of the $\xi$-term. We choose $\hbar$ since our aim is
quantization. To deal with pseudo-classical spinning particles [4, 35, 30,
26], $\hbar$ should be replaced by a characteristic action of the studied
system. Second, the symplectic form should be real. For quantization purpose,
we define the real structure on ${\mathcal{M}}$ by the involution anti-
automorphism

(2.3)
$\bar{\cdot}:{\mathcal{O}}({\mathcal{M}})\rightarrow{\mathcal{O}}({\mathcal{M}})$

equal to the identity on coordinates. In particular, we get
$\overline{\xi^{i}\xi^{j}}=\bar{\xi^{j}}\bar{\xi^{i}}=-\xi^{i}\xi^{j}$ and
$\alpha$, $\omega$ are then real.

The symplectic form $\omega$ is of degree $2$, hence $({\mathcal{M}},\omega)$
fits also into the classification of graded symplectic manifold of degree $2$,
performed in [37] by Roytenberg. The Poisson bracket associated to $\omega$
lowers the degree by $2$ and is completely determined, thanks to Leibniz
property, by the following equalities

$\\{X,Y\\}=\nabla_{[X,Y]}+\frac{\hbar}{2\mathsf{i}}R^{k}_{\leavevmode\nobreak\
lij}\xi_{k}\xi^{l}X^{i}Y^{j},\quad\\{X,\xi\\}=\nabla_{X}\xi,\quad\\{\xi,\xi^{\prime}\\}=\frac{\mathsf{i}}{\hbar}\text{}^{-1}(\xi,\xi^{\prime}),$

where $X,Y\in\mathrm{Pol}_{1}(T^{*}M)$ identify to vector fields,
$\xi,\xi^{\prime}$ are $1$-forms on $M$, $\text{}^{-1}$ is the metric induced
by  on $T^{*}M$ and $(R^{k}_{\leavevmode\nobreak\
l})_{ij}=[\nabla_{i},\nabla_{j}]$ are the components of its Riemann tensor.

###  2.2. Spinor differential operators

Let $(M,\text{})$ be a pseudo-Riemannian spin manifold of even dimension $n$.
Its spinor bundle $S$ satisfies
$\mathrm{End}S\cong\mathrm{\mathbb{C}l}(M,\text{})$, where
$\mathrm{\mathbb{C}l}(M,\text{})$ is the complex Clifford bundle of
$(M,\text{})$. We introduce the usual isomorphism

(2.4)
$\gamma:\Omega^{\mathbb{C}}(M)\rightarrow\Gamma(\mathrm{\mathbb{C}l}(M,\text{})),$

which satisfies
$\gamma(\xi)\gamma(\xi^{\prime})+\gamma(\xi^{\prime})\gamma(\xi)=-2\text{}^{-1}(\xi,\xi^{\prime})$
for any $\xi,\xi^{\prime}\in\Omega^{\mathbb{C}}_{1}(M)$. The algebra
$\mathcal{D}(M,S)$ of differential operators acting on sections of $S$ is an
algebra filtered by the order of derivations, over the subalgebra
$\Gamma(\mathrm{End}S)\cong\Gamma(\mathrm{\mathbb{C}l}(M,\text{}))$ of zeroth
order operators. The latter admits also a filtration, leading to one
filtration on each subspace $\mathcal{D}_{k}(M,S)$ of $k^{\text{th}}$ order
operators. Resorting to a spinor connection on $S$, we can then define a naive
bifiltration of $\mathcal{D}(M,S)$ by the following subspaces, indexed by
$k\in\mathbb{N}$ and $\kappa\leq n$,

$\mathcal{D}^{\nabla}_{k,\kappa}(M,S)=\text{span}\\{\gamma(\xi_{1})\cdots\gamma(\xi_{\mu})\nabla_{X_{1}}\cdots\nabla_{X_{m}}\,|\,m\leq
k\text{ and }\mu\leq\kappa\\},$

where $\xi_{i}$ pertains to $\Omega_{1}^{\mathbb{C}}(M)$ and $\nabla_{X_{i}}$
is the spinor covariant derivative, induced by the connection on $S$, along
the vector field $X_{i}$. Such a bifiltration behaves badly with respect to
the commutator, since $[\nabla_{X},\nabla_{Y}]$ is of degree $2$ in
$\gamma(\xi)$. Moreover, it depends on the choice of connection, as
$\nabla_{X}=X^{i}\partial_{i}+X^{i}\Gamma_{ij}^{k}\gamma(\xi^{j})\gamma(\xi_{k})$.
These difficulties are overcome by the following definition, introduced in
[31],

$\mathcal{D}_{k,\kappa}(M,S)=\text{span}\\{\gamma(\xi_{1})\cdots\gamma(\xi_{\mu})\nabla_{X_{1}}\cdots\nabla_{X_{m}}\,|\,m\leq
k\text{ and }2m+\mu\leq 2k+\kappa\\}.$

For such a bifiltration, if $A$ and $B$ are two operators of orders
$(k,\kappa)$ and $(k^{\prime},\kappa^{\prime})$, then their product $AB$ is of
order $(k+k^{\prime},\kappa+\kappa^{\prime})$ and their commutator $[A,B]$
splits into two terms of orders $(k+k^{\prime}-1,\kappa+\kappa^{\prime})$ and
$(k+k^{\prime},\kappa+\kappa^{\prime}-2)$. Thus, the associated bigraded
algebra, defined as the direct sum over
$(k,\kappa)\in\mathbb{N}\times\llbracket 0,n\rrbracket$ of the subspaces

${\mathrm{bigr}}_{k,\kappa}\mathcal{D}(M,S)=\mathcal{D}_{k,\kappa}(M,S)/(\mathcal{D}_{k-1,\kappa}(M,S)+\mathcal{D}_{k,\kappa-1}(M,S)),$

is a graded-commutative algebra. In addition to the usual filtration by the
order of derivations, the above bifiltration allows us to introduce another
one, given by the following increasing sequence

(2.5)
$\mathcal{D}_{[\ell]}(M,S)=\bigcup_{2k+\kappa=\ell}\mathcal{D}_{k,\kappa}(M,S).$

Thus, $\nabla_{X}$ and $\gamma(\xi)\gamma(\xi^{\prime})$ are considered of
same order. This filtration is compatible with the commutator so that the
associated graded algebra ${\mathrm{gr}}\mathcal{D}(M,S)$ is endowed with a
Poisson bracket of degree $-2$. In consequence, it is referred thereafter as
the Hamiltonian filtration.

######  Proposition 2.2.

[31] We get the following isomorphisms

$\displaystyle{\mathrm{bigr}}\mathcal{D}(M,S)$ $\displaystyle\cong$
$\displaystyle{\mathcal{O}}({\mathcal{M}})\cong\Gamma({\mathcal{S}}TM\otimes\Lambda
T^{*}M),$ $\displaystyle{\mathrm{gr}}\mathcal{D}(M,S)$ $\displaystyle\cong$
$\displaystyle{\mathcal{O}}({\mathcal{M}}),$

of respectively bigraded algebras and graded Poisson algebras.

Accordingly, we introduce the Hamiltonian principal symbol maps defined by the
following projections,

(2.6)
$\sigma_{\ell}:\mathcal{D}_{[\ell]}(M,S)\rightarrow{\mathcal{O}}_{[\ell]}({\mathcal{M}}),$

which are normalized by $\sigma_{2}(\nabla_{\partial_{i}})=p_{i}$ and
$\sigma_{1}(\gamma(\xi^{i}))=\left(\frac{\hbar}{2\mathsf{i}}\right)^{1/2}\xi^{i}$
to satisfy the usual relations:

(2.7)
$\sigma_{\ell}(A)\sigma_{\ell^{\prime}}(B)=\sigma_{\ell+\ell^{\prime}}(AB)\quad\text{and}\quad\\{\sigma_{\ell}(A),\sigma_{\ell^{\prime}}(B)\\}=\sigma_{\ell+\ell^{\prime}-2}([A,B]),$

for all differential operators $A,B$ of Hamiltonian orders $\ell$ and
$\ell^{\prime}$.

By definition
$|\Lambda|^{\lambda}:=|\Lambda^{\mathrm{top}}T^{*}M|^{\otimes\lambda}$ is the
trivial line bundle of $\lambda$-densities, the $1$-densities being the
natural objects for integration. Resorting to a pseudo-Hermitian inner product
$\langle\cdot,\cdot\rangle_{S}$ on $S$, we obtain then a pairing between
sections of $S\otimes|\Lambda|^{\lambda}$ and $S\otimes|\Lambda|^{\mu}$ if
$\lambda+\mu=1$. Consequently, the adjoint of an operator $D$ between the
latter vector bundles is defined by

(2.8) $\int_{M}\langle\phi,D\psi\rangle_{S}=\int_{M}\langle
D^{t}\phi,\psi\rangle_{S},$

for all compactly supported sections $\phi,\psi$ of
$S\otimes|\Lambda|^{\lambda}$. We choose the map $\gamma$, introduced in
(2.4), such that it intertwines the conjugation on ${\mathcal{M}}$ and the
adjoint operation on $\Gamma(\mathrm{\mathbb{C}l}(M,\text{}))$.

###  2.3. The conformal Lie algebra ${\mathfrak{g}}$

Let $(M,\text{})$ be a pseudo-Riemannian manifold of signature $(p,q)$. A
conformal Killing vector field on $M$ is a vector field $X$ whose Lie
derivative action preserves the direction of the metric, i.e. $L_{X}\text{}$
is proportional to . By definition, $(M,\text{})$ is conformally flat if
$\text{}=F\eta$ locally, where $F$ is a positive function and $\eta$ is the
flat metric of same signature $(p,q)$. Then, the Lie algebra ${\mathfrak{g}}$
of local conformal Killing vector fields of $(M,\text{})$ is of maximal
dimension and identifies to $\mathfrak{o}(p+1,q+1)$. On
$(\mathbb{R}^{p,q},F\eta)$, the Lie algebra
${\mathfrak{g}}\cong\mathfrak{o}(p+1,q+1)$ of conformal Killing vector fields
is given explicitly, in terms of its generators, by

$\displaystyle X_{i}$ $\displaystyle=$ $\displaystyle\partial_{i},$
$\displaystyle X_{ij}$ $\displaystyle=$ $\displaystyle
x_{i}\partial_{j}-x_{j}\partial_{i},$ (2.9) $\displaystyle X_{0}$
$\displaystyle=$ $\displaystyle x^{i}\partial_{i},$ $\displaystyle\bar{X}_{i}$
$\displaystyle=$ $\displaystyle
x_{j}x^{j}\partial_{i}-2x_{i}x^{j}\partial_{j},$

for $i,j=1,\ldots,n$, the indices being lowered using $\eta$. The latter
vector fields provide the local realization of ${\mathfrak{g}}$ on
$(M,\text{})$, endowed with a $|1|$-gradation via the polynomial degree of the
coefficients. With respect to the metric $\eta$, ${\mathfrak{g}}_{-1}$ is the
Lie subalgebra of translations, ${\mathfrak{g}}_{0}$ splits into the Lie
subalgebra $\mathfrak{o}(p,q)$ of rotations and its center, generated by the
Euler vector field $X_{0}$, and ${\mathfrak{g}}_{1}$ is the subspace of
conformal inversions. We denote the Lie algebra of isometries by
$\mathfrak{e}(p,q):=\mathfrak{o}(p,q)\ltimes{\mathfrak{g}}_{-1}$, and the one
of similitudes by
$\mathfrak{ce}(p,q):=\mathbb{R}X_{0}\ltimes\mathfrak{e}(p,q)$.

###  2.4. The ${\mathfrak{g}}$-module of tensorial symbols
$\mathsf{T}^{\delta}$

Let us remind that any vector field $X$ on $M$ admits a canonical lift to the
linear frame bundle via its first order jet $(\partial_{i}X^{j})$. This leads
to an action of $\mathrm{Vect}(M)$ by Lie derivatives on any associated
bundle, e.g. $\ell^{\lambda}_{X}=X^{i}\partial_{i}+\lambda\partial_{i}X^{i}$
on $\lambda$-densities. The tensor bundle ${\mathcal{S}}TM\otimes\Lambda
T^{*}M$ is also associated to the principal frame bundle, therefore it admits
an action of $\mathrm{Vect}(M)$ by Lie derivatives. Under the identification
of the latter tensors with functions on the supercotangent bundle
${\mathcal{M}}$, the Lie derivative is given by the following lift of
$\mathrm{Vect}(M)$ to ${\mathcal{M}}$

(2.10) $\mathrm{Vect}(M)\ni
X\mapsto\hat{X}:=X^{i}\partial_{i}+(\partial_{i}X^{j})(\xi^{i}\partial_{\xi^{j}}-p_{j}\partial_{p_{i}}),$

where $(x^{i},p_{i},\xi^{i})$ denotes a local natural coordinate system on
${\mathcal{M}}$ and $(\partial_{i},\partial_{p_{i}},\partial_{\xi^{i}})$ are
the corresponding partial derivatives. Clearly, this $\mathrm{Vect}(M)$-action
preserves the bigradation of the tensor algebra
$\Gamma({\mathcal{S}}TM\otimes\Lambda T^{*}M)$.

######  Definition 2.3.

We denote by $\mathbb{T}^{\delta}$ the vector bundle $\bigoplus_{\kappa\leq
n}{\mathcal{S}}TM\otimes\Lambda^{\kappa}T^{*}_{\mathbb{C}}M\otimes|\Lambda|^{\delta-\frac{\kappa}{n}}$.
The bigraded module of sections
$\mathsf{T}^{\delta}=\Gamma(\mathbb{T}^{\delta})$ is endowed with the natural
action of $X\in\mathrm{Vect}(M)$, given on the $\kappa$-component by

$\mathbb{L}_{X}^{\delta}=\hat{X}\otimes\mathrm{Id}+\mathrm{Id}\otimes\ell^{\delta-\kappa/n}_{X}.$

The elements of $\mathsf{T}^{\delta}$ will be referred to as tensorial
symbols.

The usual space of symbol of $\mathcal{D}(M,S)$ is its graded algebra for the
filtration by the order of derivations. It identifies to
$\mathrm{Pol}(T^{*}M)\otimes\Gamma(\mathrm{\mathbb{C}l}(M,\text{}))$. The
ideal of Clifford relations in the tensor bundle is stable under conformal
transformations, hence we get an action of the Lie algebra ${\mathfrak{g}}$ of
conformal Killing vector fields on $\Gamma(\mathrm{\mathbb{C}l}(M,\text{}))$.
This action preserves the filtration of $\Gamma(\mathrm{\mathbb{C}l}(M,g))$
and the corresponding action on its associated graded algebra
$\Omega^{\mathbb{C}}(M)$ is easily identified. This leads to the following
proposition, justifying the introduction of $\mathsf{T}^{\delta}$.

######  Proposition 2.4.

Let $(M,\text{})$ be a pseudo-Riemannian manifold and
$\Sigma:=\xi^{i}\partial_{\xi^{i}}$ be the Euler vector field of $T[1]M$. We
get an isomorphism of ${\mathfrak{g}}$-modules

$\mathrm{Id}\otimes|\mathrm{vol}_{\text{}}|^{\frac{\Sigma}{n}}\gamma:\mathsf{T}^{\delta}\rightarrow\Gamma({\mathcal{S}}TM\otimes|\Lambda|^{\delta})\otimes\Gamma(\mathrm{\mathbb{C}l}(M,\text{})),$

which depends only on the conformal class of .

###  2.5. The ${\mathfrak{g}}$-module of Hamiltonian symbols
$\mathsf{S}^{\delta}$

In contradistinction with the cotangent bundle case, the natural lift (2.10)
of $\mathrm{Vect}(M)$ by Lie derivatives do not lead to Hamiltonian vector
fields on ${\mathcal{M}}$. Besides, the preservation of the potential $1$-form
$\alpha$, see (2.2), is not a strong enough condition to determine a unique
lift of $X\in\mathrm{Vect}(M)$ to ${\mathcal{M}}$. In [31], we ask in addition
for preservation of the direction of the $1$-form
$\beta=\text{}_{ij}\xi^{i}dx^{j}$. Both conditions can be satisfied only for
vector fields $X\in{\mathfrak{g}}$, in that case they fix a unique lift

${\mathfrak{g}}\ni
X\mapsto\tilde{X}:=\hat{X}+\frac{\Sigma}{n}+\frac{\hbar}{2\mathsf{i}}\left(R^{k}_{\leavevmode\nobreak\
lij}\xi_{k}\xi^{l}X^{j}+\nabla_{i}(\partial_{[l}X_{k]})\xi^{k}\xi^{l}\right)\partial_{p_{i}},$

the brackets denoting skew-symmetrization. The action of $\tilde{X}$ clearly
preserves the gradation
${\mathcal{O}}({\mathcal{M}})=\bigoplus_{\ell\in\mathbb{N}}{\mathcal{O}}_{[\ell]}({\mathcal{M}})$
defined in (2.1).

######  Definition 2.5.

We denote by $\mathsf{S}^{\delta}$ the graded ${\mathfrak{g}}$-module
${\mathcal{O}}({\mathcal{M}})\otimes\Gamma(|\Lambda|^{\delta})$ of Hamiltonian
symbols, where the action of $X\in{\mathfrak{g}}$ is given by

$\mathsf{L}_{X}^{\delta}=\tilde{X}\otimes\mathrm{Id}+\mathrm{Id}\otimes\ell^{\delta}_{X},$

and the gradation is inherited from the one on ${\mathcal{O}}({\mathcal{M}})$.

For $2k+\kappa=\ell$, the spaces
$\mathsf{S}^{\delta}_{k,\kappa}:={\mathcal{O}}_{k,\kappa}({\mathcal{M}})\otimes\Gamma(|\Lambda|^{\delta})$
provide a gradation of the homogeneous component
$\mathsf{S}^{\delta}_{[\ell]}:={\mathcal{O}}_{[\ell]}({\mathcal{M}})\otimes\Gamma(|\Lambda|^{\delta})$
along the decomposition (2.1) of the latter space. But only the associated
increasing filtration by the $p$-degree,

(2.11)
$\mathsf{S}^{\delta}_{\frac{\ell-n}{2},n}\subset\mathsf{S}^{\delta}_{\frac{\ell-n}{2},n}\oplus\mathsf{S}^{\delta}_{\frac{\ell+1-n}{2},n-1}\subset\cdots\subset\mathsf{S}^{\delta}_{[\ell]},$

is preserved by the ${\mathfrak{g}}$-action $\mathsf{L}^{\delta}$ (by
convention, if an index is not a natural integer then we get the null space).
Accordingly, we can define the associated graded ${\mathfrak{g}}$-module
${\mathrm{gr}}\mathsf{S}^{\delta}$, which turns to be isomorphic to
$\mathsf{T}^{\delta}$. The principal tensorial symbol maps are defined by the
usual projections, and resorting to
${\mathrm{gr}}\mathsf{S}^{\delta}\cong\mathsf{T}^{\delta}$, they read as

(2.12) $\varepsilon_{k,\kappa}:\bigoplus_{j\leq
k,(n-\kappa)/2}\mathsf{S}^{\delta}_{k-j,\kappa+2j}\rightarrow\mathsf{T}^{\delta}_{k,\kappa}.$

###  2.6. The ${\mathfrak{g}}$-module of spinor differential operators
$\mathsf{D^{\lambda,\mu}}$

Concerning Lie derivatives, the case of the spinor bundle is more involved
than for natural bundles like $TM$. Following Kosmann, we choose
$L_{X}=\nabla_{X}+\gamma(\boldsymbol{d}X^{\flat})/2$ as Lie derivative [25],
with $X^{\flat}$ the $1$-form deduced from $X$ by the metric . This formula
gives a representation of Lie algebras only if restricted to conformal Killing
vector fields $X\in{\mathfrak{g}}$. We introduce
$L^{\lambda}_{X}:=L_{X}\otimes\mathrm{Id}+\mathrm{Id}\otimes\ell_{X}^{\lambda}$,
the induced Lie derivative on $\Gamma(S\otimes|\Lambda|^{\lambda})$.

######  Definition 2.6.

We denote by $\mathsf{D^{\lambda,\mu}}$ the ${\mathfrak{g}}$-module
$\mathcal{D}(M;S\otimes|\Lambda|^{\lambda},S\otimes|\Lambda|^{\mu})$ of
differential operators. The action of ${\mathfrak{g}}$ on one of its element
$A$ is given by the following Lie derivative

$\mathcal{L}_{X}^{\lambda,\mu}A=L^{\mu}_{X}A-AL^{\lambda}_{X}.$

The bifiltration on the underlying space of $\mathsf{D^{\lambda,\mu}}$ turns
to be preserved by the above ${\mathfrak{g}}$-action
$\mathcal{L}^{\lambda,\mu}$. Hence it induces ${\mathfrak{g}}$-modules
structure on ${\mathrm{bigr}}\mathsf{D^{\lambda,\mu}}$,
${\mathrm{gr}}\mathsf{D^{\lambda,\mu}}$ and on the usual symbol space
$\Gamma({\mathcal{S}}TM\otimes|\Lambda|^{\mu-\lambda})\otimes\Gamma(\mathrm{\mathbb{C}l}(M,g))$.
We compare them below.

######  Proposition 2.7.

For $\delta=\mu-\lambda$, we have the following isomorphisms of (bi)graded
${\mathfrak{g}}$-modules:
${\mathrm{gr}}\mathsf{D^{\lambda,\mu}}\cong\mathsf{S}^{\delta}$ and
${\mathrm{bigr}}\mathsf{D^{\lambda,\mu}}\cong{\mathrm{gr}}\mathsf{S}^{\delta}\cong\mathsf{T}^{\delta}\cong\Gamma({\mathcal{S}}TM\otimes|\Lambda|^{\mu-\lambda})\otimes\Gamma(\mathrm{\mathbb{C}l}(M,g))$.

###  2.7. Explicit formulæ for actions of ${\mathfrak{g}}$

We restrict here to the local model $(\mathbb{R}^{p,q},F\eta)$ of a
conformally flat manifold to provide explicit formulæ for
$\mathcal{L}_{X}^{\lambda,\mu}$, $\mathsf{L}_{X}^{\delta}$ and
$\mathbb{L}_{X}^{\delta}$. The supercotangent bundle admits then natural
cartesian coordinates $(x^{i},p_{i},\xi^{i})$ which transform tensorially,
$p_{i}$ identifying to $\partial_{i}$ and $\xi^{i}$ to $dx^{i}$. The potential
$1$-form $\alpha$ of ${\mathcal{M}}$ is written in terms of such coordinates
in equation (2.2). By definition, it reads simply as
$\alpha=\tilde{p}_{i}\boldsymbol{d}x^{i}+\frac{\hbar}{2\mathsf{i}}\eta_{ij}\tilde{\xi}^{i}\boldsymbol{d}\tilde{\xi}^{j}$
in Darboux coordinates $(x^{i},\tilde{p}_{i},\tilde{\xi}^{i})$. We deduce that
$(\tilde{\xi}^{i})$ form an orthonormal frame of $T^{*}M$ and that
$\tilde{p}_{i}=p_{i}+(\hbar/2\mathsf{i})\Gamma_{ij}^{k}\xi^{j}\xi_{k}$. In
terms of spinor differential operators, these coordinates satisfy
$p_{i}=\sigma_{2}(\nabla_{i})$ and
$\gamma(\xi^{i})\gamma(\xi^{j})+\gamma(\xi^{j})\gamma(\xi^{i})=-2F^{-1}\eta^{ij}$,
whereas $\tilde{p}_{i}=\sigma_{2}(\partial_{i})$ and
$\gamma(\tilde{\xi}^{i})\gamma(\tilde{\xi}^{j})+\gamma(\tilde{\xi}^{j})\gamma(\tilde{\xi}^{i})=-2\eta^{ij}$.
Using both coordinate systems, we introduce a linear isomorphism,

(2.13) $\displaystyle{\mathcal{F}}:\mathsf{T}^{\delta}$
$\displaystyle\longrightarrow$ $\displaystyle\mathsf{S}^{\delta}$
$\displaystyle P^{i_{1}\dots i_{k}}_{j_{1}\ldots
j_{\kappa}}(x)\,\xi^{j_{1}}\ldots\xi^{j_{\kappa}}p_{i_{1}}\ldots p_{i_{k}}$
$\displaystyle\longmapsto$
$\displaystyle|\mathrm{vol}_{\eta}|^{\frac{\kappa}{n}}P^{i_{1}\dots
i_{k}}_{j_{1}\ldots
j_{\kappa}}(x)\,\tilde{\xi}^{j_{1}}\ldots\tilde{\xi}^{j_{\kappa}}\tilde{p}_{i_{1}}\ldots\tilde{p}_{i_{k}},$

where $\mathrm{vol}_{\eta}$ is the canonical volume form
$dx^{1}\wedge\ldots\wedge dx^{n}$ attached to the metric $\eta$. Remark that
$|\mathrm{vol}_{\eta}|^{1/n}\tilde{\xi}^{i}=|\mathrm{vol}_{\text{}}|^{1/n}\xi^{i}$,
where $\mathrm{vol}_{\text{}}$ is the volume form associated to the metric
$\text{}=F\eta$. In addition, we introduce another linear isomorphism, the
normal ordering,

(2.14) $\displaystyle{\mathcal{N}}:\mathsf{S}^{\delta}$
$\displaystyle\longrightarrow$ $\displaystyle\mathsf{D^{\lambda,\mu}}$
$\displaystyle P^{i_{1}\ldots i_{k}}_{j_{1}\ldots
j_{\kappa}}(x)\tilde{\xi}^{j_{1}}\ldots\tilde{\xi}^{j_{\kappa}}\,\tilde{p}_{i_{1}}\ldots\tilde{p}_{i_{k}}$
$\displaystyle\longmapsto$ $\displaystyle P^{i_{1}\ldots i_{k}}_{j_{1}\ldots
j_{\kappa}}(x)\frac{\tilde{\gamma}^{j_{1}}}{\sqrt{2}}\ldots\frac{\tilde{\gamma}^{j_{\kappa}}}{\sqrt{2}}\,\frac{\hbar}{\mathsf{i}}\partial_{i_{1}}\ldots\frac{\hbar}{\mathsf{i}}\partial_{i_{k}},$

where $\tilde{\gamma}^{j}$ denotes $\gamma(\tilde{\xi}^{j})$ and
$\delta=\mu-\lambda$.

######  Proposition 2.8.

The maps ${\mathcal{F}}$ and ${\mathcal{N}}$ are isomorphisms of
$\mathfrak{ce}(p,q)$-modules, and are right inverses of respectively the
symbol maps $\varepsilon_{k,\kappa}$ on $\mathsf{T}^{\delta}_{k,\kappa}$, see
(2.12), and $\left(\frac{\hbar}{\mathsf{i}}\right)^{\ell/2}\sigma_{\ell}$ on
$\mathsf{S}^{\delta}_{[\ell]}$, see (2.6). Moreover, for any
$X\in{\mathfrak{g}}$, we have

(2.15) $\displaystyle{\mathcal{F}}^{-1}\mathsf{L}_{X}^{\delta}{\mathcal{F}}$
$\displaystyle=$
$\displaystyle\mathbb{L}_{X}^{\delta}-\frac{\hbar}{2\mathsf{i}}\xi_{k}\xi^{j}(\partial_{i}\partial_{j}X^{k})\partial_{p_{i}},$
(2.16)
$\displaystyle\qquad({\mathcal{N}}{\mathcal{F}})^{-1}\mathcal{L}_{X}^{\lambda,\mu}{\mathcal{N}}{\mathcal{F}}$
$\displaystyle=$
$\displaystyle{\mathcal{F}}^{-1}\mathsf{L}_{X}^{\delta}{\mathcal{F}}+\frac{\hbar}{4\mathsf{i}}(\partial_{j}\partial_{k}X^{i})\left(-2p_{i}\partial_{p_{j}}+\chi^{j}_{i}\right)\partial_{p_{k}}-\frac{\hbar}{\mathsf{i}}\lambda\partial_{j}(\partial_{i}X^{i})\partial_{p_{j}},$

where
$\chi^{j}_{i}=\xi^{j}\partial_{\xi^{i}}-\xi_{i}\partial_{\xi_{j}}+\frac{1}{2}\partial_{\xi_{j}}\partial_{\xi^{i}}$.

The explicit expressions of the actions $\mathsf{L}_{X}^{\delta}$ and
$\mathcal{L}_{X}^{\lambda,\mu}$ for all $X\in{\mathfrak{g}}$, are deduced from
the Lie derivatives $\mathbb{L}^{\delta}_{X}$, given below for $X$ a generator
of ${\mathfrak{g}}$, cf. (2.9),

$\displaystyle\mathbb{L}_{X_{i}}^{\delta}$ $\displaystyle=$
$\displaystyle\partial_{i},$ (2.17)
$\displaystyle\mathbb{L}_{X_{ij}}^{\delta}$ $\displaystyle=$ $\displaystyle
x_{i}\partial_{j}-x_{j}\partial_{i}-(p_{i}\partial_{p^{j}}-p_{j}\partial_{p^{i}})+\xi_{i}\partial_{\xi^{j}}-\xi_{j}\partial_{\xi^{i}},$
$\displaystyle\mathbb{L}_{X_{0}}^{\delta}$ $\displaystyle=$ $\displaystyle
x^{i}\partial_{i}-p_{i}\partial_{p_{i}}+\delta n,$
$\displaystyle\mathbb{L}^{\delta}_{\bar{X}_{i}}$ $\displaystyle=$
$\displaystyle(x_{j}x^{j}\partial_{i}-2x_{i}x^{j}\partial_{j})+(-2p_{i}x_{j}\partial_{p_{j}}+2x_{i}p_{j}\partial_{p_{j}}+2p_{k}x^{k}\partial_{p^{i}})$
$\displaystyle+2x_{j}\xi^{j}\partial_{\xi^{i}}-2\xi_{i}x^{k}\partial_{\xi^{k}}-2n\delta
x_{i}.$

###  2.8. Definitions of the ${\mathfrak{g}}$-equivariant superization and
quantization

The supercotangent bundle $({\mathcal{M}},\omega)$ of $(M,\text{})$ proves to
be the phase space for a classical spinning particle on $(M,\text{})$ [4, 35,
31]. The Hamiltonian action of ${\mathfrak{g}}$ turns its space of functions
${\mathcal{O}}({\mathcal{M}})$ into the ${\mathfrak{g}}$-module
$\mathsf{S}^{\delta}$, with $\delta=0$. It can be interpreted as the space of
classical observables for a spinning particle, whereas the space of quantum
observables of such a particle is known to be the space of differential
operators acting on spinors, or more precisely the ${\mathfrak{g}}$-module
$\mathsf{D^{\lambda,\mu}}$ for $\lambda=\mu=\frac{1}{2}$. This justifies the
name quantization in the following definition.

######  Definition 2.9.

Let $\lambda,\mu\in\mathbb{R}$ and $\delta=\mu-\lambda$. A conformally
equivariant quantization is an isomorphism of ${\mathfrak{g}}$-modules,
$\mathcal{Q}^{\lambda,\mu}:\mathsf{S}^{\delta}\rightarrow\mathsf{D^{\lambda,\mu}}$,
which is a right inverse of the principal Hamiltonian symbol maps defined in
(2.6):
$\sigma_{\ell}\circ\mathcal{Q}^{\lambda,\mu}=\left(\frac{\hbar}{\mathsf{i}}\right)^{\ell/2}\mathrm{Id}$
on $\mathsf{S}^{\delta}_{[\ell]}$ for all integers $\ell$. Moreover,
$\mathcal{Q}^{\lambda,\mu}$ preserves the usual filtration by the order of
derivations.

Such a quantization extends the geometric quantization of the supercotangent
bundle obtained in [31], which is also conformally equivariant but defined
only on symbols of Hamiltonian degree at most $2$. Moreover,
${\mathcal{Q}}^{\lambda,\lambda}$ gives rise to a deformation quantization of
$\mathsf{S}^{0}$, the star-product being defined as usual by pull-back of the
product on $\mathsf{D}^{\lambda,\lambda}$ via
${\mathcal{Q}}^{\lambda,\lambda}$. Notice that Fedosov's deformation
quantization of symplectic supermanifolds has been investigated in [7].

Regarding $\mathsf{T}^{\delta}$, this is a ${\mathfrak{g}}$-module of
geometric significance, defined as a twist by densities of the tensorial
$\mathrm{Vect}(M)$-module $\Gamma({\mathcal{S}}TM\otimes\Lambda
T^{*}\mathbb{R}^{n})$. If $\delta=0$, it contains $\mathrm{Pol}(T^{*}M)$ as a
submodule, which can be interpreted as the algebra of classical observables
for a (non-spinning) particule on $(M,\text{})$, so that the following map
deserves the name superization.

######  Definition 2.10.

A conformally equivariant superization is an isomorphism of
${\mathfrak{g}}$-modules,
$\mathfrak{S}^{\delta}:\mathsf{T}^{\delta}\rightarrow\mathsf{S}^{\delta}$,
which preserves the Hamiltonian gradation (2.1) and the filtration by the
$p$-degree (2.11). Moreover, it is a right inverse of the principal tensorial
symbol maps defined in (2.12):
$\varepsilon_{k,\kappa}\circ\mathfrak{S}^{\delta}=\mathrm{Id}$ on
$\mathsf{T}^{\delta}_{k,\kappa}$ for all integers $k$ and $\kappa\leq n$.

Let $\mathcal{E}=p_{i}\partial_{p_{i}}$ and $\Sigma=\xi^{i}\partial_{\xi^{i}}$
be the Euler operators of the vector bundles $T^{*}M$ and $T[1]M$. By
proposition 2.4, the map
$\mathcal{Q}^{\lambda,\mu}\circ\mathfrak{S}^{\delta}\circ((\hbar/\mathsf{i})^{-\mathcal{E}}\otimes(\sqrt{2})^{\Sigma}\gamma^{-1})$
is a ${\mathfrak{g}}$-module morphism

$\Gamma({\mathcal{S}}TM\otimes|\Lambda|^{\delta})\otimes\Gamma(\mathrm{End}S)\to\mathcal{D}(M;S\otimes|\Lambda|^{\lambda},S\otimes|\Lambda|^{\mu}),$

which is a right inverse of the usual principal symbol maps. Consequently, it
gives a ${\mathfrak{g}}$-equvariant quantization in the sense of [8].

##  3\. Invariant differential operators on the supercotangent bundle

In all this section we work over the local model $(\mathbb{R}^{p,q},F\eta)$ of
a conformally flat manifold, so that we get global actions of the Lie algebras
$\mathfrak{e}(p,q)\leq\mathfrak{ce}(p,q)\leq{\mathfrak{g}}$ on
$\mathsf{T}^{\delta}$. They integrate into actions of the corresponding Lie
groups: $\mathrm{E}(p,q)$ of isometries of the metric $\eta$,
$\mathrm{CE}(p,q)$ of conformal affine transformations and
$G:=\mathrm{O}(p+1,q+1)$ which acts only locally. Up to the isomorphisms
${\mathcal{F}}$ and ${\mathcal{N}}$, defined in (2.13) and (2.14), conformally
equivariant superizations and quantizations are given by
$\mathfrak{ce}(p,q)$-equivariant endomorphisms of the tensorial symbol space
$\mathsf{T}^{\delta}=\Gamma(\mathbb{T}^{\delta})$. We study here their
building blocks.

######  Definition 3.1.

We denote by $\mathcal{D}^{\delta,\delta^{\prime}}({\mathcal{M}})$ the space
of differential operators on ${\mathcal{M}}$, endowed with the natural local
$G$-action on operators from $\mathsf{T}^{\delta}$ to
$\mathsf{T}^{\delta^{\prime}}$. For $H$ a Lie subgroup of $G$, the subspace of
(locally) $H$-invariant differential operators is denoted by
$\mathcal{D}^{\delta,\delta^{\prime}}({\mathcal{M}})^{H}$.

As $G$-modules, we have
$\mathcal{D}^{\delta,\delta^{\prime}}({\mathcal{M}})\cong\mathcal{D}(\mathbb{R}^{n};\mathbb{T}^{\delta},\mathbb{T}^{\delta^{\prime}})$.
Regarding this identification, we name fiberwise operators those linear
operators which act only along the fibers of the tensor bundle $\mathbb{T}$.

###  3.1. Euclidean invariants

The superspace $\mathbb{R}^{2n|p,q}$, $m=p+q$, is endowed with a canonical
Poisson bivector
$\Pi=\sum_{a=1}^{n}\partial_{a}\wedge\partial_{a+n}+\sum_{a=2n+1}^{2n+p}\partial_{a}\wedge\partial_{a}-\sum_{a=2n+p+1}^{2n+m}\partial_{a}\wedge\partial_{a}$
in a cartesian coordinate system $(x^{a})_{a=1,\ldots,2n+m}$. The (super)
Heisenberg Lie algebra $\mathfrak{h}(2n|p,q)$ and the orthosymplectic Lie
algebra $\mathfrak{spo}(2n|p,q)$ are defined as Lie subalgebras of the Poisson
algebra of polynomial functions over $\mathbb{R}^{2n|p,q}$. Namely, the first
one is given by the constant and linear functions, and the second one by the
quadratic one. For $m=0$, we recover the usual definitions of the Heisenberg
and symplectic Lie algebras, and $\mathfrak{spo}(0|p,q)$ identifies to the
orthogonal Lie algebra $\mathfrak{o}(p,q)$.

Resorting to Weyl theory of invariants [41], we get below a slight
generalization of the Howe duality between the Lie algebras
$\mathfrak{o}(p,q)$ and $\mathfrak{spo}(2|1,1)$ in $\mathfrak{spo}(2n|n,n)$,
$n=p+q$. See e.g. [24, 28] for more informations on the latter dual pair. As
before, we denote by $(x^{i},p_{i},\xi^{i})$ the natural cartesian coordinates
on ${\mathcal{M}}$ and by $(\partial_{i},\partial_{p_{i}},\partial_{\xi^{i}})$
the associated derivatives.

######  Proposition 3.2.

Let $\mathcal{E}=p_{i}\partial_{p_{i}}$, $\Sigma=\xi^{i}\partial_{\xi^{i}}$
and $\delta\in\mathbb{R}$. The subalgebra of isometry invariant differential
operators on ${\mathcal{M}}$ satisfies
$\mathcal{D}^{\delta,\delta}({\mathcal{M}})^{\mathrm{E}(p,q)}\cong\mathfrak{U}(\mathfrak{spo}(2|1,1)\ltimes\mathfrak{h}(2|1,1))$
for $n\geq 3$. For all $n$, it is generated by the fiberwise operators

(3.18)
$\begin{array}[]{cccc}R=\eta^{ij}p_{i}p_{j},&E=\mathcal{E}+\frac{n}{2},&T=\eta_{ij}\partial_{p_{i}}\partial_{p_{j}},&\mathsf{\Sigma}=\Sigma-\frac{n}{2},\\\\[3.0pt]
\boldsymbol{\boldsymbol{Q}}=\xi^{i}p_{i},&\boldsymbol{\delta}=\eta_{ij}\xi^{i}\partial_{p_{j}},&\boldsymbol{\delta}^{*}=\eta^{ij}p_{i}\partial_{\xi^{j}},&\boldsymbol{\boldsymbol{Q}}^{*}=\partial_{\xi^{i}}\partial_{p_{i}},\end{array}$

generating the Lie algebra $\mathfrak{spo}(2|1,1)$, and by the differential
operators

(3.19)
$\begin{array}[]{ccc}G=\eta^{ij}p_{i}\partial_{j},&D=\partial_{i}\partial_{p_{i}},&L=\eta^{ij}\partial_{i}\partial_{j},\\\\[3.0pt]
\boldsymbol{d}=\xi^{i}\partial_{i},&\boldsymbol{d}^{*}=\eta^{ij}\partial_{\xi^{i}}\partial_{j},&\end{array}$

generating the Lie algebra $\mathfrak{h}(2|1,1)$.

###### Proof.

We resort to Weyl quantization of $T^{*}{\mathcal{M}}$ which is
$\mathfrak{spo}(4n|n,n)\ltimes\mathfrak{h}(4n|2p,2q)$-equivariant. It
establishes a correspondence between the $\mathfrak{e}(p,q)$-invariants of
$\mathcal{D}^{\delta,\delta}({\mathcal{M}})$ and of
$\mathrm{Pol}(T^{*}{\mathcal{M}})$, which admits
$(x^{i},\hat{x}_{i},p_{i},\hat{p}^{i},\xi^{i},\hat{\xi}_{i})$ as generators.
Regarding the action of $\mathfrak{e}(p,q)$ on ${\mathcal{M}}$, given by
(2.17), the $\mathfrak{e}(p,q)$-invariants of
$\mathrm{Pol}(T^{*}{\mathcal{M}})$ reduce to the $\mathfrak{o}(p,q)$-invariant
polynomials on the space
$(\mathbb{R}^{n})^{*}\times(\mathbb{R}^{n})^{*}\times\mathbb{R}^{n}\times\Pi\mathbb{R}^{n}\times(\Pi\mathbb{R}^{n})^{*}$
with coordinates $(\hat{x}_{i},p_{i},\hat{p}^{i},\xi^{i},\hat{\xi}_{i})$. By
the Weyl theorem [41, theorem 2.9.A, p.53] these invariants split into even
and odd invariants: the even ones are generated by the $15$ scalar products
between these five types of coordinates, and the odd ones are constructed from
the determinant and are not invariant under the group $\mathrm{O}(p,q)$. So we
are left with the even invariants. The two squares of odd variables $\xi$ and
$\hat{\xi}$ vanish and only $13$ non-vanishing scalar products remain. The $8$
ones without the variable $\hat{x}$ can be identified to a basis of the
quadratic polynomials on $T^{*}\mathbb{R}^{1|1}$, while the five other ones
can be identify to a basis of those of degrees $0$ and $1$. They correspond by
Weyl quantization to the $8$ operators (3.18) and the $5$ operators (3.19)
respectively. By equivariance property of Weyl quantization we get that the
operators (3.18) generate the Lie algebra $\mathfrak{spo}(2|1,1)$, the
operators (3.19) generate the Lie algebra $\mathfrak{h}(2|1,1)$, and together
they generate the algebra of even $\mathfrak{e}(p,q)$-invariants, or
equivalently of $\mathrm{E}(p,q)$-invariants, in
$\mathcal{D}^{\delta,\delta}({\mathcal{M}})$.

Assuming $n\geq 3$, we get $d(\hat{x}^{i}\hat{x}_{i})\wedge
d(\hat{x}^{i}p_{i})\wedge\ldots\wedge d(\xi^{i}\hat{\xi}_{i})\neq 0$. Thus the
$13$ obtained operators are algebraically independent and
$\mathcal{D}^{\delta,\delta}({\mathcal{M}})^{\mathrm{E}(p,q)}$ is then
isomorphic to the enveloping algebra
$\mathfrak{U}(\mathfrak{spo}(2|1,1)\ltimes\mathfrak{h}(2|1,1))$. ∎

All of the $13$ operators in (3.18) and (3.19) identify to well-known
operators on tensors. The three operators $R,\mathcal{E},T$, restricted to
$\Gamma({\mathcal{S}}T\mathbb{R}^{n})$, correspond to the metric, the Euler
operator and the trace. Together with $G,D,L$, which generalize the gradient,
the divergence and the Laplacian, they have been introduced in [12, 13] as
building blocks of the conformally equivariant quantization of cotangent
bundles. The operator $\Sigma$ is the Euler operator on $\Gamma(\Lambda
T^{*}\mathbb{R}^{n})$ and the remaining operators square to zero and identify
to well-known (co-)differentials. Indeed, $\boldsymbol{d},\boldsymbol{d}^{*}$
are the usual de Rham (co-)differentials on $\Gamma(\Lambda
T^{*}\mathbb{R}^{n})$ and $\boldsymbol{\delta},\boldsymbol{\delta}^{*}$ are
Koszul type (co-)differentials on the whole space
$\Gamma({\mathcal{S}}T\mathbb{R}^{n}\otimes\Lambda T^{*}\mathbb{R}^{n})$. As
for $\boldsymbol{Q},\boldsymbol{Q}^{*}$, they are (co-)differentials
introduced by Manin to give a cohomological description of the Berezinian
[29]. Notice that $\boldsymbol{Q}$ is also a function on ${\mathcal{M}}$: this
is the principal symbol of the Dirac operator. The notation $\boldsymbol{Q}$
is borrowed from physics where it plays the role of a supercharge [19]. Let us
mention that similar algebras of invariant operators on tensor (and spinors)
have been investigated in [20, 21, 22], over a background $(M,\text{})$ of
constant curvature.

The bold operators with a $*$-exponent lower the degree in $\xi$ by one, while
the other ones rise it by one. The next proposition shows that, indeed,
$\boldsymbol{\delta}^{*},\boldsymbol{Q}^{*}$ are the codifferentials of
$\boldsymbol{\delta},\boldsymbol{Q}$.

######  Proposition 3.3.

On $\mathsf{T}^{\delta}$, the operators
$\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}\boldsymbol{\delta}^{*}$,
$\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}^{*}\boldsymbol{\delta}$ and
$\frac{1}{n+\mathcal{E}-\Sigma}\boldsymbol{Q}^{*}\boldsymbol{Q}$,
$\frac{1}{n+\mathcal{E}-\Sigma}\boldsymbol{Q}\boldsymbol{Q}^{*}$ are two pairs
of complementary projections, which lead to the decompositions

$\mathsf{T}^{\delta}=\ker\boldsymbol{\delta}\oplus\ker\boldsymbol{\delta}^{*}\qquad\text{and}\qquad\mathsf{T}^{\delta}=\ker\boldsymbol{Q}\oplus\ker\boldsymbol{Q}^{*}.$

Moreover, we have $\ker\boldsymbol{\delta}=\mathrm{im}\boldsymbol{\delta}$,
$\ker\boldsymbol{\delta}^{*}=\mathrm{im}\boldsymbol{\delta}^{*}=\boldsymbol{\delta}^{*}\ker\boldsymbol{\delta}$
and $\ker\boldsymbol{Q}=\mathrm{im}\boldsymbol{Q}$,
$\ker\boldsymbol{Q}^{*}=\mathrm{im}\boldsymbol{Q}^{*}=\boldsymbol{Q}^{*}\ker\boldsymbol{Q}$.

###### Proof.

A straightforward computation leads to
$[\boldsymbol{\delta}^{*},\boldsymbol{\delta}]=\mathcal{E}+\Sigma$, hence
$\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}\boldsymbol{\delta}^{*}$ is a
projection whose complementary projection is
$\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}^{*}\boldsymbol{\delta}$.
Hence, we have
$\ker\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}\boldsymbol{\delta}^{*}=\mathrm{im}\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}^{*}\boldsymbol{\delta}$
and a complementary space in $\mathsf{T}^{\delta}$ is provided by
$\mathrm{im}\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}\boldsymbol{\delta}^{*}=\ker\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}^{*}\boldsymbol{\delta}$.

We compute now the latter four spaces. Using the above commutation relation,
we first get that
$\boldsymbol{\delta}^{*}\boldsymbol{\delta}\boldsymbol{\delta}^{*}=(\mathcal{E}+\Sigma)\boldsymbol{\delta}^{*}$.
The equalities of their kernels and of their images imply that
$\ker\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}\boldsymbol{\delta}^{*}=\ker\boldsymbol{\delta}^{*}$
and
$\mathrm{im}\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}^{*}\boldsymbol{\delta}=\mathrm{im}\boldsymbol{\delta}^{*}$.
Similarly, we have
$\boldsymbol{\delta}\boldsymbol{\delta}^{*}\boldsymbol{\delta}=(\mathcal{E}+\Sigma)\boldsymbol{\delta}$,
which leads to
$\mathrm{im}\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}\boldsymbol{\delta}^{*}=\mathrm{im}\boldsymbol{\delta}$
and
$\ker\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}^{*}\boldsymbol{\delta}=\ker\boldsymbol{\delta}$.
Combining all these equalities, we deduce the results concerning
$\boldsymbol{\delta},\boldsymbol{\delta}^{*}$ as well as the announced
decomposition of $\mathsf{T}^{\delta}$. The case of
$\boldsymbol{Q},\boldsymbol{Q}^{*}$ is analogous. ∎

###  3.2. Conformal invariants

The $\mathfrak{e}(p,q)$ action on $\mathsf{T}^{\delta}$ is independent of
$\delta$, hence the subspace of $\mathfrak{e}(p,q)$-invariants in
$\mathcal{D}^{\delta,\delta^{\prime}}({\mathcal{M}})$ identifies to the
previously determined one in $\mathcal{D}^{\delta,\delta}({\mathcal{M}})$. In
contradistinction, the action of the dilation vector field depends on the
shift of weight $\delta^{\prime}-\delta$. It turns out that all the
$\mathfrak{e}(p,q)$-invariant generators are also
$\mathfrak{ce}(p,q)$-invariants for the shift indicated in (3.20). Moreover,
the fiberwise operators are automatically invariant under conformal
inversions. The situation for the $13$ generators of
$\mathcal{D}^{\delta,\delta^{\prime}}({\mathcal{M}})^{\mathrm{E}(p,q)}$ is sum
up in the following table,

(3.20) $\begin{array}[]{|c||c|c|c|c|c|}\hline\cr\text{values of
}n(\delta^{\prime}-\delta)&-2&-1&0&1&2\\\\[3.0pt]
\hline\cr\mathfrak{ce}(p,q)\text{-invariant
operators}&T&\boldsymbol{\delta},\boldsymbol{Q}^{*}&\mathcal{E},D,\Sigma&\boldsymbol{Q},\boldsymbol{\delta}^{*},\boldsymbol{d},\boldsymbol{d}^{*}&R,G,L\\\\[3.0pt]
\hline\cr{\mathfrak{g}}\text{-invariant
operators}&T&\boldsymbol{\delta},\boldsymbol{Q}^{*}&\mathcal{E},\Sigma&\boldsymbol{Q},\boldsymbol{\delta}^{*}&R\\\\[3.0pt]
\hline\cr\end{array}$

Restricting now to $\delta=\delta^{\prime}$, we deduce that any
$\mathrm{CE}(p,q)$-invariant operator is obtained from arbitrary powers of the
operators $\mathcal{E},D,\Sigma$ and monomials of the form

(3.21)
$R^{r}\boldsymbol{Q}^{\alpha}(\boldsymbol{\delta}^{*})^{\beta}\boldsymbol{d}^{\gamma}(\boldsymbol{d}^{*})^{\gamma^{\prime}}G^{g}L^{l}\boldsymbol{\delta}^{\beta^{\prime}}(\boldsymbol{Q}^{*})^{\alpha^{\prime}}T^{t},$

where
$2r+2g+2l+\alpha+\beta+\gamma+\gamma^{\prime}-\beta^{\prime}-\alpha^{\prime}-2t=0$
and the exponents of odd operators are equal to $0$ or $1$, since they have
null square. From the table (3.20) we deduce that the fiberwise $G$-invariant
operators are of the same form but do not involve any of the $5$ operators in
(3.19). In fact, they exhaust all the $G$-invariant differential operators on
${\mathcal{M}}$. The proof is similar to the one concerning differential
operators acting on contravariant symmetric tensors
$\Gamma({\mathcal{S}}T\mathbb{R}^{n})$, provided in [11, Lemma 3.3].
Therefore, we get the following result.

######  Proposition 3.4.

The algebra of conformal invariants
$\mathcal{D}^{\delta,\delta}({\mathcal{M}})^{G}$ is generated by

(3.22)
$\mathcal{E},\Sigma,RT,\boldsymbol{Q}\boldsymbol{Q}^{*},\boldsymbol{\delta}^{*}\boldsymbol{\delta},\boldsymbol{Q}\boldsymbol{\delta},\boldsymbol{\delta}^{*}\boldsymbol{Q}^{*}.$

Moreover, $\mathcal{E}$ is in the center and
$\mathcal{E},\Sigma,RT,\boldsymbol{Q}\boldsymbol{Q}^{*}+\boldsymbol{\delta}^{*}\boldsymbol{\delta}$
generate an abelian subalgebra.

###### Proof.

As conformal invariants operators are all fiberwise, they coincide with the
$\mathrm{CE}(p,q)$-invariant fiberwise operators. We deduce from equation
(3.21) that they are generated by $\mathcal{E},\Sigma$ and
$R^{r}\boldsymbol{Q}^{\alpha}(\boldsymbol{\delta}^{*})^{\beta}\boldsymbol{\delta}^{\beta^{\prime}}(\boldsymbol{Q}^{*})^{\alpha^{\prime}}T^{t}$
for $2r+\alpha+\beta-\beta^{\prime}-\alpha^{\prime}-2t=0$. Since the exponents
of odd operators are equal to $0$ or $1$, we obtain the announced generators
in (3.22), plus $R\boldsymbol{\delta}\boldsymbol{Q}^{*}$ and
$\boldsymbol{Q}\boldsymbol{\delta}^{*}T$.

We trivially check that $\mathcal{E}$ is in the center and that $\Sigma$
commute to $RT$, $\boldsymbol{Q}\boldsymbol{Q}^{*}$ and
$\boldsymbol{\delta}^{*}\boldsymbol{\delta}$. A direct computation shows that
$\frac{1}{2}[RT,\boldsymbol{Q}\boldsymbol{Q}^{*}]=-\frac{1}{2}[RT,\boldsymbol{\delta}^{*}\boldsymbol{\delta}]=[\boldsymbol{Q}\boldsymbol{Q}^{*},\boldsymbol{\delta}^{*}\boldsymbol{\delta}]=R\boldsymbol{\delta}\boldsymbol{Q}^{*}-\boldsymbol{Q}\boldsymbol{\delta}^{*}T$.
Therefore,
$\mathcal{E},\Sigma,RT,\boldsymbol{Q}\boldsymbol{Q}^{*}+\boldsymbol{\delta}^{*}\boldsymbol{\delta}$
generate a commutative algebra. Moreover, the sum of the three commutators of
$[RT,\boldsymbol{Q}\boldsymbol{Q}^{*}]$ with $RT$,
$\boldsymbol{Q}\boldsymbol{Q}^{*}$,
$\boldsymbol{\delta}^{*}\boldsymbol{\delta}$, leads to
$R\boldsymbol{\delta}\boldsymbol{Q}^{*}+\boldsymbol{Q}\boldsymbol{\delta}^{*}T$,
hence the seven operators given in (3.22) generate indeed
$\mathcal{D}^{\delta,\delta}({\mathcal{M}})^{G}$. ∎

###  3.3. Three Casimir operators

Given a representation $\rho:{\mathfrak{g}}\rightarrow\mathrm{End}(V)$, we
remind that the Killing form $B$ of ${\mathfrak{g}}$ induces a particular
${\mathfrak{g}}$-invariant operator on $V$, called the Casimir operator. It is
defined by $C_{\rho}=B^{\alpha\beta}\rho(X_{\alpha})\rho(X_{\beta}),$ where
$B^{\alpha\beta}$ is the inverse of the Gram matrix of the Killing form in the
basis $(X_{\alpha})$ of ${\mathfrak{g}}$. Hence, for each of the three
representations of ${\mathfrak{g}}$ on $\mathsf{T}^{\delta}$,
$\mathsf{S}^{\delta}$ and $\mathsf{D^{\lambda,\mu}}$, we get a Casimir
operator. We can pull back the ones of $\mathsf{S}^{\delta}$ and
$\mathsf{D^{\lambda,\mu}}$ to $\mathsf{T}^{\delta}$, via the isomorphisms
${\mathcal{F}}$ and ${\mathcal{N}}$ defined in (2.13) and (2.14) respectively.
We get then $\mathfrak{ce}(p,q)$-invariant operators on $\mathsf{T}^{\delta}$,
which can be written in terms of the generators listed in (3.18) and (3.19).
The Casimir operator of $\mathsf{T}^{\delta}$ being ${\mathfrak{g}}$-invariant
it can more specifically be written in terms of the operators in (3.22).

######  Proposition 3.5.

Let $\delta=\mu-\lambda$. The three ${\mathfrak{g}}$-modules
$\mathsf{T}^{\delta}$, $\mathsf{S}^{\delta}$ and $\mathsf{D^{\lambda,\mu}}$
admit Casimir operators. After pull-back by the local isomorphisms
${\mathcal{F}}$ and ${\mathcal{N}}$, they read on $\mathsf{T}^{\delta}$
respectively as,

(3.23) $\displaystyle C_{\mathsf{T}}$ $\displaystyle=$
$\displaystyle\hat{C}+\Sigma(\Sigma-n)+2(\boldsymbol{Q}\boldsymbol{Q}^{*}+\boldsymbol{\delta}^{*}\boldsymbol{\delta})-2\mathcal{E},$
$\displaystyle C_{\mathsf{S}}$ $\displaystyle=$ $\displaystyle
C_{\mathsf{T}}+2\frac{\hbar}{\mathsf{i}}\boldsymbol{d}\boldsymbol{\delta},$
$\displaystyle C_{\mathsf{D}}$ $\displaystyle=$ $\displaystyle
C_{\mathsf{T}}+\frac{\hbar}{\mathsf{i}}\left(GT-2\left(\mathcal{E}+n\lambda+\frac{1}{2}\right)D+2\boldsymbol{d}\boldsymbol{\delta}+\boldsymbol{d}^{*}\boldsymbol{\delta}+\boldsymbol{d}\boldsymbol{Q}^{*}+\frac{1}{2}\boldsymbol{d}^{*}\boldsymbol{Q}^{*}\right),$

where
$\hat{C}=RT+\left[1+n(\delta-1)-\mathcal{E}\right]\mathcal{E}-n^{2}\delta(\delta-1)$
is the Casimir operator of the ${\mathfrak{g}}$-module
$\Gamma({\mathcal{S}}T\mathbb{R}^{n}\otimes|\Lambda|^{\delta})$, obtained in
[12].

###### Proof.

To compute this three Casimir operators, we use the basis of ${\mathfrak{g}}$
introduce in (2.9) and the computation of the Gram matrix of the Killing form
performed in [12], which leads to

$C_{\rho}=\frac{1}{2}\eta^{ik}\eta^{jl}\rho(X_{ij})\rho(X_{kl})-\rho(X_{0})^{2}-\frac{1}{2}\eta^{ij}\rho(X_{i})\rho(\bar{X}_{j})-\frac{1}{2}\eta^{ij}\rho(\bar{X}_{i})\rho(X_{j}),$

for any representation $\rho$ of ${\mathfrak{g}}$. Replacing $\rho$ by
successively the three representations $\mathbb{L}^{\delta}$,
$\mathsf{L}^{\delta}$ and $\mathcal{L}^{\lambda,\mu}$ and resorting to the
given explicit expressions of the actions of the generators (2.9) of
${\mathfrak{g}}$ leads to the result, after straightforward but rather lengthy
computations. ∎

##  4\. Main results

We work on a spin manifold $(M,\text{})$ of dimension $n$ and signature
$(p,q)$, that is assumed to be conformally flat except in section 4.1.

###  4.1. Irreducible decomposition of the tensorial symbol bundle

Since the $\mathrm{O}(p,q)$-invariant operators introduced in (3.18) are
fiberwise, they generalize to arbitrary pseudo-Riemannian manifold
$(M,\text{})$, up to replacing the metric $\eta$ by . Their commutation
relations, given in (A.55), remain the same.

In the case of usual differential operators, the symbols are sections of the
vector bundle ${\mathcal{S}}TM$. The decomposition of the latter into
$\mathrm{O}(p,q)$-homogeneous irreducible bundles stems from the harmonic
decomposition of its fibers, isomorphic to ${\mathcal{S}}\mathbb{R}^{n}$. It
coincides with the decomposition of ${\mathcal{S}}\mathbb{R}^{n}$ into joint
eigenspaces of the $\mathrm{O}(p,q)$-invariant operators $\mathcal{E}$ and
$RT$. For weighted spinor differential operators, the space of tensorial
symbols is $\mathsf{T}^{\delta}=\Gamma(\mathbb{T}^{\delta})$, cf. definition
2.3. The decomposition of $\mathbb{T}^{\delta}$ into
$\mathrm{O}(p,q)$-homogeneous irreducible bundles relies then on the one of
its fibers, ${\mathcal{S}}\mathbb{R}^{n}\otimes\Lambda(\mathbb{R}^{n})^{*}$,
into irreducible representations of $\mathrm{O}(p,q)$. This generalization of
harmonic decomposition has been carried out in [23]. We recover it here
independently as the decomposition into joint eigenspaces of the commuting
$\mathrm{O}(p,q)$-invariant operators
$\mathcal{E},\Sigma,RT,\boldsymbol{Q}\boldsymbol{Q}^{*}+\boldsymbol{\delta}^{*}\boldsymbol{\delta}$,
obtained in proposition 3.4. To take into account the weight $\delta$, we
rather regard these bundles as $\mathrm{CO}(p,q)$-homogeneous ones. Resorting
to previous notation, we introduce the projection $\Pi_{0}:\ker
T^{2}\rightarrow\ker T$, explicitly given by
$\mathrm{Id}-\frac{1}{4n+2\mathcal{E}}RT$, and the operator
$\boldsymbol{Q}_{0}=\Pi_{0}\circ\boldsymbol{Q}$.

######  Theorem 4.1.

The vector bundle $\mathbb{T}^{\delta}$ admits an irreducible decomposition
into $\mathrm{CO}(p,q)$-homogeneous bundles, which correspond in terms of
modules of sections to the decomposition

(4.24) $\mathsf{T}^{\delta}=\bigoplus_{k\in\mathbb{N},\\\ \kappa\leq
n}\bigoplus_{s\leq[k/2]}\bigoplus_{\alpha,\beta\in\\{0,1\\}}\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta},$

where

(4.25)
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}=R^{s}(\boldsymbol{Q}_{0})^{\alpha}(\boldsymbol{\delta}^{*})^{\beta}\left(\mathsf{T}^{\delta}_{k-2s,\kappa-\alpha+\beta}\bigcap(\ker
T\cap\ker\boldsymbol{\delta}\cap\ker\boldsymbol{Q}^{*})\right).$

Those modules are ${\mathfrak{g}}$-stable and form eigenspaces of the Casimir
operator $C_{\mathsf{T}}$ with eigenvalues

(4.26)
$\gamma_{k,\kappa,s;\,ab}=\hat{\gamma}_{k,s}+\kappa(\kappa-n)+2(\alpha+\beta-1)(k-2s)+2(\beta-\alpha)\kappa+2\alpha(n-2\beta),$

where
$\hat{\gamma}_{k,s}=2s[n+2(k-s-1)]+2k[1+n(\delta-1)-k]-n^{2}\delta(\delta-1)$
is the eigenvalue of the Casimir operator $\hat{C}$, see (3.23). The
multiplicity in the decomposition (4.24) is at most two, the only isomorphic
modules occurring via

(4.27)
$\textstyle{\mathsf{T}^{\delta}_{k,\kappa,s;10}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\boldsymbol{\delta}^{*}\boldsymbol{Q}^{*}}$$\textstyle{\mathsf{T}^{\delta}_{k,\kappa-2,s;01}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\boldsymbol{Q}\boldsymbol{\delta}}$and$\textstyle{\mathsf{T}^{\delta}_{k,\kappa,s;00}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\boldsymbol{Q}_{0}\boldsymbol{\delta}^{*}T}$$\textstyle{\mathsf{T}^{\delta}_{k,\kappa,s-1;11}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{R\boldsymbol{\delta}\boldsymbol{Q}^{*}}$

for all $k,\kappa,s$ such that source and target spaces are well-defined.

###### Proof.

We first obtain the decomposition (4.24) as the decomposition of
$\mathsf{T}^{\delta}$ into joint eigenspaces for the four commuting operators
$\mathcal{E},\Sigma,RT,\boldsymbol{Q}\boldsymbol{Q}^{*}+\boldsymbol{\delta}^{*}\boldsymbol{\delta}$.
The first three ones lead to a straightforward generalization of the usual
harmonic decomposition,

$\mathsf{T}^{\delta}=\bigoplus_{k,\kappa,s}R^{s}\left(\mathsf{T}^{\delta}_{k,\kappa}\cap\ker
T\right).$

It remains then to decompose the space $\ker T$ into eigenspaces of
$\boldsymbol{Q}\boldsymbol{Q}^{*}+\boldsymbol{\delta}^{*}\boldsymbol{\delta}$.
This is tricky since $RT$, $\boldsymbol{\delta}^{*}\boldsymbol{\delta}$ and
$\boldsymbol{Q}\boldsymbol{Q}^{*}$ do not commute. The proposition 3.3 states
that $\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}\boldsymbol{\delta}^{*}$
is a projection on $\ker\boldsymbol{\delta}$ along
$\boldsymbol{\delta}^{*}\ker\boldsymbol{\delta}$. Since
$[T,\boldsymbol{\delta}\boldsymbol{\delta}^{*}]=2\boldsymbol{\delta}\boldsymbol{Q}^{*}$,
the latter projection preserves the space $\ker\boldsymbol{Q}^{*}\cap\ker T$,
but not $\ker T$. The decomposition (4.24) relies then on the equality $\ker
T=\bigoplus_{\alpha=0,1}(\boldsymbol{Q}_{0})^{\alpha}\ker
T\cap\ker\boldsymbol{Q}^{*}$. To get it, we use the identity
$\ker\boldsymbol{Q}^{*}\cap\ker T=\boldsymbol{Q}^{*}\ker T$, deduced from
proposition 3.3, and we prove that there exists an operator $A$ such that
following exact sequence is split

(4.28)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker\boldsymbol{Q}^{*}\cap\ker
T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker
T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A\boldsymbol{Q}^{*}\quad}$$\textstyle{\boldsymbol{Q}^{*}\ker
T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\boldsymbol{Q}_{0}}$$\textstyle{0.}$

The table (A.56) leads to the equality
$\boldsymbol{Q}^{*}\boldsymbol{Q}_{0}=(n+\mathcal{E}-\Sigma)-\frac{1}{\left(n+2(\mathcal{E}-1)\right)}\boldsymbol{\delta}^{*}_{0}\boldsymbol{\delta}$
on $\boldsymbol{Q}^{*}\ker T$, so that it defines an automorphism of this
space. In consequence, this operator admits an inverse $A$, satisfying
$A\boldsymbol{Q}^{*}\boldsymbol{Q}_{0}=\mathrm{Id}$ on $\boldsymbol{Q}^{*}\ker
T$. This proves (4.24).

Straightforward computations lead to the eigenvalues of the Casimir operator
$C_{\mathsf{T}}$ on the spaces $\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$.
Restricted to such a space, the eight generators of the fiberwise
$\mathrm{O}(p,q)$-invariant operators, defined in (3.18), are either null or
have zero kernel. Since $\mathrm{O}(p,q)$ is semi-simple and
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$ is a finite dimensional
${\mathcal{C}^{\infty}}(M)$-module, the latter is the module of sections of an
$\mathrm{O}(p,q)$-homogeneous irreducible bundle on $M$. Clearly, this is also
a $\mathrm{CO}(p,q)$-homogeneous bundle. As the modules
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$ are joint eigenspaces for the
four operators
$\mathcal{E},\Sigma,RT,\boldsymbol{Q}\boldsymbol{Q}^{*}+\boldsymbol{\delta}^{*}\boldsymbol{\delta}$,
the only isomorphisms between them come from the remaining fiberwise
$\mathrm{CO}(p,q)$-invariant operators listed in Proposition 3.4. The
isomorphisms (4.27) follow. ∎

Notice that the modules of zero degree in the odd variables are of the form
$\mathsf{T}^{\delta}_{k,0,s;01}$. The eigenvalue of the Casimir operator
$C_{\mathsf{T}}$ on them reduces as expected to the one of $\hat{C}$.

###  4.2. Classification of conformally invariant operators on
$\mathsf{T}^{\delta}$

In paragraph 3.2, we have determined all the conformal invariant differential
operators acting on $\mathsf{T}^{\delta}$, over the local model
$(\mathbb{R}^{p,q},F\eta)$. We turn now to those which are not necessarily
differential. It is proved in [27] that an operator
$\Gamma({\mathcal{S}}^{k}T\mathbb{R}^{n})\to\Gamma({\mathcal{S}}^{l}T\mathbb{R}^{n})$,
with $k,l\in\mathbb{N}$, is local if it is invariant under translations and
dilation. This result extends straightforwadly to operators acting from
$\mathsf{T}^{\delta}$ to $\mathsf{T}^{\delta^{\prime}}$, meaning that
$\mathfrak{ce}(p,q)$-invariant operators are differential if restricted to
functions of bounded degree in $p$. However, they are not necessarily
differential on the whole space $\mathsf{T}^{\delta}$, like the operator
$(\mathcal{E}+1)^{-1}$. This does not impact the proof of proposition 3.2 and
such operators are generated by the $\mathfrak{e}(p,q)$-invariant operators
given in (3.18) and (3.19). To determine those which are
${\mathfrak{g}}$-invariant is more tricky, there exist indeed non fiberwise
${\mathfrak{g}}$-invariant operators.

To classify them, we restrict ourselves to irreducible homogeneous bundles.
According to the previous paragraph, we have the following commutative diagram
of ${\mathfrak{g}}$-modules

(4.29)
$\textstyle{\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\boldsymbol{\delta}^{\alpha}(\boldsymbol{Q}^{*})^{\beta}T^{s}}$$\textstyle{\mathsf{T}^{\delta^{\prime}}_{k^{\prime},\kappa^{\prime},s^{\prime};\alpha^{\prime}\beta^{\prime}}}$$\textstyle{\mathsf{T}^{\delta_{0}}_{k_{0},\kappa_{0},0;00}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{T}^{\delta^{\prime}_{0}}_{k^{\prime}_{0},\kappa^{\prime}_{0},0;00}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R^{s^{\prime}}(\boldsymbol{Q}_{0})^{\alpha^{\prime}}(\boldsymbol{\delta}^{*})^{\beta^{\prime}}}$

for well-chosen indices on the bottom part. The vertical arrows being
isomorphisms, the general classification of conformally invariant operators
acting on $\mathsf{T}^{\delta}$ boils down to the one on its irreducible
pieces of the form $\mathsf{T}^{\delta}_{k,\kappa,0;00}$. They are given by
certain linear combinations of $\mathfrak{ce}(p,q)$-invariant operators of the
type

(4.30) $\boldsymbol{d}_{\bf 0}^{\gamma^{\prime}}(\boldsymbol{d}^{*}_{\bf
0})^{\gamma}G_{\bf 0}^{g}L^{\ell}_{\bf 0}D^{d}_{\bf 0},$

where the indexes $0$ and ${\bf 0}$ denote respectively the restriction and
corestriction to $\ker T$ and to $\ker
T\cap\ker\boldsymbol{Q}^{*}\cap\ker\boldsymbol{\delta}$. We determine all such
${\mathfrak{g}}$-invariant operators below. Using their equivalent description
as morphisms of generalized Verma modules, their classification can also be
derived from the general statements in [5, 6].

######  Theorem 4.2.

Let $k,k^{\prime}$ be non-negative integers, $0\leq\kappa,\kappa^{\prime}\leq
n$, $\delta,\delta^{\prime}\in\mathbb{R}$, and let us define
$j=n(\delta^{\prime}-\delta)$. The space of conformally invariant operators
$\mathcal{D}(\mathbb{T}^{\delta}_{k,\kappa,0;00},\mathbb{T}^{\delta^{\prime}}_{k^{\prime},\kappa^{\prime},0;00})^{G}$
over $(\mathbb{R}^{p,q},F\eta)$ is either trivial or of dimension $1$,
generated by

  * •

$D_{\bf 0}^{d}$ if $k^{\prime}-k=-d$, $\kappa^{\prime}=\kappa$, $j=0$ and
$\delta=1+\frac{2k-d}{n}$,

  * •

$G_{\bf 0}^{g}$ if $k^{\prime}-k=g$, $\kappa^{\prime}=\kappa$, $j=2g$ and
$\delta=-\frac{g}{n}$,

  * •

$\mathcal{L}_{\ell}$ if $k^{\prime}=k$, $\kappa^{\prime}=\kappa$, $j=2\ell$
and $\delta=\frac{1}{2}+\frac{k-\ell}{n}$,

  * •

$\boldsymbol{d}_{\bf 0}$ if $k^{\prime}=k$, $\kappa^{\prime}-\kappa=1$, $j=1$
and $\delta=\frac{k+\kappa}{n}$,

  * •

$\boldsymbol{d}^{*}_{\bf 0}$ if $k^{\prime}=k$, $\kappa^{\prime}-\kappa=-1$,
$j=1$ and $\delta=1+\frac{k-\kappa}{n}$,

where the operator $\mathcal{L}_{\ell}$ is of the form
$\sum_{\gamma=0,1}a_{\varepsilon,j}\boldsymbol{d}_{\bf
0}^{\varepsilon}(\boldsymbol{d}^{*}_{\bf
0})^{\varepsilon}\sum_{j=0}^{\ell}G_{\bf 0}^{j}L_{\bf
0}^{\ell-\varepsilon-j}D_{\bf 0}^{j}$ for certain coefficients
$a_{\varepsilon,j}\in\mathbb{R}$. If $n$ is even, we get also

  * •

$\boldsymbol{d}_{\bf 0}\mathcal{L}_{\ell}$ if $\kappa=n/2+\ell$,
$k^{\prime}=k$, $\kappa^{\prime}-\kappa=1$, $j=2\ell+1$ and
$\delta=\frac{1}{2}+\frac{k-\ell}{n}$,

  * •

$\boldsymbol{d}^{*}_{\bf 0}\mathcal{L}_{\ell}$ if $\kappa=n/2-\ell$,
$k^{\prime}=k$, $\kappa^{\prime}-\kappa=-1$, $j=2\ell+1$ and
$\delta=\frac{1}{2}+\frac{k-\ell}{n}$.

  * •

$\boldsymbol{d}_{\bf 0}\boldsymbol{d}^{*}_{\bf 0}$ if $\kappa=n/2+1$,
$k^{\prime}=k$, $\kappa^{\prime}=\kappa$, $j=2$ and
$\delta=\frac{1}{2}+\frac{k-1}{n}$,

  * •

$\boldsymbol{d}^{*}_{\bf 0}\boldsymbol{d}_{\bf 0}$ if $\kappa=n/2-1$,
$k^{\prime}=k$, $\kappa^{\prime}=\kappa$, $j=2$ and
$\delta=\frac{1}{2}+\frac{k-1}{n}$.

###### Proof.

We follow the proof given by the author in [33], in the case of the cotangent
bundle. Let
$A\in\mathcal{D}(\mathbb{T}^{\delta}_{k,\kappa,0;00},\mathbb{T}^{\delta^{\prime}}_{k^{\prime},\kappa^{\prime},0;00})^{G}$.
According to (4.30), it is of the form $A=\boldsymbol{d}_{\bf
0}^{\gamma^{\prime}}(\boldsymbol{d}_{\bf 0}^{*})^{\gamma}G_{\bf
0}^{g}\mathcal{L}_{\ell}D_{\bf 0}^{d}$ for a certain operator
$\mathcal{L}_{\ell}$ satisfying
$\mathcal{L}_{\ell}=a_{\varepsilon,j}\boldsymbol{d}_{\bf
0}^{\varepsilon}(\boldsymbol{d}_{\bf 0}^{*})^{\varepsilon}G_{\bf 0}^{j}L_{\bf
0}^{\ell-\varepsilon-j}D_{\bf 0}^{j}$, where $g,d,\ell$ are integers,
$\gamma,\gamma^{\prime}=0,1$, and $a_{\varepsilon,j}$ are reals indexed by
$\varepsilon=0,1$ and $j=0,\ldots,\ell$. Moreover, we can ask for $a_{0,0}\neq
0$ so that $\ell$ is optimal.

We reduce now further the expression of $A$ by its invariance under conformal
inversions. The action of $\bar{X}_{i}$ on the powers of the five operators
entering into $A$ is computed in the appendix, it follows that

(4.31) $[A,\mathbb{L}_{\bar{X}_{i}}^{*}]\in
E_{D}\partial_{p_{i}}\oplus\Pi_{\bf 0}p_{i}E_{G}\oplus
E_{\boldsymbol{d}^{*}}\Pi_{\bf 0}\partial_{\xi^{i}}\oplus\Pi_{\bf
0}\xi_{i}E_{\boldsymbol{d}}\oplus\partial_{i}E_{L},$

where $\Pi_{\bf 0}$ is the projection onto the space $\ker
T\cap\ker\boldsymbol{Q}^{*}\cap\ker\boldsymbol{\delta}$, and $E_{G}$, $E_{D}$,
$E_{L}$, $E_{\boldsymbol{d}}$, $E_{\boldsymbol{d}^{*}}$ are vector spaces
generated by the five operators $\boldsymbol{d}_{0}$,
$\boldsymbol{d}^{*}_{0}$, $G_{0}$, $D_{0}$, $L_{0}$. The independence of the
monomials in $A$ with different powers of $L_{0}$ together with the vanishing
of the five components of $[A,\mathbb{L}_{\bar{X}_{i}}^{*}]$ lead then to the
result. E.g. the vanishing of the component in $E_{D}\partial_{p_{i}}$ of the
higher degree term in $L_{\bf 0}$ of $[A,L_{X_{i}}^{*}]$ reads as

$[D_{\bf 0}^{d},L_{X_{i}}^{*}]=0,$

and from the relations (A.54) we deduce that necessarily
$\delta=1+\frac{2k-d-1}{n}$ if $d\neq 0$. Along the same reasoning we get
that: $G_{\bf 0}^{g}$, $\boldsymbol{d}_{\bf 0}^{\gamma^{\prime}}$ and
$(\boldsymbol{d}^{*}_{\bf 0})^{\gamma}$ are ${\mathfrak{g}}$-invariant.
Therefore, we have $\delta+\frac{2l}{n}=\frac{1-g}{n}$ if $g\neq 0$,
$\delta+\frac{2(\ell+g)}{n}=1+\frac{k-\kappa}{n}$ if $\gamma=1$ and
$\delta+\frac{2(\ell+g)+\gamma}{n}=\frac{k+\kappa}{n}$ if $\gamma^{\prime}=1$.
As the operator $A$ is ${\mathfrak{g}}$-invariant, the same holds for
$\mathcal{L}_{\ell}$. Since the component in $\partial_{i}E_{L}$ of the higher
degree term in $L_{0}$ of $[\mathcal{L}_{\ell},L_{X_{i}}^{*}]$ vanishes,
equations (A.54) lead us to $\delta=\frac{1}{2}+\frac{k-\ell}{n}$ if $\ell\neq
0$. Then, straightforward but lengthy computations show that there exist
unique reals $a_{\varepsilon,j}$ such that $\mathcal{L}_{\ell}$ is conformally
invariant.

If $n$ is odd, the five values found for $\delta$ are incompatible two by two,
so only one of the five exponents can be non zero. If $n$ is even, the values
of $\delta$ for two non-vanishing exponents among $\gamma$, $\gamma^{\prime}$,
$\ell$ can be compatible for constrained value of $\kappa$. The result
follows. ∎

The theorem being of local nature and depending only of the conformal class of
the metric, it extends straightforwardly to conformally flat manifolds
$(M,\text{})$. The local expressions of the ${\mathfrak{g}}$-invariant
operators remain the same if we choose a conformal coordinate system, but to
obtain their covariant formulæ is a non trivial task.

###  4.3. Existence and uniqueness of the ${\mathfrak{g}}$-equivariant
quantization and superization

In the seminal paper [12], existence and uniqueness of the conformally
equivariant quantization of cotangent bundles was proved, resorting to
diagonalization of the Casimir operators of the modules of differential
operators and of their symbols. Thanks to theorem 4.1, we can apply the same
method to prove existence and uniqueness of equivariant superization and
quantization of supercotangent bundles, introduced in definitions 2.9 and
2.10. This is our main result.

######  Theorem 4.3.

Let $(M,\text{})$ be a conformally flat manifold. There exist two sets
$I^{\mathfrak{S}},I^{\mathcal{Q}}\subset\mathbb{Q}$ such that:

  1. (1)

conformally equivariant superization
$\mathfrak{S}^{\delta}:\mathsf{T}^{\delta}\rightarrow\mathsf{S}^{\delta},$
exists and is unique if $\delta\not\in I^{\mathfrak{S}}$,

  2. (2)

conformally equivariant quantization
$\mathcal{Q}^{\lambda,\mu}:\mathsf{S}^{\delta}\rightarrow\mathsf{D^{\lambda,\mu}},$
exists and is unique if $\delta\not\in I^{{\mathcal{Q}}}$.

Moreover, if $\lambda+\mu=1$, the quantization intertwines conjugation (2.3)
and adjoint operation (2.8), namely
$\mathcal{Q}^{\lambda,\mu}(\bar{\cdot})=\mathcal{Q}^{\lambda,\mu}(\cdot)^{t}$.

###### Proof.

The two results can be proved in the same way, following [12]. We focus on the
second one, as we will provide a stronger statement for superization in
theorem 5.1.

Existence and uniqueness of $\mathcal{Q}^{\lambda,\mu}$ is a result of local
nature so that we can work over the local conformal model
$(\mathbb{R}^{p,q},F\eta)$, with $F$ a positive function. We denote by
$\widetilde{C_{\mathsf{S}}}={\mathcal{F}}\circ
C_{\mathsf{S}}\circ{\mathcal{F}}^{-1}$ the Casimir operator of
$\mathsf{S}^{\delta}$ and by
$\widetilde{C_{\mathsf{D}}}={\mathcal{N}}\circ{\mathcal{F}}\circ
C_{\mathsf{D}}\circ{\mathcal{F}}^{-1}\circ{\mathcal{N}}^{-1}$ the one of
$\mathsf{D^{\lambda,\mu}}$. They are computed in proposition 3.5. If a
conformally equivariant quantization $\mathcal{Q}^{\lambda,\mu}$ exists then
it intertwines the two former Casimir operators, i.e.
$\mathcal{Q}^{\lambda,\mu}\circ\widetilde{C_{\mathsf{S}}}=\widetilde{C_{\mathsf{D}}}\circ\mathcal{Q}^{\lambda,\mu}$.
As a consequence, each eigenvector of $\widetilde{C_{\mathsf{S}}}$ is mapped
by $\mathcal{Q}^{\lambda,\mu}$ to an eigenvector of $\tilde{C_{\mathsf{D}}}$
of same eigenvalue and same principal symbol.

######  Lemma 4.4.

Let $k$ be a positive integer and $N$ be an operator on $\mathsf{T}^{\delta}$
which lowers by one the degree in $p$. For $\delta\in\mathbb{R}$ out of a
finite subset of $\mathbb{Q}$, to each eigenvector
$P_{k}\in\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$ of $C_{\mathsf{T}}$
corresponds a unique eigenvector of $C_{\mathsf{T}}+N$ of the form $P_{k}+P$
with $P$ of degree less than $k$.

###### Proof.

We use the notation of the lemma's statement. By hypothesis, we have the
equality $(C_{\mathsf{T}}+N)(P_{k}+P)=\gamma(P_{k}+P)$ for $\gamma$ a real
number. This implies that $\gamma=\gamma_{k,\kappa,s;\alpha\beta}$, the
eigenvalue of $C_{\mathsf{T}}$ on
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$, and $P=P_{k-1}+\ldots+P_{0}$
with $P_{l}$ of degree $l$ in $p$, satisfying
$NP_{l}=(\gamma_{k,\kappa,s;\alpha\beta}-C_{\mathsf{T}})P_{l-1}$. Then, $P$ is
uniquely determined by $P_{k}$, as soon as $\gamma_{k,\kappa,s;\alpha\beta}$
is disctinct of the eigenvalues of $C_{\mathsf{T}}$ on $\bigoplus_{0\leq l\leq
k-1}\mathsf{T}^{\delta}_{l}$. In view of their expressions (4.26), this is
true except for a finite number of rational values of $\delta$. ∎

The lemma can be applied to the Casimir operators $C_{\mathsf{S}}$ and
$C_{\mathsf{D}}$. Starting with an eigenvector $P_{k}$ of $C_{\mathsf{T}}$ we
get unique eigenvectors $P^{\mathsf{S}}$ and $P^{\mathsf{D}}$ of
$C_{\mathsf{S}}$ and $C_{\mathsf{D}}$. In consequence, a conformally
equivariant quantization should satisfy
$\mathcal{Q}^{\lambda,\mu}:{\mathcal{F}}(P^{\mathsf{S}})\mapsto{\mathcal{N}}\circ{\mathcal{F}}(P^{\mathsf{D}})$.
Since the Casimir operator $C_{\mathsf{T}}$ is diagonalizable on the natural
symbol space $\mathsf{T}^{\delta}$, this specifies a unique quantization map,
which turns out to be a ${\mathfrak{g}}$-equivariant quantization.

Let $\lambda+\mu=1$. As the linear map ${\mathcal{N}}$ intertwines conjugation
and the adjoint operation, the map
$P\mapsto\mathcal{Q}^{\lambda,\mu}(\bar{P})^{t}$ is a right inverse to the
principal symbol map $\sigma$ on homogeneous symbols. Moreover, we easily
check that the actions of ${\mathfrak{g}}$ on $\mathsf{S}^{\delta}$ and
$\mathsf{D^{\lambda,\mu}}$ commute with the conjugation when they are pulled-
back to $\mathsf{T}^{\delta}$. Consequently, the map defined above is a
${\mathfrak{g}}$-equivariant quantization and by uniqueness it is equal to
$\mathcal{Q}^{\lambda,\mu}$. We deduce that $\mathcal{Q}^{\lambda,\mu}$
intertwines conjugation and adjoint operation. ∎

This theorem extends to the supercotangent bundle the main result of [12],
giving existence and uniqueness of the conformally equivariant quantization on
the cotangent bundle. If $\delta\notin I^{\mathfrak{S}}\cup I^{\mathcal{Q}}$,
we can compose the two previous maps and get existence and uniqueness of a
${\mathfrak{g}}$-equivariant map
$\mathsf{T}^{\delta}\rightarrow\mathsf{D^{\lambda,\mu}}$ which is a right
inverse of the usual symbol map. As explained in Paragraph 2.8, this gives a
particular instance of the general result in [8] about AHS-equivariant
quantization.

The significance of the subsets of exceptional values of the weight $\delta$
in the context of equivariant quantization has been revealed in [33],
following previous results in [38]. They correspond to the existence of
${\mathfrak{g}}$-invariant operators on the initial module, which are not
fiberwise. The proof in [33] generalizes straightforwardly to this context.

######  Theorem 4.5.

[33] The ${\mathfrak{g}}$-equivariant superization exists and is unique on
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$ if and only if there is no non-
trivial ${\mathfrak{g}}$-invariant operator from this module to
$\bigoplus_{1\leq\ell<k}\mathsf{T}^{\delta}_{k-\ell,\kappa+2\ell}$.
The ${\mathfrak{g}}$-equivariant quantization exists and is unique on
$\mathsf{S}^{\delta}_{k,\kappa,s;\alpha\beta}$ if and only if there is no non-
trivial ${\mathfrak{g}}$-invariant operator from this module to
$\bigoplus_{\tiny\begin{array}[]{c}0\leq\ell<k\\\
0\leq\ell^{\prime}\leq\ell,\kappa\end{array}}\mathsf{S}^{\delta}_{k-\ell,\kappa-2\ell^{\prime}}$.

From the Theorem 4.2, providing the classification of conformally invariant
operators on $\mathsf{T}^{\delta}$, we easily deduce the following corollary.

######  Corollary 4.6.

The subsets of critical values $I^{\mathfrak{S}}$ and $I^{{\mathcal{Q}}}$ for
the conformally equivariant superization and quantization are included into
$\frac{1}{n}\mathbb{N}^{*}$.

##  5\. Explicit formulæ for the ${\mathfrak{g}}$-equivariant quantization and
superization

###  5.1. Local formulæ

To write down explicit local formulæ for the maps $\mathfrak{S}^{\delta}$ and
$\mathcal{Q}^{\lambda,\mu}$, we restrict to the local model
$(\mathbb{R}^{p,q},F\eta)$. Thus, we can use the Euclidean invariants (for the
metric $\eta$) classified in section 3. The obtained formulæ being the only
conformally equivariant ones, they are globally defined as a whole on
arbitrary conformally flat manifolds $(M,\text{})$, but their building blocks
are only locally defined, as they require the flat metric $\eta$.

The superization is easier to deal with, and we obtain an explicit formula for
it on any component $\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$ of
$\mathsf{T}^{\delta}$. We write the formula in a shape which illustrates
Theorem 4.5. As previously, the $0$ index denotes restriction and
corestriction of operators to the space $\ker T$ and $\Pi_{0}$ is the
projection onto that space.

######  Theorem 5.1.

The conformally equivariant superization exists on
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$ if and only if $\delta\notin
I^{\mathfrak{S}}_{k,\kappa,s;\alpha\beta}$, given below. Then it is unique,
and reads as

$\displaystyle\mathfrak{S}^{\delta}$ $\displaystyle=$
$\displaystyle{\mathcal{F}}\circ\bigg{(}\mathrm{Id}+\frac{\hbar}{\mathsf{i}}c_{\boldsymbol{d}}$
$\displaystyle\times$
$\displaystyle\Big{[}\boldsymbol{d}\boldsymbol{\delta}+c_{D}\,R^{s}\boldsymbol{Q}\boldsymbol{\delta}(D+\frac{1}{k+\kappa-n\delta}\boldsymbol{d}\boldsymbol{Q}^{*})T^{s}+c_{G}\,R^{s-1}\boldsymbol{Q}\boldsymbol{\delta}(G_{0}+\frac{1}{k+\kappa-n\delta}\boldsymbol{d}\boldsymbol{\delta}^{*}_{0})T^{s}\Big{]}\bigg{)}.$

Denoting by
$\rho_{k,s}=\prod_{s^{\prime}=1}^{s}2s^{\prime}(n+2(k-2s+s^{\prime}-1))$ the
eigenvalue of $R^{s}T^{s}$ on $\mathsf{T}^{\delta}_{k,\kappa,s}$ and by
$a_{k,s}=\frac{n+2(k-s)}{n+2(k-s+1)}$ an auxiliary factor, we have

(5.32)
$\left\\{\begin{array}[]{l}c_{\boldsymbol{d}}=-\frac{1}{k+\kappa+1-n\delta},\\\\[6.0pt]
c_{D}=\frac{1}{(2(k-s-1)+n(1-\delta))\rho_{k,s}a_{k,s}},\\\\[6.0pt]
c_{G}=\frac{2s}{(2s-n\delta)\rho_{k,s}}.\end{array}\right.$

The non-existence cases $\delta\in I^{\mathfrak{S}}_{k,\kappa,s;\alpha\beta}$
correspond to the existence of ${\mathfrak{g}}$-invariant operators with
source space $\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$. They are
classified below

(5.33)
$\begin{array}[]{|c||c|c|c|c|}\hline\cr\delta&{\mathfrak{g}}-\text{invariant
operator}&\text{source space
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$}\\\\[4.0pt]
\hline\cr\frac{k+\kappa+1}{n}&[\boldsymbol{d}\boldsymbol{\delta}+c_{D}\ldots+c_{G}\ldots]&\alpha\beta\neq
10,\kappa\leq(n-2),1\leq k\\\\[4.0pt]
\hline\cr\frac{2s}{n}&R^{s-1}\boldsymbol{Q}\boldsymbol{\delta}(G_{0}+\frac{1}{k+\kappa-n\delta}\boldsymbol{d}\boldsymbol{\delta}^{*}_{0})T^{s}&\alpha=0,1\leq
s,\kappa\leq(n-2)\\\\[4.0pt] \hline\cr
1+\frac{2(k-s-1)}{n}&R^{s}\boldsymbol{Q}\boldsymbol{\delta}(D+\frac{1}{k+\kappa-n\delta}\boldsymbol{d}\boldsymbol{Q}^{*})T^{s}&\beta=1,\kappa\leq(n-2),2\leq
k\\\\[4.0pt]
\hline\cr\frac{k+\kappa}{n}&R^{s}\boldsymbol{Q}\boldsymbol{\delta}\boldsymbol{d}\boldsymbol{Q}^{*}T^{s}&\alpha\beta=11,1\leq\kappa\leq(n-1),2\leq
k\\\\[4.0pt]
&R^{s-1}\boldsymbol{Q}\boldsymbol{\delta}\boldsymbol{d}\boldsymbol{\delta}^{*}T^{s}&\alpha\beta=00,1\leq
s,1\leq\kappa\leq(n-1)\\\\[4.0pt] \hline\cr\end{array}$

###### Proof.

The ${\mathfrak{g}}$-equivariance of $\mathfrak{S}^{\delta}$ reads as
$\mathfrak{S}^{\delta}\circ\mathbb{L}_{X}^{\delta}=\mathsf{L}_{X}^{\delta}\circ\mathfrak{S}^{\delta}$
for all $X\in{\mathfrak{g}}$. As ${\mathcal{F}}$ is a section of the principal
tensorial symbol map $\varepsilon_{k,\kappa}$ which is
$\mathfrak{ce}(p,q)$-invariant, we have
$\mathfrak{S}^{\delta}={\mathcal{F}}\circ(\mathrm{Id}+N)$ with $N$ a
$\mathfrak{ce}(p,q)$-invariant operator from
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}$ to
$\bigoplus_{1\leq\ell<k}\mathsf{T}^{\delta}_{k-\ell,\kappa+2\ell}$. In view of
the form (3.21) of those invariant operators, $N$ is a linear combination of
the following linearly independent operators

$\boldsymbol{d}\boldsymbol{\delta},\quad
R^{s}\boldsymbol{Q}\boldsymbol{\delta}DT^{s},\quad
R^{s-1}\boldsymbol{Q}\boldsymbol{\delta}G_{0}T^{s}\quad
R^{s}\boldsymbol{Q}\boldsymbol{\delta}\boldsymbol{d}\boldsymbol{Q}^{*}T^{s},\quad
R^{s-1}\boldsymbol{Q}\boldsymbol{\delta}\boldsymbol{d}\boldsymbol{\delta}^{*}_{0}T^{s}.$

Indeed, there is no $\mathfrak{ce}(p,q)$-invariant operator lowering the
degree in $\xi$ by $4$, and above are all those which lower the degree in
$\xi$ by $2$ and rise the degree in $p$ by one.

Using the comparison (2.15) between the action of ${\mathfrak{g}}$ on
$\mathsf{S}^{\delta}$ and $\mathsf{T}^{\delta}$, the
${\mathfrak{g}}$-equivariance of $\mathfrak{S}^{\delta}$ reads then as

(5.34)
$[N,\mathbb{L}^{\delta}_{\bar{X}_{i}}]=-2\frac{\hbar}{\mathsf{i}}\xi_{i}\boldsymbol{\delta},$

for all $i=1,\ldots,n$. The actions of the conformal inversion $\bar{X}_{i}$
on the operators $G,D,\boldsymbol{d},\boldsymbol{d}^{*}$ are given in (A.53).
From them and the conformal invariance of $R^{s}T^{s}$,
$\boldsymbol{Q}\boldsymbol{\delta}$, we deduce that, on
$\mathsf{T}^{\delta}_{k,\kappa,s}$,

$\displaystyle[\boldsymbol{d}\boldsymbol{\delta},\mathbb{L}^{\delta}_{\bar{X}_{i}}]$
$\displaystyle=$ $\displaystyle
2(k+\kappa+1-n\delta)\xi_{i}\boldsymbol{\delta}+\frac{2}{\rho_{k,s}}R^{s}\boldsymbol{Q}\boldsymbol{\delta}\partial_{p_{i}}T^{s}+\frac{4s}{\rho_{k,s}}R^{s}\boldsymbol{Q}\boldsymbol{\delta}p_{i}T^{s},$
$\displaystyle\left[R^{s}\boldsymbol{Q}\boldsymbol{\delta}DT^{s},\mathbb{L}^{\delta}_{\bar{X}_{i}}\right]$
$\displaystyle=$ $\displaystyle
2(2(k-s-1)+n(1-\delta))R^{s}\boldsymbol{Q}\boldsymbol{\delta}\partial_{p_{i}}T^{s}+2R^{s}\boldsymbol{Q}\boldsymbol{\delta}\xi_{i}\boldsymbol{Q}^{*}T^{s},$
$\displaystyle\left[R^{s-1}\boldsymbol{Q}\boldsymbol{\delta}G_{0}T^{s},\mathbb{L}^{\delta}_{\bar{X}_{i}}\right]$
$\displaystyle=$ $\displaystyle
2(2s-n\delta)R^{s-1}\boldsymbol{Q}\boldsymbol{\delta}\Pi_{0}p_{i}T^{s}-2R^{s-1}\boldsymbol{Q}\boldsymbol{\delta}\xi_{i}\boldsymbol{\delta}^{*}T^{s}.$

The result follows then from the relation
$R\partial_{p_{i}}T^{s}+2sp_{i}T^{s}=2s\Pi_{0}p_{i}T^{s}+(a_{k,s})^{-1}R\partial_{p_{i}}T^{s}$.
∎

The conformally equivariant superization admits also a global formula on
$\mathsf{T}^{\delta}$ given below, as a drawback it does not allow to link
easily the critical weights with conformally invariant operators.

######  Proposition 5.2.

On $\mathsf{T}^{\delta}$, the conformally equivariant superization is given by

(5.35)
$\mathfrak{S}^{\delta}={\mathcal{F}}\circ\bigg{(}\mathrm{Id}+\frac{\hbar}{\mathsf{i}}C_{\boldsymbol{d}}\Big{[}\boldsymbol{d}\boldsymbol{\delta}+\boldsymbol{Q}\boldsymbol{\delta}(n\delta\mathbb{D}-\mathbb{G}T)C_{G}\Big{]}\bigg{)},$

where

(5.36)
$\left\\{\begin{array}[]{l}(C_{\boldsymbol{d}})^{-1}=-(\mathcal{E}+\Sigma-n\delta),\\\\[6.0pt]
(C_{G})^{-1}=RT+n\delta\big{(}2(\mathcal{E}-1)+n(1-\delta)\big{)},\end{array}\right.\qquad\qquad\left\\{\begin{array}[]{l}\mathbb{D}=D+\frac{1}{\mathcal{E}+\Sigma+1-n\delta}\boldsymbol{d}\boldsymbol{Q}^{*},\\\\[6.0pt]
\mathbb{G}=G+\frac{1}{\mathcal{E}+\Sigma+1-n\delta}\boldsymbol{d}\boldsymbol{\delta}^{*}.\end{array}\right.$

###### Proof.

Taking advantage of the proof of the previous theorem, we just have to show
that

$N=\frac{\hbar}{\mathsf{i}}C_{\boldsymbol{d}}\Big{[}\boldsymbol{d}\boldsymbol{\delta}+\boldsymbol{Q}\boldsymbol{\delta}(n\delta\mathbb{D}-\mathbb{G}T)C_{G}\Big{]},$

satisfies the relation (5.34). This follows from the next computations on
$\mathsf{T}^{\delta}$, relying again on (A.53) and the conformal invariance of
$\boldsymbol{Q}\boldsymbol{\delta}$,

$\displaystyle[\boldsymbol{d}\boldsymbol{\delta},\mathbb{L}_{\bar{X}_{i}}]$
$\displaystyle=$ $\displaystyle
2(\mathcal{E}+\Sigma-n\delta)\xi_{i}\boldsymbol{\delta}+\boldsymbol{Q}\boldsymbol{\delta}\partial_{p_{i}},$
$\displaystyle[\boldsymbol{Q}\boldsymbol{\delta}\mathbb{D},\mathbb{L}_{\bar{X}_{i}}]$
$\displaystyle=$ $\displaystyle
2\boldsymbol{Q}\boldsymbol{\delta}\big{(}\partial_{p_{i}}(2(\mathcal{E}-1)+n(1-\delta))-p_{i}T\big{)},$
$\displaystyle[\boldsymbol{Q}\boldsymbol{\delta}\mathbb{G}T,\mathbb{L}_{\bar{X}_{i}}]$
$\displaystyle=$ $\displaystyle
2\boldsymbol{Q}\boldsymbol{\delta}\big{(}-n\delta
p_{i}T+\partial_{p_{i}}RT\big{)}.$

∎

We turn now to the conformally equivariant quantization. Explicit computations
are much harder and we restrict ourselves to $\mathsf{S}^{\delta}_{\leq 1}$,
the space of symbols of first order in $p$.

######  Theorem 5.3.

Let $\lambda,\mu\in\mathbb{R}$ such that $\mu-\lambda=\delta$. For
${n\delta\neq 1,\ldots,n+1}$, there exists a unique conformally equivariant
quantization $\mathcal{Q}^{\lambda,\mu}:\mathsf{S}^{\delta}_{\leq
1}\rightarrow\mathsf{D}^{\lambda,\mu}_{1}$, which reads as

(5.37)
$\mathcal{Q}^{\lambda,\mu}={\mathcal{N}}\circ{\mathcal{F}}\circ\left(\mathrm{Id}+\frac{\hbar}{\mathsf{i}}\Big{[}c_{D}(\Sigma)\,D+c_{\boldsymbol{\delta}}(\Sigma)\,\boldsymbol{d}^{*}\boldsymbol{\delta}+c_{\boldsymbol{d}}(\Sigma)\,\boldsymbol{d}\boldsymbol{Q}^{*}+c_{*}(\Sigma)\,\boldsymbol{d}^{*}\boldsymbol{Q}^{*}\Big{]}\right)\circ{\mathcal{F}}^{-1},$

the coefficients depending on the odd Euler operator
$\Sigma=\xi^{i}\partial_{\xi^{i}}$ as follows

(5.38)
$\left\\{\begin{array}[]{l}c_{D}(\Sigma)=\frac{2n\lambda+1}{2n(1-\delta)+2},\\\\[6.0pt]
c_{\boldsymbol{\delta}}(\Sigma)=\frac{n(1-\delta-2\lambda)}{(\Sigma-n(1-\delta))(2n(1-\delta)+2)},\\\\[6.0pt]
c_{\boldsymbol{d}}(\Sigma)=-\frac{n(1-\delta-2\lambda)}{(\Sigma-n\delta)(2n(1-\delta)+2)},\\\\[6.0pt]
c_{*}(\Sigma)=\frac{1}{4(\Sigma-n(1-\delta))}.\end{array}\right.$

The remaining critical cases, $n\delta=1,\ldots n+1$, correspond to the
existence of a ${\mathfrak{g}}$-invariant operator on certain subspaces
$\mathsf{T}^{\delta}_{1,\kappa,0;\alpha\beta}$ as below

(5.39)
$\begin{array}[]{|c||c|c|c|c|}\hline\cr\delta&{\mathfrak{g}}-\text{invariant
operator}&\text{source space
$\mathsf{T}^{\delta}_{1,\kappa,0;\alpha\beta}$}&\lambda\text{ s.t.
}{\mathcal{Q}}^{\lambda,\lambda+\delta}\text{ exists}\\\\[4.0pt]
\hline\cr\frac{1}{n}&\boldsymbol{d}\boldsymbol{Q}^{*}&\kappa=1,\alpha=1&\lambda=\frac{n-1}{2n}\\\\[4.0pt]
&\boldsymbol{d}^{*}\boldsymbol{\delta}&\kappa=n-1,\alpha\beta=01&\lambda=\frac{n-1}{2n}\\\\[4.0pt]
\hline\cr\frac{\ell}{n},&\boldsymbol{d}\boldsymbol{Q}^{*}&\kappa=\ell,\alpha=1&none\\\\[4.0pt]
2\leq\ell\leq
n&\boldsymbol{d}^{*}\boldsymbol{\delta}&\kappa=n-\ell,\alpha\beta=01&none\\\\[4.0pt]
&\boldsymbol{d}^{*}\boldsymbol{Q}^{*}&\kappa=n-\ell+2,\alpha\beta=10&none\\\\[4.0pt]
\hline\cr\frac{n+1}{n}&D+\frac{1}{\Sigma-n-1}\boldsymbol{d}\boldsymbol{Q}^{*}-\frac{1}{\Sigma+1}\boldsymbol{d}^{*}\boldsymbol{\delta}&\kappa=0,\ldots,n&\lambda=\frac{-1}{2n}\\\\[4.0pt]
\hline\cr\end{array}$

For the two critical values $\delta=\frac{1}{n},\frac{n+1}{n}$, the
quantization ${\mathcal{Q}}^{\lambda,\lambda+\delta}$ exists for a unique
value of $\lambda$ and is given by, respectively,

$\displaystyle{\mathcal{Q}}^{\frac{n-1}{2n},\frac{n+1}{2n}}$ $\displaystyle=$
$\displaystyle{\mathcal{N}}\circ{\mathcal{F}}\circ\left(\mathrm{Id}+\frac{\hbar}{\mathsf{i}}\Big{[}c_{D}(\Sigma)\,D+c_{*}(\Sigma)\,\boldsymbol{d}^{*}\boldsymbol{Q}^{*}\Big{]}\right)\circ{\mathcal{F}}^{-1},$
$\displaystyle{\mathcal{Q}}^{\frac{-1}{2n},\frac{2n+1}{2n}}$ $\displaystyle=$
$\displaystyle{\mathcal{N}}\circ{\mathcal{F}}\circ\left(\mathrm{Id}+\frac{\hbar}{\mathsf{i}}\Big{[}c_{\boldsymbol{\delta}}(\Sigma)\,\boldsymbol{d}^{*}\boldsymbol{\delta}+c_{\boldsymbol{d}}(\Sigma)\,\boldsymbol{d}\boldsymbol{Q}^{*}+c_{*}(\Sigma)\,\boldsymbol{d}^{*}\boldsymbol{Q}^{*}\Big{]}\right)\circ{\mathcal{F}}^{-1}.$

It is unique on
$\mathfrak{S}^{\delta}(\mathsf{T}^{\delta}_{1,\kappa,0;\alpha\beta})$, up to
prospective ${\mathfrak{g}}$-invariant operators on that space, obtained from
the list above via the superization $\mathfrak{S}^{\delta}$.

###### Proof.

The ${\mathfrak{g}}$-equivariance of ${\mathcal{Q}}^{\lambda,\mu}$ reads as
$\mathcal{Q}^{\lambda,\mu}\circ\mathsf{L}_{X}^{\delta}=\mathcal{L}_{X}^{\lambda,\mu}\circ\mathcal{Q}^{\lambda,\mu}$
for all $X\in{\mathfrak{g}}$. As ${\mathcal{N}}$ and ${\mathcal{F}}$ are
sections of the principal Hamiltonian and tensorial symbol maps which are
$\mathfrak{ce}(p,q)$-invariant, we have
$\mathcal{Q}^{\lambda,\mu}={\mathcal{N}}\circ{\mathcal{F}}\circ(\mathrm{Id}+N)$
with $N$ a $\mathfrak{ce}(p,q)$-invariant operator from
$\mathsf{T}^{\delta}_{1,\kappa,0;\alpha\beta}$ to
$\bigoplus_{\kappa^{\prime}<\kappa+2}\mathsf{T}^{\delta}_{0,\kappa^{\prime}}$.
In view of the form (3.21) of those invariant operators, $N$ is a linear
combination of the following linearly independent operators,

$D,\quad\boldsymbol{d}^{*}\boldsymbol{\delta},\quad\boldsymbol{d}\boldsymbol{Q}^{*},\quad\boldsymbol{d}^{*}\boldsymbol{Q}^{*},$

as all the other ones carry at least two derivations in the $p$ variables, or
do not lower the Hamiltonian order, like $\boldsymbol{d}\boldsymbol{\delta}$.
In consequence, $\mathcal{Q}^{\lambda,\mu}$ is of the form specified by
(5.37).

Using the comparisons (2.15) and (2.16) between the action of ${\mathfrak{g}}$
on $\mathsf{T}^{\delta}$, $\mathsf{S}^{\delta}$ and
$\mathsf{D^{\lambda,\mu}}$, the ${\mathfrak{g}}$-equivariance of
$\mathcal{Q}^{\lambda,\mu}$ reads then as

(5.40)
$[N,\mathbb{L}^{\delta}_{\bar{X}_{i}}]=\frac{\hbar}{\mathsf{i}}\big{(}2n\lambda\partial_{p_{i}}+\xi^{i}\boldsymbol{Q}^{*}-\boldsymbol{\delta}\partial_{\xi_{i}}+\frac{1}{2}\partial_{\xi_{i}}\boldsymbol{Q}^{*}\big{)},$

for all $i=1,\ldots,n$. From the computations (A.53) in appendix we deduce the
following equalities in $\mathsf{T}^{\delta}_{1,\kappa}$,

$\displaystyle\left[\boldsymbol{d}^{*}\boldsymbol{\delta},\mathbb{L}_{\bar{X}_{i}}^{\delta}\right]$
$\displaystyle=$ $\displaystyle
2(\Sigma-n(1-\delta))(\boldsymbol{\delta}\partial_{\xi^{i}}-\partial_{p^{i}}),$
$\displaystyle\left[\boldsymbol{d}\boldsymbol{Q}^{*},\mathbb{L}_{\bar{X}_{i}}^{\delta}\right]$
$\displaystyle=$ $\displaystyle 2(\Sigma-n\delta)\xi_{i}\boldsymbol{Q}^{*},$
$\displaystyle\left[\boldsymbol{d}^{*}\boldsymbol{Q}^{*},\mathbb{L}_{\bar{X}_{i}}^{\delta}\right]$
$\displaystyle=$ $\displaystyle
2(-\Sigma+n(1-\delta))\partial_{\xi^{i}}\boldsymbol{Q}^{*},$
$\displaystyle\left[D,\mathbb{L}_{\bar{X}_{i}}^{\delta}\right]$
$\displaystyle=$ $\displaystyle
2\boldsymbol{\delta}\partial_{\xi^{i}}-2\xi_{i}\boldsymbol{Q}^{*}+2n(1-\delta)\partial_{p^{i}}.$

By substitution in the ${\mathfrak{g}}$-equivariance condition (5.40), this
determines a unique operator $N$, as specified by (5.38). The remaining
statements of the theorem easily follow. ∎

######  Remark 5.4.

From the table (5.33), we deduce that the conformally equivariant superization
$\mathfrak{S}^{\delta}$ exists on each source space of the
${\mathfrak{g}}$-invariant operators listed in (5.39). Consequently, they can
be transported as ${\mathfrak{g}}$-invariant operators on
$\mathsf{S}^{\delta}$, in accordance with theorem 4.5.

######  Remark 5.5.

Each of the ${\mathfrak{g}}$-invariant operators in tables (5.33) and (5.39)
can be restricted and corestricted as an operator:
$\mathsf{T}^{\delta}_{k,\kappa,s;\alpha\beta}\rightarrow\mathsf{T}^{\delta}_{k^{\prime},\kappa^{\prime},s^{\prime};\alpha^{\prime}\beta^{\prime}}$.
Using the fiberwise invariant operators like in expression (4.30), it
corresponds to one of the ${\mathfrak{g}}$-invariant operator entering the
classification of Theorem 4.2. For table (5.33), they are respectively
$\boldsymbol{d}_{\bf 0},G_{\bf 0},D_{\bf 0},\boldsymbol{d}_{\bf 0}$ and for
table (5.39) $\boldsymbol{d}_{\bf 0},\boldsymbol{d}^{*}_{\bf
0},\boldsymbol{d}_{\bf 0},\boldsymbol{d}^{*}_{\bf 0},\boldsymbol{d}^{*}_{\bf
0},D_{\bf 0}$.

###  5.2. Covariant formulæ and the curved case

We would like now to devise formulæ in covariant terms for the equivariant
superization and quantization and extend them over arbitrary conformal
manifolds. In [13], this was done in a conformally invariant fashion for the
conformally equivariant quantization of cotangent bundles. The situation here
is more involved, and only the composition
$\mathcal{Q}^{\lambda,\mu}\circ\mathfrak{S}^{\delta}$ admits a conformally
invariant prolongation.

Starting with a pseudo-Riemannian spin manifold $(M,\text{})$ endowed with the
Levi-Civita connection, we get covariant derivations on all associated bundles
to the principal bundle of spin frames. For $X\in\mathrm{Vect}(M)$, we denote
by $\nabla^{\lambda}_{X}$ the one acting on
$\Gamma(S\otimes|\Lambda|^{\lambda})$, and by $\partial^{\nabla}_{X}$ the one
acting on $\mathsf{T}^{\delta}=\Gamma(\mathbb{T}^{\delta})$. The latter
derivative should be interpreted as the horizontal covariant derivative on the
supercotangent bundle ${\mathcal{M}}$. It allows to generalize the $13$
invariant operators introduced in Proposition 3.2 as global operators over any
pseudo-Riemannian manifold $(M,\text{})$, replacing the flat metric $\eta$ by
and the derivatives $\partial_{i}$ by the covariant ones
$\partial^{\nabla}_{i}$. Their commutation relations remain the same than in
the flat case, see (A.55), except between the five ones containing covariant
derivatives. Accordingly, we denote them with a superscript $\nabla$,

(5.41) $G^{\nabla}=\text{}^{ij}p_{i}\partial^{\nabla}_{j},\quad
D^{\nabla}=\partial_{p_{i}}\partial^{\nabla}_{i},\quad
L^{\nabla}=\partial^{\nabla}_{i}\text{}^{ij}\partial^{\nabla}_{j},\quad\boldsymbol{d}^{\nabla}=\xi^{i}\partial^{\nabla}_{i},\quad\boldsymbol{d}^{*\nabla}=\text{}^{ij}\partial_{\xi^{i}}\partial^{\nabla}_{j}.$

With such a modification, the formulæ (5.35) and (5.37) provide covariant
expressions which are proved to coincide, in the conformally flat case, with
the conformally equivariant superization and quantization, respectively.

######  Proposition 5.6.

Let $(M,\text{})$ be a pseudo-Riemannian spin manifold,
$P\in\mathsf{T}^{\delta}$, $P_{1}\in\mathsf{S}^{\delta}_{1}$ and
$\lambda,\mu\in\mathbb{R}$ such that $\mu-\lambda=\delta$. Using notation of
(5.36) and (5.38), we define the applications

(5.42)
$\mathfrak{S}^{\delta}_{\nabla}(P)=|\mathrm{vol}_{\text{}}|^{\frac{\Sigma}{n}}\left(P+\frac{\hbar}{\mathsf{i}}C_{\boldsymbol{d}}\Big{[}\boldsymbol{d}^{\nabla}\boldsymbol{\delta}+\boldsymbol{Q}\boldsymbol{\delta}(n\delta\mathbb{D^{\nabla}}-\mathbb{G^{\nabla}}T)C_{G}\Big{]}P\right)$

and

(5.43) $\displaystyle{\mathcal{Q}}^{\lambda,\mu}_{\nabla}(P_{1})$
$\displaystyle=$
$\displaystyle\frac{\hbar}{\mathsf{i}(\sqrt{2})^{\Sigma}}\,\gamma(P_{1}^{i})\nabla^{\lambda}_{i}$
$\displaystyle+$
$\displaystyle\frac{\hbar}{\mathsf{i}(\sqrt{2})^{\Sigma}}\,\gamma\Big{(}c_{D}(\Sigma)\,D^{\nabla}P_{1}+c_{\boldsymbol{\delta}}(\Sigma)\,\boldsymbol{d}^{*\nabla}\boldsymbol{\delta}P_{1}+c_{\boldsymbol{d}}(\Sigma)\,\boldsymbol{d}^{\nabla}\boldsymbol{Q}^{*}P_{1}+c_{*}(\Sigma)\,\boldsymbol{d}^{*\nabla}\boldsymbol{Q}^{*}P_{1}\Big{)}.$

These maps coincide with, respectively, the conformally equivariant
superization and quantization if $(M,\text{})$ is conformally flat.

###### Proof.

Let $(M,\text{})$ be a conformally flat manifold and $x$ a point in $M$. We
prove that $\mathfrak{S}^{\delta}_{\nabla}(P)$ and $\mathfrak{S}^{\delta}(P)$
are equal at $x$. By conformal flatness of , there exists a neighborhood of
$x$ and conformal coordinates such that $\text{}_{ij}=F\eta_{ij}$ for $F$ a
positive function and $\eta$ the flat metric. Moreover, up to a conformal
change of coordinates we can assume that all the first derivatives of $F$ at
$x$ vanish. This is a classical result, which can be derived from the
infinitesimal action of inversions $L_{\bar{X}_{i}}\eta=-4x_{i}\eta$. Using
such a coordinate system, we can write down both
$\mathfrak{S}^{\delta}_{\nabla}(P)$ and $\mathfrak{S}^{\delta}(P)$ at $x$.
They coincide since the covariant formula (5.42) involves only first order
derivations. The same argument applies for the quantization maps. ∎

######  Theorem 5.7.

Let $(M,\text{})$ be a pseudo-Riemannian spin manifold,
$P\in\mathsf{T}_{1}^{\delta}$ and $\lambda,\mu\in\mathbb{R}$ such that
$\mu-\lambda=\delta$. The following map

$\displaystyle\mathsf{Q}^{\lambda,\mu}_{\nabla}(P)$ $\displaystyle=$
$\displaystyle\frac{\hbar}{\mathsf{i}(\sqrt{2})^{\Sigma}}\,|\mathrm{vol}_{\text{}}|^{\frac{\Sigma}{n}}\gamma(P^{i})\nabla^{\lambda}_{i}$
$\displaystyle+$
$\displaystyle\frac{\hbar}{\mathsf{i}(\sqrt{2})^{\Sigma}}\,|\mathrm{vol}_{\text{}}|^{\frac{\Sigma}{n}}\gamma\Big{(}c_{0}\,\boldsymbol{d}^{\nabla}\boldsymbol{\delta}P+c_{1}\,D^{\nabla}P+c_{2}\,\boldsymbol{d}^{*\nabla}\boldsymbol{\delta}P+c_{3}\,\boldsymbol{d}^{\nabla}\boldsymbol{Q}^{*}P+c_{4}\,\boldsymbol{d}^{*\nabla}\boldsymbol{Q}^{*}P\Big{)},$

is conformally invariant, i.e. depends only on the conformal class of the
metric , if and only if the real coefficients $c_{0},c_{1},c_{2},c_{3},c_{4}$
are such that $\mathsf{Q}^{\lambda,\mu}_{\nabla}$ is equal to the composition
${\mathcal{Q}}_{\nabla}^{\lambda,\mu}\circ\mathfrak{S}^{\delta}_{\nabla}$. But
neither $\mathfrak{S}^{\delta}_{\nabla}$ nor
${\mathcal{Q}}_{\nabla}^{\lambda,\mu}$ are conformally invariant.

###### Proof.

We suppose that  and $\hat{\text{}}$ are two metrics conformally related
through $\hat{\text{}}=F\text{}$, with $F$ a positive function on $M$. The
Christoffel symbols of their Levi-Civita connections satisfy

$\hat{\Gamma}_{ij}^{k}-\Gamma_{ij}^{k}=\frac{1}{2F}\left(F_{i}\delta^{k}_{j}+F_{j}\delta^{k}_{i}-F^{k}\text{}_{ij}\right),$

where $F_{i}=\partial_{i}F$ and $F^{k}=\text{}^{ik}\partial_{i}F$. From the
usual expressions of the spinor and the horizontal covariant derivatives in
terms of Christoffel symbols, we deduce

$\displaystyle\hat{\nabla}^{\lambda}_{j}-\nabla^{\lambda}_{j}$
$\displaystyle=$
$\displaystyle-\frac{1}{8F}[\gamma_{j},\gamma^{i}]F_{i}-\frac{n\lambda}{2F}F_{j},$
$\displaystyle\partial^{\hat{\nabla}}_{i}-\partial^{\nabla}_{i}$
$\displaystyle=$
$\displaystyle\frac{1}{2F}\left(F_{i}\mathcal{E}+F_{k}p_{i}\partial_{p_{k}}-F^{j}p_{j}\partial_{p^{i}}\right)-\frac{1}{2F}\left(F_{k}\xi^{k}\partial_{\xi^{i}}-F^{j}\xi_{i}\partial_{\xi^{j}}\right)-\frac{n\delta}{2F}F_{i},$

the latter acting on $\mathsf{T}^{\delta}$. From the conformal invariance of
$|\mathrm{vol}_{\text{}}|^{\frac{\Sigma}{n}}\gamma$, and of
$\boldsymbol{\delta}$, $\boldsymbol{Q}^{*}$ as operators from
$\mathsf{T}^{\delta}$ to $\mathsf{T}^{\delta-1/n}$, we deduce the expression
of each of the six terms involved in the map
$\mathsf{Q}^{\lambda,\mu}_{\hat{\nabla}}-\mathsf{Q}^{\lambda,\mu}_{\nabla}$:

$\displaystyle|\mathrm{vol}_{\text{}}|^{\frac{\Sigma}{n}}\gamma(P^{j})(\hat{\nabla}^{\lambda}_{j}-\nabla^{\lambda}_{j})$
$\displaystyle=$
$\displaystyle-|\mathrm{vol}_{\text{}}|^{\frac{\Sigma}{n}}\gamma\left(\big{[}\boldsymbol{\delta}\xi^{i}-\frac{1}{2}\xi^{i}\boldsymbol{Q}^{*}+\frac{1}{2}\boldsymbol{\delta}\partial_{\xi_{i}}+\frac{1}{4}\boldsymbol{Q}^{*}\partial_{\xi_{i}}+n\lambda\partial_{p_{i}}\big{]}P\right)\frac{F_{i}}{2F},$
$\displaystyle
c_{0}\,\left(\boldsymbol{d}^{\hat{\nabla}}-\boldsymbol{d}^{\nabla}\right)\boldsymbol{\delta}$
$\displaystyle=$ $\displaystyle
c_{0}\left((\Sigma-n\delta)\xi^{i}\boldsymbol{\delta}\right)\frac{F_{i}}{2F},$
$\displaystyle c_{1}\left(D^{\hat{\nabla}}-D^{\nabla}\right)$ $\displaystyle=$
$\displaystyle
c_{1}\left(\boldsymbol{\delta}\partial_{\xi_{i}}-\xi^{i}\boldsymbol{Q}^{*}+n(1-\delta)\partial_{p_{i}}\right)\frac{F_{i}}{2F},$
$\displaystyle
c_{2}\,\left(\boldsymbol{d}^{*\hat{\nabla}}-\boldsymbol{d}^{*\nabla}\right)\boldsymbol{\delta}$
$\displaystyle=$ $\displaystyle
c_{2}\left((\Sigma-n(1-\delta))\boldsymbol{\delta}\partial_{\xi_{i}}\right)\frac{F_{i}}{2F},$
$\displaystyle
c_{3}\,\left(\boldsymbol{d}^{\hat{\nabla}}-\boldsymbol{d}^{\nabla}\right)\boldsymbol{Q}^{*}$
$\displaystyle=$ $\displaystyle
c_{3}\left((\Sigma-n\delta)\xi^{i}\boldsymbol{Q}^{*}\right)\frac{F_{i}}{2F},$
$\displaystyle
c_{4}\,\left(\boldsymbol{d}^{*\hat{\nabla}}-\boldsymbol{d}^{*\nabla}\right)\boldsymbol{Q}^{*}$
$\displaystyle=$ $\displaystyle
c_{4}\left((\Sigma-n(1-\delta))\boldsymbol{Q}^{*}\partial_{\xi^{i}}\right)\frac{F_{i}}{2F}.$

The first equality holds for all $P\in\mathsf{T}^{\delta}_{1}$ whereas the
other ones should be understood as equality of operators acting on
$\mathsf{T}^{\delta}_{1}$. The equality
$\mathsf{Q}^{\lambda,\mu}_{\hat{\nabla}}(P)-\mathsf{Q}^{\lambda,\mu}_{\nabla}(P)=0$,
for all $P\in\mathsf{T}^{\delta}_{1}$, is then equivalent to a linear system
in the five coefficients entering the definition of
$\mathsf{Q}^{\lambda,\mu}_{\nabla}$. Solving this system leads exactly to
$\mathsf{Q}^{\lambda,\mu}_{\nabla}={\mathcal{Q}}^{\lambda,\mu}_{\nabla}\circ\mathfrak{S}^{\delta}_{\nabla}$,
as wanted.

Since $c_{0}=c_{\boldsymbol{d}}(\Sigma)=\frac{-1}{\Sigma-n\delta}$, the
preceding computations lead to

$\mathfrak{S}^{\delta}_{\hat{\nabla}}(P)=F^{\Sigma/2}\left(\mathfrak{S}^{\delta}_{\nabla}(P)-\frac{\hbar}{\mathsf{i}}\frac{F_{j}}{2F}\xi^{j}\boldsymbol{\delta}P\right),$

and then neither $\mathfrak{S}^{\delta}_{\nabla}$ nor
${\mathcal{Q}}^{\lambda,\mu}_{\nabla}$ are conformally invariant. ∎

Let $A$ be a first order differential operator acting on the tensor bundle
${\mathcal{S}}TM\otimes\Lambda T^{*}M$, for $(M,\text{})$ a conformally flat
manifold. Then, ${\mathfrak{g}}$-equivariance of $A$ implies its conformal
invariance, whenever the ${\mathfrak{g}}$-actions are the natural ones on
tensors [17]. But the Hamiltonian action is not of this type, as seen in
(2.15). This explains the lack of conformal invariance of the conformally
equivariant superization.

##  6\. Symmetries of spinning particles

We classify all the symmetries of free massless spinning particles over
conformally flat manifolds, both in the classical and quantum cases. They
arise via the conformally equivariant superization and quantization of
conformal Killing hook tensors, that we introduce over an arbitrary pseudo-
Riemannian manifold.

###  6.1. Reminder on the non-spinning case

A classical particle on a pseudo-Riemannian manifold $(M,\text{})$ admits the
cotangent bundle $T^{*}M$ as phase space. In the free case, its motion is
described as the Hamiltonian flow of $R=\text{}^{ij}p_{i}p_{j}$, which
projects onto the geodesic flow of $(M,\text{})$. The symmetries, or conserved
quantities, of such a system are the functions $K\in\mathrm{Pol}(T^{*}M)$
which Poisson commute with $R$. If the particle is in addition massless, it
moves along the null cone, characterized by $R=0$. Then, the symmetries are
functions $K$ which are required to Poisson commute with $R$ on the null cone,
i.e. such that $\\{R,K\\}\in(R)$ where $(R)$ denotes the ideal generated by
$R$ in $\mathrm{Pol}(T^{*}M)$. Moreover, the functions in $(R)$ are considered
as trivial symmetries since they vanish on the null cone. Through the
identification of symmetric tensors $\Gamma({\mathcal{S}}TM)$ and functions
$\mathrm{Pol}(T^{*}M)$, we get the following classical result, where $T$
denotes the operator $\text{}_{ij}\partial_{p_{i}}\partial_{p_{j}}$ and
$\Pi_{0}$ the projection on traceless tensors.

######  Proposition 6.1.

The symmetries of a free massless particle on $(M,\text{})$ are given by
conformal Killing symmetric tensors. The latter are symmetric tensors $K$ of
order $k\in\mathbb{N}$, characterized by one of the three equivalent
conditions

$\left\\{\begin{array}[]{l}\\{R,K\\}\in(R)\text{ and }TK=0,\\\
\Pi_{0}G^{\nabla}K=0,\\\ \Pi_{0}\nabla_{(i_{0}}K_{i_{1}\cdots
i_{k})}=0,\end{array}\right.$

where the round brackets denote symmetrization.

Roughly speaking, a quantum massless free particle is given by an harmonic
function of the conformal Laplacian
$\Delta=\nabla_{i}\text{}^{ij}\nabla_{j}+\frac{n-2}{4(n-1)}\mathrm{Scal}$,
where $\mathrm{Scal}$ denotes the scalar curvature. The latter is a
conformally invariant operator if considered as an element of the space
$\mathcal{D^{\lambda,\mu}}:=\mathcal{D}(M;|\Lambda|^{\lambda},|\Lambda|^{\mu})$
for $\lambda=\frac{n-2}{2n}$, $\mu=\frac{n+2}{2n}$. Following Eastwood, we
introduce higher symmetries of $\Delta$.

######  Definition 6.2.

Let $\lambda=\frac{n-2}{2n}$, $\mu=\frac{n+2}{2n}$ and let
$(\Delta)=\\{A\Delta|\,A\in\mathcal{D}^{\mu,\lambda}\\}$ be the left ideal
generated by the conformal Laplacian in $\mathcal{D}^{\lambda,\lambda}$. A
higher symmetry of $\Delta$ is a class of differential operators
$[D_{1}]\in\mathcal{D}^{\lambda,\lambda}/(\Delta)$, such that $\Delta
D_{1}=D_{2}\Delta$, for some $D_{2}\in\mathcal{D}^{\mu,\mu}$.

Such symmetries preserve the kernel of $\Delta$ and are quantum analogs of the
conformal Killing symmetric tensors. Indeed, denoting by ${\mathcal{K}}_{0}$
the space of conformal Killing symmetric tensors and by ${\mathcal{A}}_{0}$
the space of higher symmetries of $\Delta$, we have the following theorem.

######  Theorem 6.3.

[16, 32] Let $(M,\text{})$ be a conformally flat manifold. The conformally
equivariant quantization descends to an isomorphism of
${\mathfrak{g}}$-modules
${\mathcal{Q}}^{\lambda,\lambda}:{\mathcal{K}}_{0}\rightarrow{\mathcal{A}}_{0}$,
mapping conformal Killing symmetric tensors to higher symmetries of $\Delta$.

###  6.2. Conformal Killing hook tensors

We call hook tensors the tensors whose symmetry is described by a hook Young
diagram. They correspond to elements in the tensor algebra $\mathsf{T}^{0}$
and are referred as $(k,\kappa)$-tensors if they lie in
$\mathsf{T}^{0}_{k,\kappa}$. Thus, $(k,0)$-tensors are symmetric and
$(0,\kappa)$-tensors are skew-symmetric. We define in this section the notion
of conformal Killing hook tensors, so that it generalizes both conformal
Killing symmetric tensors and conformal Killing forms, defined in the usual
way below.

######  Definition 6.4.

A skew-symmetric tensor $\eta$ of order $\kappa$ is conformal Killing, if it
satisfies

$\nabla_{X}\eta=\frac{1}{\kappa+1}\left\langle
X,\mathbf{d}\eta\right\rangle+\frac{1}{n-\kappa+1}X^{\flat}\wedge\mathbf{d}^{*}\eta,$

for all $X\in\mathrm{Vect}(M)$. Moreover, $\mathbf{d}^{*}$ and $\mathbf{d}$
denote the de Rham (co-)differentials, and $X^{\flat}$ is the dual $1$-form of
$X$ through the metric .

We would like to characterize the space of conformal Killing forms as the
kernel of a conformally invariant operator. In addition to the invariant
operators introduced in table (A.52) and their curved analogs (5.41), we
resort to the conformally invariant projections $\Pi_{\bf
0}:\mathsf{T}^{\delta}\rightarrow\bigoplus_{k,\kappa}\mathsf{T}^{\delta}_{k,\kappa,0;00}$
and
$\Pi_{01}:\mathsf{T}^{\delta}\rightarrow\bigoplus_{k,\kappa}\mathsf{T}^{\delta}_{k,\kappa,0;01}$.
The images of both projections are respectively equal to the space
$\ker\boldsymbol{\delta}\cap\ker\boldsymbol{Q}^{*}\cap\ker T$ and its image by
$\boldsymbol{\delta}^{*}$.

######  Proposition 6.5.

The operator $\Pi_{\bf 0}\circ
G^{\nabla}:\mathsf{T}^{-\frac{1}{n}}\rightarrow\mathsf{T}^{\frac{1}{n}}$ is
conformally invariant and satisfies $\boldsymbol{\delta}^{*}\Pi_{\bf
0}G^{\nabla}=\Pi_{01}G^{\nabla}\boldsymbol{\delta}^{*}$. On the space $\ker
T\cap\ker\boldsymbol{Q}^{*}\cap\ker\boldsymbol{\delta}$, it is equal to

(6.44) $\Pi_{\bf 0}\circ
G^{\nabla}=G^{\nabla}-\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}^{*}\boldsymbol{d}^{\nabla}-\frac{1}{n+\mathcal{E}-\Sigma}\boldsymbol{Q}\boldsymbol{d}^{*\nabla}-\frac{1}{n+2\mathcal{E}-4}RD^{\nabla},$

and it vanishes else, i.e. on all the spaces
$\mathsf{T}^{-\frac{1}{n}}_{k,\kappa,s;\alpha\beta}$ with $s\neq 0$ or
$\alpha\beta\neq 00$. The kernel of
$\Pi_{01}G^{\nabla}\boldsymbol{\delta}^{*}$ restricted to
$\mathsf{T}^{-\frac{1}{n}}_{0,\kappa,0;00}$ is the space of conformal Killing
$\kappa$-forms.

###### Proof.

The operator $\Pi_{\bf 0}\circ G^{\nabla}$ is ${\mathfrak{g}}$-invariant in
the conformally flat case, as proved in theorem 4.2. Moreover, it is the
composition of the Levi-Civita covariant derivative with the projection on
some irreducible homogeneous subbundle, so that, by Fegan's work [17], it is
invariant by a conformal change of metric over a general manifold
$(M,\text{})$.

Since $G^{\nabla}$ commutes with $\boldsymbol{\delta}^{*}$, the equality
$\boldsymbol{\delta}^{*}\Pi_{\bf 0}=\Pi_{01}\boldsymbol{\delta}^{*}$ leads to
$\boldsymbol{\delta}^{*}\Pi_{\bf
0}G^{\nabla}=\Pi_{01}G^{\nabla}\boldsymbol{\delta}^{*}$. Moreover,
$G^{\nabla}$ commutes with $R$ and $\boldsymbol{Q}$, so that the vanishing of
$\Pi_{\bf 0}\circ G^{\nabla}$ on
$\mathsf{T}^{-\frac{1}{n}}_{k,\kappa,s;\alpha\beta}$ if $s\neq 0$ or
$\alpha\beta\neq 00$ is obvious. In the remaining case, the expression of
$\Pi_{\bf 0}\circ G^{\nabla}$ follows from the vanishing of its composition
with $T$, $\boldsymbol{Q}^{*}$ and $\boldsymbol{\delta}$. The coefficients are
computed thanks to the table (A.55) of commutation relations.

From proposition 3.3, we deduce the equality
$\ker\boldsymbol{\delta}^{*}\cap\ker\boldsymbol{\delta}=\\{0\\}$. Therefore
the kernel of $\Pi_{01}G^{\nabla}\boldsymbol{\delta}^{*}$ on
$\mathsf{T}^{-\frac{1}{n}}_{0,\kappa,0;00}$ is equal to the one of $\Pi_{\bf
0}\circ G^{\nabla}$. Let $\eta\in\mathsf{T}^{-\frac{1}{n}}_{0,\kappa,0;00}$,
and $X\in\mathrm{Vect}(M)$. We get

$\text{}_{ij}X^{i}\partial_{p_{j}}(\Pi_{\bf 0}\circ
G^{\nabla}\eta)=\nabla_{X}\eta-\frac{1}{\mathcal{E}+\Sigma}X^{i}\partial_{\xi^{i}}\boldsymbol{d}^{\nabla}\eta-\frac{1}{n+\mathcal{E}-\Sigma}\text{}_{ij}X^{i}\xi^{j}\boldsymbol{d}^{*\nabla}\eta.$

Since $\boldsymbol{d}^{*\nabla}$ and $\boldsymbol{d}^{\nabla}$ identify to de
Rham (co-)differentials on differential forms, $\eta$ has to be a conformal
Killing form to lie in the kernel of $\Pi_{\bf 0}\circ G^{\nabla}$. As $X$ is
arbitrary, the converse statement clearly holds. ∎

######  Definition 6.6.

Let $k\in\mathbb{N}^{*}$ and $\kappa\in\llbracket 0,n\rrbracket$. A conformal
Killing hook tensor of degree $(k,\kappa)$ is an element in $\ker
T\cap\ker\boldsymbol{Q}^{*}\cap\boldsymbol{\delta}^{*}=\bigoplus_{k,\kappa}\mathsf{T}^{0}_{k,\kappa,0;01}$,
which lies in the kernel of $\Pi_{01}\circ G^{\nabla}$.

From the proposition 6.1, we deduce that conformal Killing $(k,0)$-tensors are
usual conformal Killing symmetric tensors of order $k$. The proposition 6.5
shows that conformal Killing $(1,\kappa-1)$-tensors identify to conformal
Killing $\kappa$-forms through the operator $\boldsymbol{\delta}^{*}$.
Therefore, the latter definition extends both notions. As it is the kernel of
a first order operator, the space ${\mathcal{K}}:=\ker(\Pi_{01}\circ
G^{\nabla})$ of conformal Killing hook tensors is a subalgebra of
$\mathsf{T}^{0}$.

###  6.3. Pseudo-classical spinning particles

The phase space of a spinning particle on a pseudo-Riemannian manifold
$(M,\text{})$ is its supercotangent bundle $({\mathcal{M}},\omega)$, the spin
being represented by a quadratic function in the Grassmann variables,
$S=S_{ij}\xi^{i}\xi^{j}$. It automatically transforms in the right way under
the action of the orthogonal group. This Hamiltonian model is equivalent to
the well-known Lagrangian one developed in [4], it is the classical
counterpart of the quantum description of spinning particles as spinor fields
[31]. In the free case, the equations of motion of the particle are again
given by the Hamiltonian flow of $R$. Denoting by $x$ the parameterized
trajectory, we get in particular

$\displaystyle\nabla_{\dot{x}}\dot{x}^{i}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\text{}^{il}(R^{a}_{\leavevmode\nobreak\
bkl}\xi_{a}\xi^{b})\dot{x}^{k},$ $\displaystyle\nabla_{\dot{x}}\xi^{a}$
$\displaystyle=$ $\displaystyle 0,$

where $(R^{a}_{\leavevmode\nobreak\ bkl})$ denote the components of the
Riemann tensor. The second equation shows that the spin is parallely
transported, whereas the first one is an analog of Papapetrou's equation [34],
which generalizes the geodesic equation for an extended object with spin. The
deviation from geodesic motion is due to a coupling between the curvature and
the spin. Generally, the particle's spin is spacelike, that is expressed by
$p_{i}\xi^{i}=0$. See [35, 26] for further informations.

The classical counterpart of the Dirac operator is its principal Hamiltonian
symbol, given by $\boldsymbol{Q}=p_{i}\xi^{i}$, which Poisson squares to
$R=\\{\boldsymbol{Q},\boldsymbol{Q}\\}$. Thus, all symmetries of the
Hamiltonian flow of $\boldsymbol{Q}$ are also symmetries for the one of $R$.
Following [19], we call them supercharges.

######  Definition 6.7.

A supercharge is an element $K$ of ${\mathcal{O}}({\mathcal{M}})$ such that
$\\{\boldsymbol{Q},K\\}=0$. A conformal supercharge is an element $K$ of
${\mathcal{O}}({\mathcal{M}})$ such that
$\\{\boldsymbol{Q},K\\}\in(\boldsymbol{Q},R)$, where $(\boldsymbol{Q},R)$ is
the ideal in ${\mathcal{O}}({\mathcal{M}})$ generated by $\boldsymbol{Q}$ and
$R$.

Among the conformal supercharges stand all the elements in the ideal
$(\boldsymbol{Q},R)$, which are considered as trivial supercharges since they
vanish if $\boldsymbol{Q}=0=R$. Consequently, the relevant space of conformal
supercharges is the Poisson algebra

${\mathcal{S}}{\mathcal{C}}:=\\{K\in{\mathcal{O}}({\mathcal{M}})|\\{\boldsymbol{Q},K\\}\in(\boldsymbol{Q},R)\\}/(\boldsymbol{Q},R),$

which identifies to a symplectic reduction of the Poisson algebra
${\mathcal{O}}({\mathcal{M}})$.

We classify below all the conformal supercharges in a close spirit to
proposition 6.1. This generalize [19, 39], where supercharges are built from
Killing forms.

######  Theorem 6.8.

If $(M,\text{})$ is a conformally flat manifold, the conformally equivariant
superization induces an isomorphism of ${\mathfrak{g}}$-modules

(6.45)
$\mathfrak{S}^{0}:{\mathcal{K}}\longrightarrow{\mathcal{S}}{\mathcal{C}},$

mapping conformal Killing hook tensors to conformal supercharges. On general
manifolds, for $K\in{\mathcal{K}}$, we have the following equivalence

(6.46)
$\mathfrak{S}^{0}_{\nabla}(K)\in{\mathcal{S}}{\mathcal{C}}\Longleftrightarrow\Pi_{01}(\boldsymbol{d}^{\nabla})^{2}K=0.$

This condition reads explicitly as:
$\Pi_{01}\left(\xi^{k}\xi^{l}W^{a}_{\leavevmode\nobreak\
bkl}p_{a}\partial_{p_{b}}K\right)=0$, with $W$ the Weyl tensor.

###### Proof.

Let $K\in\mathsf{T}^{0}$. If $K$ factorizes through $R$ or $\boldsymbol{Q}$,
then so does $\mathfrak{S}^{0}_{\nabla}(K)$ according to (5.42) and we get a
trivial symmetry. Hence, according to the decomposition (4.25), we get that
$K\in\ker T\cap\ker\boldsymbol{Q}^{*}$. We compute

$\displaystyle\\{\boldsymbol{Q},\mathfrak{S}^{0}_{\nabla}(K)\\}$
$\displaystyle=$
$\displaystyle\big{(}\boldsymbol{d}^{\nabla}-\frac{\mathsf{i}}{\hbar}\boldsymbol{\delta}^{*}\big{)}\circ\big{(}\mathrm{Id}-\frac{\hbar}{\mathsf{i}}\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{d}^{\nabla}\boldsymbol{\delta}\big{)}(K)+H,$
$\displaystyle=$
$\displaystyle-\frac{\mathsf{i}}{\hbar}\boldsymbol{\delta}^{*}K+\left(\boldsymbol{d}^{\nabla}+\frac{1}{\mathcal{E}+\Sigma}\boldsymbol{\delta}^{*}\boldsymbol{d}^{\nabla}\boldsymbol{\delta}-\frac{\hbar}{\mathsf{i}}\frac{1}{\mathcal{E}+\Sigma}(\boldsymbol{d}^{\nabla})^{2}\boldsymbol{\delta}\right)(K)+H,$
$\displaystyle=$
$\displaystyle-\frac{\mathsf{i}}{\hbar}\boldsymbol{\delta}^{*}K+\frac{\boldsymbol{\delta}}{\mathcal{E}+\Sigma}\Big{(}G^{\nabla}-\boldsymbol{d}^{\nabla}\boldsymbol{\delta}^{*}-\frac{\hbar}{\mathsf{i}}(\boldsymbol{d}^{\nabla})^{2}\Big{)}(K)+H,$

with $H\in(R,\boldsymbol{Q})$. By the proposition 3.3, we have
$\mathrm{im}\boldsymbol{\delta}\cap\mathrm{im}\boldsymbol{\delta}^{*}=\\{0\\}$.
Therefore, $\mathfrak{S}^{0}_{\nabla}(K)$ is a conformal supercharge if and
only if $\boldsymbol{\delta}^{*}K\in(R,\boldsymbol{Q})$ and
$\boldsymbol{\delta}\Big{(}G^{\nabla}-\boldsymbol{d}^{\nabla}\boldsymbol{\delta}^{*}-\frac{\hbar}{\mathsf{i}}(\boldsymbol{d}^{\nabla})^{2}\Big{)}(K)\in(R,\boldsymbol{Q})$.
In view of the commutation relations given in table (A.55), the first equation
reads as $\boldsymbol{\delta}^{*}K=0$, which is automatic if
$K\in{\mathcal{K}}$. Assuming that $\boldsymbol{\delta}^{*}K=0$, the second
equation is equivalent to
$\Pi_{01}G^{\nabla}K=\frac{\hbar}{\mathsf{i}}\Pi_{01}(\boldsymbol{d}^{\nabla})^{2}K$.
In the conformally flat case the right hand side vanishes, and we get the