# Higher symmetries of the Laplacian via quantization

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.

We develop a new approach, based on quantization methods, to study higher
symmetries of invariant differential operators. We focus here on conformally
invariant powers of the Laplacian over a conformally flat manifold, and
recover results of Eastwood, Leistner, Gover and Šilhan.

In particular, conformally equivariant quantization establishes a clear
correspondence between hamiltonian symmetries of the null geodesic flow and
the algebra of higher symmetries of the conformal Laplacian. Resorting to
symplectic reduction, this leads to a quantization of the minimal nilpotent
coadjoint orbit of the conformal group and allows to identify the latter
algebra of symmetries in terms of the Joseph ideal. By the way, we obtain a
tangential star-product for a family of coadjoint orbits of the conformal
group.

###### Key words and phrases:

Symmetry algebra, Laplacian, Quantization, Conformal geometry, Minimal
nilpotent orbit, Symplectic reduction.

###### 2010 Mathematics Subject Classification:

58J10, 53A30, 70S10, 17B08, 53D20, 53D55

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

##  1\. Introduction

The higher symmetries of a differential operator $P$ are the differential
operators $D_{1}$ satisfying $PD_{1}=D_{2}P$ for some differential operator
$D_{2}$. They clearly form an algebra and preserve the kernel of $P$. The
first order ones define a Lie algebra ${\mathfrak{g}}$ generalizing the
invariance Lie algebra of $P$. The determination of the space of higher
symmetries of $P$, together with its algebra and ${\mathfrak{g}}$-module
structure, is of interest from at least two point of view: the integrability
of the equation $P\phi=0$, with $\phi$ in the source space of $P$, and the
representation theory of the ${\mathfrak{g}}$-module $\ker P$.

The first example which deserves to be investigated is certainly the conformal
(or Yamabe) Laplacian, that we consider on a conformally flat manifold
$(M,\mathrm{g})$ of arbitrary signature $(p,q)$ and dimension $n=p+q\geq 3$.
It writes as
$\Delta=\nabla_{i}\,\mathrm{g}^{ij}\nabla_{j}+\frac{n-2}{4(n-1)}R$, where
$\nabla$ is the Levi-Civita connection and $R$ the scalar curvature, and it
sends $\lambda$\- to $\mu$-densities, with $\lambda=\frac{n-2}{2n}$,
$\mu=\frac{n+2}{2n}$. On one hand, $\Delta\phi=0$ is the most basic wave
equation, describing e.g. a free massless quantum particle on
$(M,\mathrm{g})$. Its integration has been achieved in various contexts
resorting to symmetries, and we highlight here only two seminal works. On
$\mathbb{R}^{3}$, Boyer, Kalnins and Miller have classified all the second
order symmetries of the Laplacian [10], which allows them to get all the
possible coordinates systems separating the equation $\Delta\phi=0$. For
curved but Ricci-flat manifolds, Carter has built up a way to get symmetries,
by quantization of Killing tensors [13], and has successfully used it to
integrate on Kerr space-time the wave equation $(\Delta+m^{2})\phi=0$, with a
mass term $m\in\mathbb{R}$. On the other hand, the first order symmetries of
the conformal Laplacian are given by

$\Delta(X+\lambda\mathrm{Div}X)=(X+\mu\mathrm{Div}X)\Delta,$

where $\mathrm{Div}$ is the divergence operator and
$X\in{\mathfrak{g}}\simeq\mathrm{o}(p+1,q+1)$ is a conformal Killing vector.
This provides a representation of ${\mathfrak{g}}$ on $\ker\Delta$ which
integrates, if $M=\mathbb{S}^{p}\times\mathbb{S}^{q}$ and $p+q$ even, to a
unitary irreducible representation of the Lie group $G=\mathrm{O}(p+1,q+1)$
[5]. This is the intensively studied minimal representation of $G$, see e.g.
[27, 28]. It is linked to the minimal nilpotent coadjoint orbit
$\mathcal{O}_{00}$ of $G$ through the Joseph ideal [25], which appears as the
kernel of the representation of $\mathfrak{U}({\mathfrak{g}})$, the universal
enveloping algebra of ${\mathfrak{g}}$, on $\ker\Delta$. Nevertheless, this
representation cannot be obtained via the Kirillov’s orbit method [26] since
$\mathcal{O}_{00}$ admits no invariant polarization [43]. In the complex
setting, the alternative deformation program has been applied successfully by
Arnal, Benamor and Cahen, who have proved existence and uniqueness of a graded
${\mathfrak{g}}$-equivariant star product on regular functions on
$\mathcal{O}_{00}^{\mathbb{C}}$ [1]. From this star product, Astashkevich and
Brylinski have built a unitary irreducible representation of $G^{\mathbb{C}}$
[2], but the way to recover the real representation of $G$ on $\ker\Delta$ is
unclear.

Resorting to conformal ambient space, Eastwood has been able to determine the
space of higher symmetries of the conformal Laplacian [19]. The latter
corresponds as a ${\mathfrak{g}}$-module to the space of conformal Killing
tensors, which are the Hamiltonian symmetries of the null geodesic flow.
Besides, as an algebra, it identifies to a quotient
$\mathfrak{U}({\mathfrak{g}})/J$ of the universal enveloping algebra.
Surprisingly, this ideal $J$ coincides with the previously mentioned Joseph
ideal [22]. The correspondence obtained by Eastwood might then be interpreted
as a quantization of the minimal coadjoint orbit of $G$.

Our aim is precisely to prove that the results of Eastwood stem from the now
well-developed theory of equivariant quantization of cotangent bundles [17, 9,
33, 12], and that after a basic symplectic reduction, this quantization
provides a supplement to the orbit method in the case of the minimal nilpotent
coadjoint orbit of $G$. In particular, it induces on that orbit the star
product of Arnal, Benamor and Cahen. The present approach can be generalized
to number of cases, indeed, equivariant quantization is available for any
$|1|$-graded parabolic geometry and for differential operators acting on any
irreducible natural bundles [12]. To illustrate its efficiency, we deal with
the determination of higher symmetries of the conformal powers of the
Laplacian, denoted $\Delta^{\ell}$. Thus, we recover recent results of Gover
and Šilhan [40] obtained via tractor calculus and highly non-trivial
computations. Let us now detail the content of this article.

In Section $2$, we introduce our main tools, namely the classification of
conformally invariant operators on symbols [21, 4, 35], the conformally
equivariant quantization [17] and the induced star product on symbols [16].

In Section $3$ lies our first main result. We characterize the space
$\mathcal{A}^{\lambda,\ell}$ of higher symmetries of $\Delta^{\ell}$ and the
space $\mathcal{K}^{\ell}$ of $\ell$-generalized conformal Killing tensors
[36] as kernels of conformally invariant operators. Then, we prove that
conformally equivariant quantization maps one operator to the other, and then
one space to the other. This correspondence can be made explicit, since
formulae have been derived for this quantization [18, 32], even in the curved
case [37, 39].

In section $4$, we first introduce algebras $\mathcal{K}$ and
$\mathcal{A}^{\lambda}$ generated by ${\mathfrak{g}}$, the Lie algebra of
conformal vector fields. The spaces of symmetries $\mathcal{K}^{\ell}$ and
$\mathcal{A}^{\lambda,\ell}$ are obtained as quotients of the latter. Then, we
describe all the coadjoint orbits of $G$ in the image of the moment map
$\mu:T^{*}\mathbb{R}^{p+1,q+1}\rightarrow{\mathfrak{g}}^{*}$ as symplectic
reductions of the source manifold. Their algebras of regular functions are
determined, and two of them identify to $\mathcal{K}$ and $\mathcal{K}^{1}$.
As a consequence, we get an explicit description of the symmetry algebras
$\mathcal{A}^{\lambda,\ell}$. In particular, the algebra $\mathcal{K}^{1}$ is
associated to the minimal nilpotent coadjoint orbit $\mathcal{O}_{00}$ and the
corresponding symmetry algebra $\mathcal{A}^{\lambda,1}$ of the Laplacian is
isomorphic to the universal enveloping algebra of ${\mathfrak{g}}$ factored by
the Joseph ideal. Finally, we build a star product on each coadjoint orbit in
the image of $\mu$. In particular, the conformally equivariant quantization
induces the star-product of Arnal, Benamor and Cahen [1] on $\mathcal{O}_{00}$
and furnishes a representation of it on $\ker\Delta$, which integrates into
the minimal representation of $G$ if $M=\mathbb{S}^{p}\times\mathbb{S}^{q}$
and $p+q\geq 4$ is even.

##  2\. Conformal geometry of differential operators and of their symbols

We introduce in this section the basic notions that we need and the two key
facts leading to our main theorem in the next section, namely: the existence
and uniqueness of the conformally equivariant quantization [17] and the
classification of the conformally invariant operators on the space of symbols
[35].

###  2.1. Actions of vector fields

Let $M$ be a smooth manifold. We start with the definitions of the algebra
$\mathcal{D}(M)$ of differential operators on $M$ and of its algebra of
symbols ${\mathcal{S}}(M)$. The first one is the union of the spaces
$\mathcal{D}_{k}(M)$ of differential operators of order $k$, defined as the
spaces of operators $A$ on ${\mathcal{C}}^{\infty}(M)$ satisfying
$[\cdots[A,f_{0}],\cdots],f_{k}]=0$ for all functions
$f_{0},\ldots,f_{k}\in{\mathcal{C}}^{\infty}(M)$. The second one,
${\mathcal{S}}(M)$, is its canonically associated graded algebra, defined by
${\mathcal{S}}(M)=\bigoplus_{k=0}^{\infty}\mathcal{D}_{k}(M)/\mathcal{D}_{k-1}(M)$.
It identifies to the algebra of functions on $T^{*}M$ which are polynomial in
the fibers, the grading corresponding to the polynomial degree. The canonical
projection
$\sigma_{k}:\mathcal{D}_{k}(M)\rightarrow\mathcal{D}_{k}(M)/\mathcal{D}_{k-1}(M)$
is called the principal symbol map.

We are interested in the action of the Lie algebra $\mathrm{Vect}(M)$ of
vector fields on these both algebras. The diffeomorphisms of $M$ lift
canonically into automorphisms of ${\mathrm{GL}}(M)$, the principal bundle of
linear frames over $M$. Consequently, they act canonically on sections of
every associated bundles to ${\mathrm{GL}}(M)$. The corresponding
infinitesimal actions of $\mathrm{Vect}(M)$ are given by Lie derivatives. In
particular, we get a $\mathrm{Vect}(M)$-module structure on the spaces of
symbols ${\mathcal{S}}(M)$ and of $\lambda$-densities
${\mathcal{F}}^{\lambda}:=\Gamma(|\Lambda^{n}T^{*}M|^{\otimes\lambda})$, with
$\lambda\in\mathbb{R}$. Via a global section $|\mathrm{vol}|$, the latter
identifies to the module $({\mathcal{C}}^{\infty}(M),\ell^{\lambda})$, endowed
with the $\mathrm{Vect}(M)$-action
$\ell_{X}^{\lambda}=X+\lambda\mathrm{Div}(X)$, the operator $\mathrm{Div}$
being the divergence with respect to $|\mathrm{vol}|$. Therefore, the
$\mathrm{Vect}(M)$-module $\mathcal{D^{\lambda,\mu}}$ of differential
operators from $\lambda$\- to $\mu$-densities identifies to
$(\mathcal{D}(M),\mathcal{L}^{\lambda,\mu})$, with
$\mathcal{L}_{X}^{\lambda,\mu}A=\ell_{X}^{\mu}A-A\ell_{X}^{\lambda},$ for all
$X\in\mathrm{Vect}(M)$ and $A\in\mathcal{D}(M)$. This adjoint action preserves
the filtration of $\mathcal{D}(M)$, hence the algebra of symbols inherits of a
$\mathrm{Vect}(M)$-action compatible with the grading which coincides with the
one by Lie derivative on
${\mathcal{S}}(M)\otimes_{{\mathcal{C}^{\infty}}(M)}{\mathcal{F}}^{\delta}$,
for $\delta=\mu-\lambda$. We denote by ${\mathcal{S}}^{\delta}$ the obtained
module of symbols and by ${\mathcal{S}}^{\delta}_{k}$ the submodule of
homogeneous symbols of degree $k$.

###  2.2. Conformal Lie algebra

A conformal structure on a smooth manifold $M$ is given by the equivalence
class $[\mathrm{g}]$ of a pseudo-Riemannian metric $\mathrm{g}$, where two
metrics $h$ and $\mathrm{g}$ are considered equivalent if $h=F\mathrm{g}$ for
some positive function $F\in{\mathcal{C}^{\infty}}(M)$. The signature $(p,q)$
of the metric $\mathrm{g}$ is an invariant of the conformal structure, and to
each signature corresponds a canonical flat model $(\mathbb{R}^{p,q},[\eta])$,
with $\eta=\mathbb{I}_{p}\otimes-\mathbb{I}_{q}$. If there exists an atlas
$(U_{i},\phi_{i})$ on $(M,[\mathrm{g}])$ such that the pull-back by every
chart $\phi_{i}$ of the canonical flat conformal structure coincides with the
restriction of $[\mathrm{g}]$ to $U_{i}$, the conformal manifold
$(M,[\mathrm{g}])$ is said to be conformally flat. The vector fields that
preserve a conformal class $[\mathrm{g}]$ are called conformal Killing vector
fields, they satisfy $L_{X}\mathrm{g}=F\mathrm{g}$, with $L_{X}\mathrm{g}$ the
Lie derivative of $\mathrm{g}$ along $X$ and $F$ a function. They form a sheaf
of Lie algebras, and if $(M,[\mathrm{g}])$ is conformally flat of dimension
$p+q\geq 3$, it is locally isomorphic to ${\mathfrak{g}}=\mathrm{o}(p+1,q+1)$
the conformal Lie algebra of $(\mathbb{R}^{p,q},[\eta])$.

An important example of conformally flat manifold is
$\mathbb{S}^{p}\times\mathbb{S}^{q}$, viewed as a homogeneous space of
$G=\mathrm{O}(p+1,q+1)$. Starting from the isometric action of $G$ on the
pseudo-Euclidean space $\mathbb{R}^{p+1,q+1}$, we get an action of $G$ on the
space of isotropic half-lines, which identifies naturally to the manifold
$\mathbb{S}^{p}\times\mathbb{S}^{q}$. Via this construction, the flat metric
on $\mathbb{R}^{p+1,q+1}$ induces a conformally flat structure on
$\mathbb{S}^{p}\times\mathbb{S}^{q}$, preserved by the $G$-action.

###  2.3. Conformal invariants

Over a conformally flat manifold $(M,[\mathrm{g}])$, a conformal invariant in
a $\mathrm{Vect}(M)$-module is an invariant element under the action of
conformal Killing vector fields. Using local conformal coordinates $(x^{i})$,
which are such that $\mathrm{g}_{ij}=F\eta_{ij}$ for a positive function $F$,
the classification of conformal invariants over $(M,[\mathrm{g}])$ amounts to
the one of ${\mathfrak{g}}$-invariants over the flat space. For modules of
tensors, this can be performed thanks to the Weyl’s theory of invariants [42]
and for modules of differential operators this can be done resorting to the
results on morphisms of Verma modules obtained in [6, 7]. However, all results
presented here can be derived from the more elementary Weyl’s theory of
invariants.

First, we provide the well-known classification of the conformal invariants of
the $\mathrm{Vect}(M)$-modules of differential operators
$\mathcal{D^{\lambda,\mu}}$ and of symbols ${\mathcal{S}}^{\delta}$. We write
them in terms of the local conformal coordinates $(x^{i},p_{i})$ on $T^{*}M$,
and of their corresponding derivatives $(\partial_{i},\partial_{p_{i}})$.

######  Proposition 2.1.

On a conformally flat manifold $(M,[\mathrm{g}])$, the conformal invariants of
$({\mathcal{S}}^{\delta})_{\delta\in\mathbb{R}}$ and
$(\mathcal{D^{\lambda,\mu}})_{\lambda,\mu\in\mathbb{R}}$ are given locally, up
to a multiplicative constant, by

  * •

$\mathrm{R}^{\ell}\in{\mathcal{S}}^{\frac{2\ell}{n}}$ for $\ell\in\mathbb{N}$,

  * •

$\Delta^{\ell}\in\mathcal{D^{\lambda,\mu}}$ for $\ell\in\mathbb{N}$ and
$\lambda=\frac{n-2\ell}{2n}$, $\mu=\frac{n+2\ell}{2n}$,

where $\mathrm{R}=\eta^{ij}p_{i}p_{j}$, and
$\Delta=\eta^{ij}\partial_{i}\partial_{j}$ is the conformal Laplacian.

We refer to [24] and references inside for global expressions of the conformal
powers of the Laplacian. Since the principal symbol map is
$\mathrm{Vect}(\mathbb{R}^{n})$-equivariant, conformally invariant
differential operators give rise to conformally invariant symbols, but the
fact that they are in correspondence is remarkable.

Second, we present the classification of the conformally invariant
differential operators on the space of symbols, as it appears in [35]. It
relies on the harmonic decomposition of the ${\mathfrak{g}}$-module of
symbols. Namely, we have
${\mathcal{S}}^{\delta}=\bigoplus_{k,s\in\mathbb{N},\,2s\leq
k}{\mathcal{S}}^{\delta}_{k,s}$, where ${\mathcal{S}}^{\delta}_{k,s}$ is the
submodule of homogeneous symbols of degree $k$ of the form $P=R^{s}Q$ with $Q$
a traceless symbol, i.e. $TQ=0$ for
$T=\eta_{ij}\partial_{p_{i}}\partial_{p_{j}}$.

######  Theorem 2.2.

[35] Let $k\geq 2s$ and $k^{\prime}\geq 2s^{\prime}$ be integers, and
$\delta,\delta^{\prime}\in\mathbb{R}$. The space of conformal invariant
differential operators:
${\mathcal{S}}^{\delta}_{k,s}\rightarrow{\mathcal{S}}^{\delta^{\prime}}_{k^{\prime},s^{\prime}}$,
is either trivial or of dimension $1$. In the latter case
$j=\frac{n}{2}(\delta^{\prime}-\delta)$ is an integer and the space is
generated by

  * •

$R^{s^{\prime}}D^{d}T^{s}$, if $s^{\prime}-s=j$, $k-k^{\prime}=d-2j$ and
$\delta=1+\frac{2(k-s)-d-1}{n}$,

  * •

$R^{s^{\prime}}G_{0}^{g}T^{s}$, if $g+s^{\prime}-s=j$,
$k-k^{\prime}=s-s^{\prime}-j$ and $\delta=\frac{2s+1-g}{n}$,

  * •

$R^{s^{\prime}}\mathcal{L}_{\ell}T^{s}$, if $\ell+s^{\prime}-s=j$,
$k-k^{\prime}=2(\ell-j)$ and $\delta=\frac{1}{2}+\frac{k-\ell}{n}$,

where, locally, $D=\partial_{i}\partial_{p_{i}}$ is the divergence operator,
$G_{0}$ is the composition of the gradient operator
$G=\eta^{ij}p_{i}\partial_{j}$ with the projection on traceless symbols and
$\mathcal{L}_{\ell}=\Delta^{\ell}+a_{1}GD\Delta^{\ell-1}+\cdots+a_{\ell}G^{\ell}D^{\ell}$
for given real coefficients $a_{1},\ldots,a_{\ell}$.

Global expression for divergence and gradient operators can be find in [15],
and we refer to [45] for $\mathcal{L}_{1}$.

###  2.4. Conformally equivariant quantization

Let $\lambda,\mu\in\mathbb{R}$ and $\delta=\mu-\lambda$. We call quantization
the linear isomorphisms
${\mathcal{Q}}:{\mathcal{S}}^{\delta}\rightarrow\mathcal{D^{\lambda,\mu}}$
which are right inverses of the principal symbol map on homogeneous symbols.
If the both $\mathrm{Vect}(M)$-modules of symbols and of differential
operators are indeed isomorphic as vector spaces by the very definition of
symbols, they are not as $\mathrm{Vect}(M)$-modules [29, 30]. A natural
question is whether they are isomorphic as modules over given Lie subalgebras
of $\mathrm{Vect}(M)$, and whether this extra structure allows to single out
one quantization. This has been studied in various contexts under the name
equivariant quantization, and one of the fundamental result is the following.

######  Theorem 2.3.

[17] On a conformally flat manifold, there exists a unique conformally
equivariant quantization for generic values of $\delta=\mu-\lambda$, i.e. a
unique ${\mathfrak{g}}$-module morphism
$\mathcal{Q}^{\lambda,\mu}:{\mathcal{S}}^{\delta}\rightarrow\mathcal{D^{\lambda,\mu}}$
which is a right inverse of the principal symbol map on homogeneous symbols.

The exceptional values of $\delta$ leading to a non-unique or a non-existing
conformally equivariant quantization have been classified in [39, 35]. They
can be obtained via Theorem 2.2 and the following result.

######  Proposition 2.4.

[35] The conformally equivariant quantization exists and is unique on
${\mathcal{S}}^{\delta}_{k,s}$ if and only if there is no conformally
invariant differential operators from ${\mathcal{S}}^{\delta}_{k,s}$ to
${\mathcal{S}}^{\delta}$.

We will need a slightly stronger result than theorem 2.3, straightforwardly
deduced from [17].

######  Proposition 2.5.

Let $F$ be a ${\mathfrak{g}}$-submodule of ${\mathcal{S}}^{\delta}$. For a
generic shift $\delta$, e.g. $\delta=0$, there exists a unique morphism of
${\mathfrak{g}}$-modules from $F$ to $\mathcal{D^{\lambda,\mu}}$, which is a
right inverse of the principal symbol map on homogeneous symbols.

Explicit formulae are available for the conformally equivariant quantization.
As an example, we recall the one obtained by Radoux [37] on the space
${\mathcal{S}}^{\delta}_{*,0}=\bigoplus_{k\in\mathbb{N}}{\mathcal{S}}^{\delta}_{k,0}$
of traceless symbols. It relies on the divergence operator
$D=\partial_{i}\partial_{p_{i}}$ and the normal ordering
${\mathcal{N}}:P^{i_{1}\cdots i_{k}}(x)p_{i_{1}}\cdots p_{i_{k}}\mapsto
P^{i_{1}\cdots i_{k}}(x)\partial_{i_{1}}\cdots\partial_{i_{k}}$.

######  Proposition 2.6.

[37] Let $\delta\notin\\{1+\frac{2k-1-m}{n}|\,m=1,\ldots,k\\}$. On the space
${\mathcal{S}}^{\delta}_{k,0}$ of traceless symbols of degree $k$, the
conformally equivariant quantization is given by

(2.1)
$\mathcal{Q}^{\lambda,\mu}={\mathcal{N}}\circ\left(\sum_{m=0}^{k}c^{k}_{m}D^{m}\right),$

with $c^{k}_{0}=1$ and
$c^{k}_{m}=\frac{k-m+n\lambda}{m(2k-m-1+n(1-\delta))}\,c^{k}_{m-1},$ for
$m=1,\ldots,k$.

In the general case, including symbols with non-vanishing trace, fully
explicit formulae are known only for symbols up to the order $3$ in momenta
variables $p$, they are available in conformal coordinates [31] as well as in
covariant terms [18, 32]. Finally, let us mention that Šilhan has obtained the
expression of the conformally equivariant quantization in the curved case on
all symbols, resorting to tractor calculus [39].

###  2.5. Conformally equivariant graded star product

Let us start with standard definitions. The algebra of symbols
${\mathcal{S}}^{0}$ is commutative and graded, moreover, as a subalgebra of
${\mathcal{C}^{\infty}}(T^{*}M)$, it carries a Poisson bracket denoted by
$\\{\cdot,\cdot\\}$. A graded (or homogeneous) star product on
${\mathcal{S}}^{0}$ is an associative $\mathbb{C}[[\hbar]]$-linear product
$\star$ on ${\mathcal{S}}^{0}\otimes\mathbb{C}[[\hbar]]$, with $\hbar$ a
formal parameter. For $P,Q\in{\mathcal{S}}^{0}$, it is of the form $P\star
Q=\sum_{m\in\mathbb{N}}\left(\mathsf{i}\hbar\right)^{m}B_{m}(P,Q)$ and
satisfies:

  1. (1)

$B_{0}(P,Q)=PQ$,

  2. (2)

$B_{1}(P,Q)-B_{1}(Q,P)=\\{P,Q\\}$,

  3. (3)

for all integers $k,l,m$,
$B_{m}:{\mathcal{S}}^{0}_{k}\otimes{\mathcal{S}}^{0}_{l}\rightarrow{\mathcal{S}}^{0}_{k+l-m}$
is a bidifferential operator.

A frequently required extra property is the symmetry (or parity) of the star
product, namely $B_{m}(P,Q)=(-1)^{m}B_{m}(Q,P)$ for all integers $m$, or
equivalently $\overline{P\star Q}=\overline{Q}\star\overline{P}$, where
$\overline{\cdot}$ is the complex conjugation.

One source of star products on ${\mathcal{S}}^{0}$ is quantizations. Let us
introduce two maps, the linear map
$\Im:{\mathcal{S}}^{0}\rightarrow{\mathcal{S}}^{0}\otimes\mathbb{C}[[\hbar]]$
defined by $(\mathsf{i}\hbar)^{k}\mathrm{Id}$ on ${\mathcal{S}}^{0}_{k}$ and
the $\mathbb{C}[[\hbar]]$-linear extension of some quantization
${\mathcal{Q}}\otimes\mathrm{Id}:{\mathcal{S}}^{0}\otimes\mathbb{C}[[\hbar]]\rightarrow\mathcal{D}^{\lambda,\lambda}\otimes\mathbb{C}[[\hbar]]$.
Clearly, the composition
${\mathcal{Q}}_{\hbar}=({\mathcal{Q}}\otimes\mathrm{Id})\circ\Im$ gives rise
to a graded star product on ${\mathcal{S}}^{0}$ as the pull back by
${\mathcal{Q}}_{\hbar}$ of the composition of differential operators,

(2.2) $P\star
Q={\mathcal{Q}}_{\hbar}^{-1}({\mathcal{Q}}_{\hbar}(P)\circ{\mathcal{Q}}_{\hbar}(Q)).$

Moreover, for $\lambda=\frac{1}{2}$, this star product is symmetric if and
only if the quantization satisfies
${\mathcal{Q}}_{\hbar}(\overline{P})={\mathcal{Q}}_{\hbar}(P)^{*}$ for all
$P\in{\mathcal{S}}^{0}$. Here, the superscript ∗ denotes the adjoint operation
with respect to the Hermitian product on complex compactly supported half-
densities, given by $(\phi,\psi)=\int_{M}\overline{\phi}\psi$.

The action of $X\in\mathrm{Vect}(M)$ on
${\mathcal{S}}^{0}\subset{\mathcal{C}^{\infty}}(T^{*}M)$ is given by the
Hamiltonian vector field $\\{\mu_{X},\cdot\\}$ where $\mu_{X}=X^{i}p_{i}$.
Thanks to the star product on ${\mathcal{S}}^{0}$, we can define a new action
of $\mathrm{Vect}(M)$ on ${\mathcal{S}}^{0}$ via the star bracket, i.e.
$X\in\mathrm{Vect}(M)$ acts on $P\in{\mathcal{S}}^{0}$ by
$[\mu_{X},P]_{\star}=\mu_{X}\star P-P\star\mu_{X}$. The star product is said
conformally equivariant (or strongly ${\mathfrak{g}}$-invariant) if both
induced ${\mathfrak{g}}$-actions coincide, namely
$[\mu_{X},P]_{\star}=\mathsf{i}\hbar\\{\mu_{X},P\\}$ for all
$X\in{\mathfrak{g}}$. As one can expect, conformally equivariant quantizations
gives rise to ${\mathfrak{g}}$-equivariant star products.

######  Proposition 2.7.

[17, 16] Let $(M,[\mathrm{g}])$ be a conformally flat manifold. The star
product $\star_{\lambda}$ induced by the conformally equivariant quantization
${\mathcal{Q}}^{\lambda,\lambda}$ via equation (2.2) is a graded
${\mathfrak{g}}$-equivariant star product on ${\mathcal{S}}^{0}$. It is
symmetric if and only if $\lambda=\frac{1}{2}$.

It is easy to prove that all graded ${\mathfrak{g}}$-equivariant star products
on ${\mathcal{S}}^{0}$ arise in that way.

##  3\. Classification of the higher symmetries of the conformal powers of the
Laplacian

The aim of this section is to show how conformally equivariant quantization
sheds new light on the determination of higher symmetries of conformal
Laplacian, initiated by Eastwood [19] and pursued in [20] and [40] for
conformal powers of the Laplacian, in the conformally flat case. In all this
section we work over a conformally flat manifold $(M,[\mathrm{g}])$ of
dimension $n\geq 3$ and $\Delta^{\ell}$ denotes the conformal
$\ell^{\text{th}}$ power of the Laplacian, pertaining to
$\mathcal{D^{\lambda,\mu}}$ for values of the weights henceforth fixed to
$\lambda=\frac{n-2\ell}{2n}$, $\mu=\frac{n+2\ell}{2n}$.

###  3.1. Definition of higher symmetries of $\Delta^{\ell}$

There is a number of different notions of symmetries for a differential
operator $P$ on $M$. The most basic one is given by vector fields
$X\in\mathrm{Vect}(M)$ preserving the considered operator: $[P,X]=0$. We can
broaden this notion by considering rather differential operators
$D\in\mathcal{D}(M)$ commuting to $P$. Such symmetries preserve obviously the
eigenspaces of $P$. Here we are interested in the more general notion of
higher symmetries, which act on the kernel of $P$. We give their definitions
for $P=\Delta^{\ell}$.

######  Definition 3.1.

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

This definition does not depend on the chosen representative since
$\Delta^{\ell}(A\Delta^{\ell})=(\Delta^{\ell}A)\Delta^{\ell}$ for any
differential operator $A$. Resorting to conformal coordinates, higher
symmetries prove to be locally the same on flat and conformally flat
manifolds, but global existence can nevertheless be problematic in this more
general setting. We do not address this issue and work only locally. The
higher symmetries of $\Delta^{\ell}$ form then an algebra
$\mathcal{A}^{\lambda,\ell}$, which is characterized as the kernel of the
conformally invariant map

$\displaystyle\mathrm{QHS}:\mathcal{D}^{\lambda,\lambda}/(\Delta^{\ell})$
$\displaystyle\rightarrow$
$\displaystyle\mathcal{D}^{\lambda,\mu}/(\Delta^{\ell})$ (3.3)
$\displaystyle\left[D\right]$ $\displaystyle\mapsto$
$\displaystyle[\Delta^{\ell}D]$

where $\mathrm{QHS}$ stands for Quantum Higher Symmetries. Let us mention that
the left ideal generated by $\Delta^{\ell}$ can be defined in the modules
$\mathcal{D}^{\lambda,\lambda^{\prime}}$ for any $\lambda^{\prime}$, and in
particular for $\lambda^{\prime}=\mu$, as the subspace
$(\Delta^{\ell})=\\{A\Delta^{\ell}|\,A\in\mathcal{D}^{\mu,\lambda^{\prime}}\\}$.

######  Example 3.2.

The higher symmetries of $\Delta^{\ell}$ given by first order differential
operators are the constants, acting by multiplication as zero order
differential operators, and the Lie derivatives $\ell_{X}^{\lambda}$ for
$X\in{\mathfrak{g}}$. In accordance with Proposition 2.1, we have indeed
$\Delta^{\ell}\ell_{X}^{\lambda}=\ell_{X}^{\mu}\Delta^{\ell}$.

###  3.2. Symmetries of the null geodesic flow and generalizations

Using the properties of the principal symbol map $\sigma$, the equality
$[D,\Delta]=A\Delta$ defining higher symmetries of the Laplacian implies
$\\{\sigma(D),R\\}=\sigma(A)R$, where $\\{\cdot,\cdot\\}$ denotes the
canonical Poisson bracket on $T^{*}M$. Consequently, the principal symbol of
higher symmetries of the Laplacian is a constant of motion for the null
geodesic flow. The latter are known to be given by conformal Killing tensors,
which are symmetric tensors and identify to symbols of weight $0$. In the
following, round bracket denotes symmetrization of indices, $(\partial_{i})$
the partial derivatives associated to local conformal coordinates $(x^{i})$
and $L$ is an arbitrary tensor.

######  Definition 3.3.

A conformal Killing $k$-tensor $K$ is defined equivalently as

  * •

A symmetric traceless tensor of order $k$ s.t.
$\partial_{(i_{0}}K_{i_{1}\cdots
i_{k})}=\mathrm{g}_{(i_{0}i_{1}}L_{i_{2}\cdots i_{k})}$,

  * •

A traceless symbol of degree $k$ satisfying $\\{R,K\\}=fR$ for some function
$f$,

  * •

A traceless symbol of degree $k$ in the kernel of $G_{0}$.

As for higher symmetries, we are not concerned by global existence questions
and work locally. For $k=1$, we recover the notion of conformal Killing
vectors which identifies to the Lie algebra ${\mathfrak{g}}$. The conformal
Killing tensors of higher orders correspond to transformations of the phase
space $T^{*}M$ not preserving the configuration manifold $M$. Resorting to
Theorem 2.2, the operator
$G_{0}:{\mathcal{S}}^{0}_{k,0}\rightarrow{\mathcal{S}}^{\frac{2}{n}}_{k+1,0}$
is conformally invariant for every $k$, and this is the only one on
${\mathcal{S}}^{0}_{k,0}$. Hence, the space of conformal Killing tensors of a
given order is an indecomposable ${\mathfrak{g}}$-module. Since it is finite
dimensional, the semi-simplicity of ${\mathfrak{g}}$ implies that it is an
irreducible ${\mathfrak{g}}$-module. We can generalize this picture to tensors
(or symbols) with trace, using the conformal invariance of $G_{0}^{2s+1}T^{s}$
on ${\mathcal{S}}^{0}_{k,s}$. The following definition is due to Nikitin and
Prilipko [36].

######  Definition 3.4.

A $s$-generalized conformal Killing $k$-tensor $K$ is defined equivalently as

  * •

A symmetric traceless tensor of order $(k-2s)$ s.t.
$\partial_{(i_{0}}\cdots\partial_{i_{2s}}K_{i_{2s+1}\cdots
i_{k})}=\mathrm{g}_{(i_{0}i_{1}}L_{i_{2}\cdots i_{k})}$,

  * •

A symbol $R^{s}K\in{\mathcal{S}}^{0}_{k,s}$ which is in the kernel of
$G_{0}^{2s+1}T^{s}$.

The equivalence of the two assertions relies on the equality
$T^{s}(R^{s}K)=cK$, where $c$ is a constant. Again, the space of
$s$-generalized conformal Killing tensors of order $k$ is an irreducible
${\mathfrak{g}}$-module. We denote this subspace of ${\mathcal{S}}^{0}_{k,s}$
by $\mathcal{K}_{k,s}$ and we introduce the spaces of classical higher
symmetries $\mathcal{K}^{\ell}=\bigoplus_{s=0}^{\ell-1}\mathcal{K}_{*,s}$,
with $\mathcal{K}_{*,s}=\bigoplus_{k\geq 2s}\mathcal{K}_{k,s}$, as well as
their limit $\mathcal{K}=\bigoplus_{s\in\mathbb{N}}\mathcal{K}_{*,s}$.

###  3.3. From classical to quantum symmetries

Gover and Šilhan have proved that the two types of symmetries defined in the
two previous paragraphs are in bijection [40]. We propose a completely
different and far less technical proof, showing that the latter correspondence
is given by equivariant quantization. We assume that $\ell\in\mathbb{N}^{*}$,
$\lambda=\frac{n-2\ell}{2n}$ and $\mu=\frac{n+2\ell}{2n}$.

######  Theorem 3.5.

The conformally equivariant quantization descends to an isomorphism of
${\mathfrak{g}}$-modules
${\mathcal{Q}}^{\lambda,\lambda}:\mathcal{K}^{\ell}\rightarrow\mathcal{A}^{\lambda,\ell}$,
identifying higher symmetries of $\Delta^{\ell}$ with $s$-generalized
conformal Killing tensors for $s<\ell$. Moreover, every local section $P$ of
$\mathcal{K}$ satisfies
$\Delta^{\ell}{\mathcal{Q}}^{\lambda,\lambda}(P)={\mathcal{Q}}^{\mu,\mu}(P)\Delta^{\ell}$.

###### Proof.

The conformal invariance of $\Delta^{\ell}\in\mathcal{D^{\lambda,\mu}}$ and
the uniqueness of the conformally equivariant quantization on the
${\mathfrak{g}}$-module $(R^{\ell})$ lead to the following factorization:
${\mathcal{Q}}^{\lambda,\lambda^{\prime}}(PR^{\ell})={\mathcal{Q}}^{\mu,\lambda^{\prime}}(P)\Delta^{\ell}$,
for any symbol $P$ and any generic $\lambda^{\prime}\in\mathbb{R}$.
Consequently, we get an isomorphism of ${\mathfrak{g}}$-modules on the
quotient spaces
${\mathcal{Q}}^{\lambda,\lambda^{\prime}}:\bigoplus_{s=0}^{\ell-1}{\mathcal{S}}^{\lambda^{\prime}-\lambda}_{*,s}\rightarrow\mathcal{D}^{\lambda,\lambda^{\prime}}/(\Delta^{\ell})$,
where
${\mathcal{S}}^{\delta}_{*,s}=\bigoplus_{k\in\mathbb{N}}{\mathcal{S}}^{\delta}_{k,s}$.
We keep notation ${\mathcal{Q}}^{\lambda,\lambda^{\prime}}$ in the latter case
for simplicity. According to Theorem 2.2 and Proposition 2.4,
${\mathcal{Q}}^{\lambda,\lambda}$ always exists but not
$\mathcal{Q}^{\lambda,\mu}$, which exists if $n$ is odd or
$3\ell\leq\frac{n}{2}+1$.

The idea of the proof is to use the conformally equivariant quantization to
identify the kernel $\mathcal{A}^{\lambda,\ell}$ of the operator
$\mathrm{QHS}$, see (3.3), to the one of an operator $\mathrm{CHS}$ on
symbols, its name standing for Classical Higher Symmetries.

In the easier case where $n$ is odd or $3\ell\leq\frac{n}{2}+1$,
$\mathcal{Q}^{\lambda,\mu}$ exists and the latter operator is determined by
the following commutative diagram of ${\mathfrak{g}}$-modules

(3.4)
$\textstyle{\mathcal{D}^{\lambda,\lambda}/(\Delta^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{QHS}}$$\textstyle{\mathcal{D}^{\lambda,\mu}/(\Delta^{\ell})}$$\textstyle{\bigoplus_{s=0}^{\ell-1}{\mathcal{S}}^{0}_{*,s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\scriptstyle{\mathrm{CHS}}$$\textstyle{\bigoplus_{s=0}^{\ell-1}{\mathcal{S}}^{\frac{2\ell}{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\mu}}$

Consequently, $\mathrm{CHS}$ is a conformally invariant operator and resorting
to the classification given in Theorem 2.2, it is necessarily equal on
${\mathcal{S}}^{0}_{*,s}$ to $R^{\ell-s-1}G_{0}^{2s+1}T^{s}$, up to a
multiplicative constant. This constant cannot be zero since $\mathrm{QHS}$
does not vanish on the image of ${\mathcal{S}}^{0}_{*,s}$. Hence, by
Definition 3.4, the kernel of $\mathrm{QHS}$ is isomorphic to the space
$\mathcal{K}^{\ell}$ of $s$-generalized conformal Killing tensors for $0\leq
s<\ell$.

In the other case, the conformally invariant operator $\mathrm{CHS}$ is
defined via the following commutative diagram of ${\mathfrak{g}}$-modules

(3.5)
$\textstyle{\mathcal{D}^{\lambda,\lambda}_{k}/(\Delta^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{QHS}}$$\textstyle{\mathcal{D}^{\lambda,\mu}_{k^{\prime}}/(\Delta^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{k^{\prime}}}$$\textstyle{{\mathcal{S}}^{0}_{k,s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\scriptstyle{\mathrm{CHS}}$$\textstyle{{\mathcal{S}}^{\frac{2\ell}{n}}_{k^{\prime}},}$

where $\sigma_{k^{\prime}}$ is the principal symbol map and
$k^{\prime}\in\mathbb{N}$ is taken as small as possible, so that
$\mathrm{CHS}$ does not vanish. According to Theorem 2.2 it is proportional to
$R^{\ell-s-1}G_{0}^{2s+1}T^{s}$ or to
$R^{\ell+s-k-\frac{n}{2}}\mathcal{L}_{k+\frac{n}{2}}T^{s}$, and the second
case can occur only if $n/2+(k-s)\leq\ell$. Contrary to
$\mathcal{Q}^{\lambda,\mu}$, the principal symbol map is not an isomorphism,
hence, a priori, we just get that the space
${\mathcal{Q}}^{\lambda,\lambda}(\ker\mathrm{CHS})$ contains the kernel of
$\mathrm{QHS}$. If $\mathrm{CHS}$ is again proportional to
$R^{\ell-s-1}G_{0}^{2s+1}T^{s}$, we obtain the converse inclusion by
irreducibility of the kernel of $\mathrm{CHS}$. Else, $\mathrm{CHS}$ is
proportional to $R^{\ell+s-k-\frac{n}{2}}\mathcal{L}_{k+\frac{n}{2}}T^{s}$ and
we have $k\leq\ell+s-n/2$. According to proposition 2.4,
$\mathcal{Q}^{\lambda,\mu}$ is then well-defined on
${\mathcal{S}}_{k^{\prime\prime}}^{\frac{2\ell}{n}}$ for $k^{\prime\prime}\leq
k^{\prime}$ and we are back to the situation in (3.4). Thus, the kernels of
$\mathrm{CHS}$ and $\mathrm{QHS}$ are always in correspondence via
${\mathcal{Q}}^{\lambda,\lambda}$. We deduce that
${\mathrm{gr}}\mathcal{A}^{\lambda,\ell}\simeq\ker\mathrm{CHS}$ is a
subalgebra of ${\mathcal{S}}^{0}/(R^{\ell})$, with first degree elements given
by ${\mathfrak{g}}$. If its intersection with ${\mathcal{S}}^{0}_{k,s}$ is the
kernel of $R^{\ell+s-k-\frac{n}{2}}\mathcal{L}_{k+\frac{n}{2}}T^{s}$, then it
is infinite dimensional and so will be its intersection with
${\mathcal{S}}^{0}_{m}$ for $m\geq k$. However, for $m$ big enough, we have
proved that the kernel of $\mathrm{CHS}$ in ${\mathcal{S}}^{0}_{m,s}$ is the
one of $R^{\ell-s-1}G_{0}^{2s+1}T^{s}$, which is finite dimensional. In
conclusion, $\mathrm{CHS}$ is proportional to $R^{\ell-s-1}G_{0}^{2s+1}T^{s}$
on any ${\mathcal{S}}^{0}_{k,s}$ and we get the desired correspondence between
$\mathcal{K}^{\ell}$ and $\mathcal{A}^{\lambda,\ell}$ in all cases.

Now, we can define on $\mathcal{D}^{\lambda,\lambda}$ a new conformally
invariant operator
$\mathrm{QHS}_{0}:D\mapsto\Delta^{\ell}D-{\mathcal{Q}}^{\mu,\mu}\circ({\mathcal{Q}}^{\lambda,\lambda})^{-1}(D)\Delta^{\ell}$.
If $\mathcal{Q}^{\lambda,\mu}$ exists, the commutative diagram

$\textstyle{\mathcal{D}^{\lambda,\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{QHS}_{0}}$$\textstyle{\mathcal{D}^{\lambda,\mu}}$$\textstyle{{\mathcal{S}}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{{\mathcal{S}}^{\frac{2\ell}{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\mu}}$

leads to a non-vanishing conformally invariant operator on
${\mathcal{S}}^{0}_{*,s}$ for every $s$. According to Theorem 2.2, it is
proportional to the same operator $\mathrm{CHS}$ as before if $s<\ell-1$ and
this is the null operator else. We conclude that, in particular,
$\Delta^{\ell}{\mathcal{Q}}^{\lambda,\lambda}(P)={\mathcal{Q}}^{\mu,\mu}(P)\Delta^{\ell}$
for any $P\in\mathcal{K}$. The proof in the remaining case is analogous. ∎

######  Remark 3.6.

For $\lambda=\frac{n-2}{2n}$, classical and quantum symmetries for the
equations of motion of a free massless particle correspond to each other via
${\mathcal{Q}}^{\lambda,\lambda}$: $\\{R,K\\}\propto
R\Longleftrightarrow[\Delta,{\mathcal{Q}}^{\lambda,\lambda}(K)]=A\Delta$.

######  Remark 3.7.

We can obtain explicit expressions for the higher symmetries of
$\Delta^{\ell}$ via the formulae for the conformally equivariant quantization
given in (2.1) for $\ell=1$ or in [39] for the general case. The obtained
differential operators admit analogs in the curved case, which are not
necessarily higher symmetries anymore. E.g. all the conformal Killing
$2$-tensors do not give rise to higher symmetries of the Laplacian in general
[34].

##  4\. Algebras of symmetries: geometric realizations and deformations

The aim of this section is to provide geometric interpretations to the
algebras of classical and quantum symmetries, as well as identifying the star
product induced by the composition of quantum symmetries. In all this section
we work over a conformally flat manifold $(M,[\mathrm{g}])$ of dimension
$n\geq 3$.

###  4.1. Algebras of symmetries are generated by ${\mathfrak{g}}$

Let us give a brief reminder on universal enveloping algebra
$\mathfrak{U}({\mathfrak{g}})$ and symmetric algebra
$\mathrm{S}({\mathfrak{g}})$ of an arbitrary Lie algebra ${\mathfrak{g}}$.
From the tensor algebra of ${\mathfrak{g}}$, they inherit respectively a
filtration $\\{\mathfrak{U}_{k}({\mathfrak{g}})\\}_{k}$ and a grading
$\mathrm{S}({\mathfrak{g}})=\bigoplus_{k}\mathrm{S}_{k}({\mathfrak{g}})$ such
that
${\mathrm{gr}}\,\mathfrak{U}({\mathfrak{g}})\simeq\mathrm{S}({\mathfrak{g}})$.
Consequently, the canonical projections
$\mathfrak{U}_{k}({\mathfrak{g}})\rightarrow\mathfrak{U}_{k}({\mathfrak{g}})/\mathfrak{U}_{k-1}({\mathfrak{g}})$
define a principal symbol map, whose right inverses are called quantizations
of $\mathrm{S}({\mathfrak{g}})$. The symmetrization map
$\mathrm{Sym}:\mathrm{S}({\mathfrak{g}})\rightarrow\mathfrak{U}({\mathfrak{g}})$
defined by

$\mathrm{Sym}:X_{i_{1}}\cdots
X_{i_{k}}\mapsto\frac{1}{k!}\sum_{\tau\in\mathfrak{S}_{k}}X_{\tau(i_{1})}\cdots
X_{\tau(i_{k})}$

is known to define a ${\mathfrak{g}}$-equivariant quantization of
$\mathrm{S}({\mathfrak{g}})$ for the canonical extensions of the adjoint
action of ${\mathfrak{g}}$ to $\mathrm{S}({\mathfrak{g}})$ and
$\mathfrak{U}({\mathfrak{g}})$. Any other ${\mathfrak{g}}$-equivariant
quantization is then of the form $\Phi=\mathrm{Sym}\circ\phi$, with
$\phi=\mathrm{Id}+N$ and $N$ a ${\mathfrak{g}}$-equivariant map on
$\mathrm{S}({\mathfrak{g}})$ lowering the degree. Analogously to the case of
symbols, a ${\mathfrak{g}}$-equivariant graded star product $\star_{\Phi}$ can
be obtained on $\mathrm{S}({\mathfrak{g}})$, as the pull-back of the product
on $\mathfrak{U}({\mathfrak{g}})\otimes\mathbb{C}[[\hbar]]$ by the map
$\Phi_{\hbar}=(\Phi\otimes\mathrm{Id})\circ\Im$, where
$\Im:\mathrm{S}({\mathfrak{g}})\rightarrow\mathrm{S}({\mathfrak{g}})\otimes\mathbb{C}[[\hbar]]$
is the linear map defined by $(\mathsf{i}\hbar)^{k}\mathrm{Id}$ on
$\mathrm{S}_{k}({\mathfrak{g}})$. Denoting by $\tau$ and $\gamma$ the anti-
automorphisms of $\mathfrak{U}({\mathfrak{g}})$ and
$\mathrm{S}({\mathfrak{g}})$ defined by $-\mathrm{Id}$ on ${\mathfrak{g}}$,
the symmetry of the star product on $\mathrm{S}({\mathfrak{g}})$ is equivalent
to $\Phi_{\hbar}(\bar{\cdot})=\tau\circ\Phi_{\hbar}(\cdot)$, or simply
$\Phi\circ\gamma=\tau\circ\Phi$.

We return to the case ${\mathfrak{g}}=\mathrm{o}(p+1,q+1)$. Via the defining
universal properties of the both algebras $\mathrm{S}({\mathfrak{g}})$ and
$\mathfrak{U}({\mathfrak{g}})$, the pull-back defined by the moment map
$\mu^{*}:{\mathfrak{g}}\rightarrow{\mathcal{S}}^{0}_{1}$ and the Lie
derivative
$\ell^{\lambda}:{\mathfrak{g}}\rightarrow\mathcal{D}^{\lambda,\lambda}_{1}$
extend to algebra morphisms
$\mu^{*}:\mathrm{S}({\mathfrak{g}})\rightarrow{\mathcal{S}}^{0}$ and
$\ell^{\lambda}:\mathfrak{U}({\mathfrak{g}})\rightarrow\mathcal{D}^{\lambda,\lambda}$.

######  Theorem 4.1.

Let $\lambda\in\mathbb{R}$. The space $\mathcal{K}$ of classical symmetries is
equal to the algebra
$\mu^{*}(\mathrm{S}({\mathfrak{g}}))\simeq\mathrm{S}({\mathfrak{g}})/I$, with
$I$ a graded ideal of $\mathrm{S}({\mathfrak{g}})$. Its image by the
conformally equivariant quantization, namely
$\mathcal{A}^{\lambda}:={\mathcal{Q}}^{\lambda,\lambda}(\mathcal{K})$, is an
algebra satisfying
$\mathcal{A}^{\lambda}=\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))\simeq\mathfrak{U}({\mathfrak{g}})/J^{\lambda}$,
with $J^{\lambda}$ a filtered ideal such that
${\mathrm{gr}}\,J^{\lambda}\simeq I$. The star product $\star_{\lambda}$
induced by ${\mathcal{Q}}^{\lambda,\lambda}$ on ${\mathcal{S}}^{0}$ restricts
then to a ${\mathfrak{g}}$-equivariant graded star product on $\mathcal{K}$.

Moreover, the conformally equivariant quantization of $\mathcal{K}$ lifts to a
${\mathfrak{g}}$-equivariant quantization $\Phi^{\lambda}$ of
$\mathrm{S}({\mathfrak{g}})$, such that the following diagram commutes

(4.6)
$\textstyle{\mathrm{S}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{*}}$$\scriptstyle{\Phi^{\lambda}}$$\textstyle{\mathfrak{U}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{\lambda}}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{\mathcal{A}^{\lambda}}$

###### Proof.

We start with proving $\mathcal{K}=\mu^{*}(\mathrm{S}({\mathfrak{g}}))$. Since
they are ${\mathfrak{g}}$-modules, this amounts to proving that
$\mu^{*}(\mathrm{S}({\mathfrak{g}}))\cap{\mathcal{S}}^{0}_{k,s}=\mathcal{K}_{k,s}$
for all $k,s$. Let $\ell\in\mathbb{N}^{*}$. The space
$\mathcal{A}^{\lambda,\ell}$ is a subalgebra of
$\mathcal{D}^{\lambda,\lambda}/(\Delta^{\ell})$ containing
$\ell^{\lambda}({\mathfrak{g}})$. As the principal symbol map commutes with
the product, we get that $\mathcal{K}^{\ell}$ is a subalgebra of
${\mathcal{S}}^{0}/(R^{\ell})$ containing $\mu^{*}({\mathfrak{g}})$. Hence,
$\mathcal{K}$ is a subalgebra of ${\mathcal{S}}^{0}$, which contains the
algebra $\mu^{*}(\mathrm{S}({\mathfrak{g}}))$ generated by the conformal
Killing vector fields, in particular
$\mu^{*}(\mathrm{S}({\mathfrak{g}}))\cap{\mathcal{S}}^{0}_{k,s}\subset\mathcal{K}_{k,s}$
for all $k,s$. Using the elements generated by constant Killing vector fields,
we get that $\mu^{*}(\mathrm{S}({\mathfrak{g}}))\cap{\mathcal{S}}^{0}_{k,s}$
is non-empty and since $\mathcal{K}_{k,s}$ is irreducible, we are done.

Now, we prove that
$\mathcal{A}^{\lambda}=\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))$. By semi-
simplicity of ${\mathfrak{g}}$, its finite dimensional representations are
completely reducibles. In particular, for any $k$,
$\mathcal{K}\cap{\mathcal{S}}^{0}_{k}$ can be viewed as a submodule of
$\mathrm{S}_{k}({\mathfrak{g}})$, leading to the decomposition
$\mathrm{S}({\mathfrak{g}})\simeq\mathcal{K}\oplus I$ of the symmetric
algebra. In other words, $\mu^{*}$ admits a ${\mathfrak{g}}$-equivariant
section. Using the embedding of $\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))$
into $\mathcal{D}^{\lambda,\lambda}$, we get then the following diagram of
${\mathfrak{g}}$-modules

$\textstyle{\mathrm{S}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Sym}}$$\textstyle{\mathfrak{U}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{\lambda}}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}^{\lambda,\lambda}}$

Each arrow in the latter diagram is ${\mathfrak{g}}$-equivariant and preserves
the principal symbol. Hence, uniqueness of ${\mathcal{Q}}^{\lambda,\lambda}$
on the ${\mathfrak{g}}$-module $\mathcal{K}$ implies that the diagram is
commutative, proving
$\mathcal{A}^{\lambda}=\ell^{\lambda}(\mathfrak{U}({\mathfrak{g}}))$.

Since $\mu^{*}$ respects the grading, its kernel $I$ is a graded ideal, and
since $\ell^{\lambda}$ preserves the filtration, its kernel $J^{\lambda}$ is
filtered. Using the commutativity of the following diagram,

$\textstyle{\mathfrak{U}_{k}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}^{\lambda}\cap\mathcal{D}^{\lambda,\lambda}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{S}_{k}({\mathfrak{g}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}_{k},}$

where the vertical arrows denote principal symbol maps, we get that
${\mathrm{gr}}\,J^{\lambda}=I$.

We have proved $\mathrm{S}({\mathfrak{g}})\simeq\mathcal{K}\oplus I$, and
along the same line we get
$\mathfrak{U}({\mathfrak{g}})\simeq\mathcal{A}^{\lambda}\oplus J^{\lambda}$.
Using again the semi-simplicity of ${\mathfrak{g}}$, the isomorphism
$J^{\lambda}_{k}/J^{\lambda}_{k-1}\simeq I_{k}$ leads to
$J_{k}^{\lambda}\simeq I_{k}\oplus J^{\lambda}_{k-1}$. Thus, it exists an
isomorphism of ${\mathfrak{g}}$-modules between $I$ and $J^{\lambda}$, inverse
to the symbol map. Together with the previous decomposition of
$\mathrm{S}({\mathfrak{g}})$ and $\mathfrak{U}({\mathfrak{g}})$, this ensures
the existence of the quantization $\Phi^{\lambda}$ and the commutativity of
the diagram (4.6). ∎

The proof shows that
$\mathcal{K}_{k,s}=\mu^{*}(\mathrm{S}({\mathfrak{g}}))\cap{\mathcal{S}}^{0}_{k,s}$.
Thus, on conformally flat manifolds, the $s$-generalized conformal Killing
$k$-tensors are algebraically generated from the conformal Killing vectors.
This widely generalizes a result in [38], stating the same fact for second
order conformal Killing tensors.

######  Corollary 4.2.

Let $\ell\in\mathbb{N}^{*}$ and $\lambda=\frac{n-2\ell}{2n}$. The following
diagram of ${\mathfrak{g}}$-modules is commutative

(4.7)
$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{\mathcal{A}^{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}^{\ell}=\mathcal{K}/(R^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{Q}}^{\lambda,\lambda}}$$\textstyle{\mathcal{A}^{\lambda}/(\Delta^{\ell})=\mathcal{A}^{\lambda,\ell}}$

The algebra of $s$-generalized conformal Killing tensors, for $s<\ell$,
satisfies $\mathcal{K}^{\ell}\simeq\mathrm{S}({\mathfrak{g}})/I^{\ell}$, with
$I^{\ell}$ the ideal generated by $I$ and $(\mu^{*})^{-1}(R^{\ell})$, and the
algebra of higher symmetries of $\Delta^{\ell}$ satisfies
$\mathcal{A}^{\lambda,\ell}\simeq\mathfrak{U}({\mathfrak{g}})/J^{\lambda,\ell}$
with $J^{\lambda,\ell}$ the ideal generated by $J^{\lambda}$ and
$(\ell^{\lambda})^{-1}(\Delta^{\ell})$. The star product $\star_{\lambda}$
projects accordingly onto $\mathcal{K}^{\ell}$, defining there a graded
${\mathfrak{g}}$-equivariant star product.

###### Proof.

By definition $\mathcal{K}^{\ell}=\mathcal{K}/(R^{\ell})$ and by Theorem 3.5
we have
${\mathcal{Q}}^{\lambda,\lambda}(\mathcal{K}^{\ell})=\mathcal{A}^{\lambda,\ell}$.
Adding that $R^{\ell}\in\mu^{*}(\mathrm{S}({\mathfrak{g}}))$ and
${\mathcal{Q}}^{\lambda,\lambda}((R^{\ell}))=(\Delta^{\ell})$, we get the
announced commutative diagram. The remaining results trivially follow. ∎

###  4.2. A family of coadjoint orbits of $\mathrm{O}(p+1,q+1)$

We restrict in this paragraph to the case where $M$ is the homogeneous space
$\mathbb{S}^{p}\times\mathbb{S}^{q}$ of the conformal group
$G=\mathrm{O}(p+1,q+1)$. The latter acts linearly and in a Hamiltonian way on
$T^{*}\mathbb{R}^{p+1,q+1}$, and forms a Howe dual pair with
$\mathrm{SL}(2,\mathbb{R})$, its centralizer in the symplectic linear group
$\mathrm{Sp}(2n+2,\mathbb{R})$. More precisely, they form a symplectic dual
pair in the sense of [3]. Denoting their moment maps by

$\mu:T^{*}\mathbb{R}^{p+1,q+1}\longrightarrow{\mathfrak{g}}^{*}\qquad\text{and}\qquad
J:T^{*}\mathbb{R}^{p+1,q+1}\longrightarrow\mathrm{sl}(2,\mathbb{R})^{*},$

the symplectic reduction of $T^{*}\mathbb{R}^{p+1,q+1}$ with respect to
$\mathrm{SL}(2,\mathbb{R})$ for different regular values of the moment map $J$
gives rise to (finite coverings of) all the coadjoint orbits in the image of
$\mu$. We rather describe them as symplectic reductions at $0$ with respect to
Lie subgroups of $\mathrm{SL}(2,\mathbb{R})$ generated by the flow of
Hamiltonian functions in
$J^{*}\big{(}\mathrm{S}(\mathrm{sl}(2,\mathbb{R}))\big{)}$, i.e. polynomial
functions in $x^{2}=\eta_{AB}x^{A}x^{B}$, $xp=x^{A}p_{A}$ and
$p^{2}=\eta^{AB}p_{A}p_{B}$, where $(x^{A},p_{A})$ are cartesian coordinates
on $T^{*}\mathbb{R}^{p+1,q+1}$. Important such functions are given by the
Casimir elements of ${\mathfrak{g}}$ and $\mathrm{sl}(2,\mathbb{R})$ in
${\mathcal{C}^{\infty}}(T^{*}\mathbb{R}^{p+1,q+1})$. They are equal to
$C=(xp)^{2}-x^{2}p^{2}$ and $C/4$ respectively, if we define the Killing form
by the map $(X,Y)\mapsto\frac{1}{2}\mathrm{Tr}(\rho(X)\rho(Y))$ with $\rho$
their standard representation.

We denote by $\left\langle f_{1},\ldots,f_{k}\right\rangle$ the Lie group
generated by the flow of Hamiltonian functions
$f_{1},\ldots,f_{k}\in{\mathcal{C}^{\infty}}(T^{*}\mathbb{R}^{p+1,q+1})$ and
by $T^{*}\mathbb{R}^{p+1,q+1}//\left\langle f_{1},\ldots,f_{k}\right\rangle$
the corresponding symplectic quotient at $0$. If those functions are linearly
closed under the Poisson bracket, the latter space is then the quotient of the
zero locus of $f_{1},\ldots,f_{k}$ by their Hamiltonian flows. By the Marsden-
Weinstein theorem, this quotient space is a symplectic manifold if $0$ is a
regular value of the involved Hamiltonian functions. E.g., we have

(4.8) $T^{*}\left(\mathbb{R}^{p+1,q+1}\setminus\\{0\\}\right)//\left\langle
xp,x^{2}\right\rangle\simeq T^{*}M.$

Notice that $T^{*}M$ splits in three stable submanifolds under the Hamiltonian
$G$-action, according to the sign of the norm of the covectors, with
straightforward notations: $T^{*}M=T^{*}_{+}M\sqcup T^{*}_{0}M\sqcup
T^{*}_{-}M$.

######  Theorem 4.3.

Let $p,q\geq 1$, $n\geq 3$ and $P(\alpha,\beta)$ be the space of planes in
$\mathbb{R}^{p+1,q+1}$ of signature $(\alpha,\beta)$. The coadjoint orbits of
$G$ in the image of $\mu$ are classified along four families:

  1. (1)

the one parameter family of semi-simple orbits $\mathcal{O}_{a^{+}}$ and
$\mathcal{O}_{a^{-}}$ for $a\in\mathbb{R}^{*}_{+}$ such that

$\textstyle{T^{*}\mathbb{R}^{p+1,q+1}//\left\langle
xp,C-a\right\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\qquad\mathbb{Z}_{2}}$$\textstyle{\mathcal{O}_{a^{+}}\sqcup\mathcal{O}_{a^{-}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq\quad}$$\textstyle{P(2,0)\sqcup
P(0,2),}$

  2. (2)

the one parameter family of semi-simple orbits $\mathcal{O}_{a}$ for
$a\in\mathbb{R}^{*}_{-}$ such that

$\textstyle{T^{*}\mathbb{R}^{p+1,q+1}//\left\langle
xp,C-a\right\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\qquad\mathbb{Z}_{2}}$$\textstyle{\mathcal{O}_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq\quad}$$\textstyle{P(1,1),}$

  3. (3)

the two nilpotent orbits $\mathcal{O}_{0^{+}}$ and $\mathcal{O}_{0^{-}}$ such
that

$\textstyle{T^{*}_{+}M\sqcup
T^{*}_{-}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{Z}_{2}}$$\textstyle{\mathcal{O}_{0^{+}}\sqcup\mathcal{O}_{0^{-}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{R}^{*}\quad}$$\textstyle{P(1,0)\sqcup
P(0,1),}$

  4. (4)

The minimal nilpotent orbit $\mathcal{O}_{00}$ such that

$\textstyle{(T^{*}M\setminus M)//\left\langle
R\right\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\qquad\mathbb{Z}_{2}}$$\textstyle{\mathcal{O}_{00}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{R}^{*}}$$\textstyle{P(0,0).}$

All the arrows denote $G$-equivariant coverings, whose fibers are indicated as
superscript. The first ones are symplectomorphisms.

###### Proof.

Through the $G$-module isomorphisms
$\Lambda^{2}\mathbb{R}^{p+1,q+1}\simeq{\mathfrak{g}}\simeq{\mathfrak{g}}^{*}$,
coadjoint orbits are identified to $G$-orbits in the space of bivectors,
endowed with the natural $G$-action. The moment map $\mu$ is then given by
$T^{*}\mathbb{R}^{p+1,q+1}\ni(u,v)\mapsto u\wedge v$, and valued in the space
of simple bivectors $\mathrm{Bv}=\\{u\wedge
v|\,u,v\in\mathbb{R}^{p+1,q+1}\\}$. Our key tool is the $G$-equivariant
projection of $\mathrm{Bv}$ on the Grassmannian $\mathrm{Gr}(2,n+2)$ of planes
in $\mathbb{R}^{p+1,q+1}$. This is encompass in the following sequence of
$G$-spaces:

(4.9)
$\textstyle{T^{*}\mathbb{R}^{p+1,q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\mathrm{SL}(2,\mathbb{R})}$$\textstyle{\mathrm{Bv}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{R}^{*}\quad\qquad}$$\textstyle{\mathrm{Gr}(2,n+2)\cup\\{0\\},}$

where the superscripts denote the fibers of the coverings over
$\mathrm{Gr}(2,n+2)$. The moment map preserves the Poisson structure, hence a
$G$-stable submanifold of $T^{*}\mathbb{R}^{p+1,q+1}$ projects onto coadjoint
orbits of $G$, which themselves project onto $G$-orbits of
$\mathrm{Gr}(2,n+2)\cup\\{0\\}$. Thanks to the Witt Theorem, the latter are
known to be $\\{0\\}$ and the $6$ spaces $P(\alpha,\beta)$ of planes of given
signature $(\alpha,\beta)$ for the induced metric. E.g., the $G$-stable
embedding of $T^{*}_{+}M\sqcup T^{*}_{-}M$ into $T^{*}\mathbb{R}^{p+1,q+1}$
projects onto coadjoint orbits, which project onto the orbits $P(1,0)\sqcup
P(0,1)$ in $\mathrm{Gr}(2,n+2)$. The fibers of these two projections are
easily computed. In the three other cases, the zero locus of the given
Hamiltonian functions have the announced images in $\mathrm{Gr}(2,n+2)$. Their
Hamiltonian flows act only in the fibers of $\mu$, so the map $\mu$ descends
to the symplectic quotients. The fibers are proved to reduce to
$\mathbb{Z}_{2}$, resorting to the explicit expressions of the one parameter
groups generated by $xp$, $C$ and $R$. Namely, they respectively are
$(u,v)\mapsto(tu,t^{-1}v)$, $(u,v)\mapsto(u+(tv^{2})v,v-(tu^{2})u)$ and
$(u,v)\mapsto(u+tv,v)$ for $t\in\mathbb{R}^{*}$. We end with the four
sequences $(1)$, $(2)$, $(3)$, and $(4)$. There, a unique coadjoint orbit lies
over each orbit in $\mathrm{Gr}(2,n+2)$, since the action of the group $G$ is
transitive in the fibers of each arrow. For a proof of the minimality of
$\mathcal{O}_{00}$ we refer to [44]. ∎

The two last points in the latter theorem combine, according to Cordani [14],
to provide a conformal regularization by $T^{*}M$ of the cone
$\mathcal{O}_{0^{+}}\cup\mathcal{O}_{00}\cup\mathcal{O}_{0^{-}}$, with
singularity in $\mathcal{O}_{00}$.

######  Remark 4.4.

The used symplectic reductions of $T^{*}\mathbb{R}^{p+1,q+1}$ correspond
clearly to symplectic reduction with respect to $\mathrm{SL}(2,\mathbb{R})$
at, respectively, the points $(0,\sqrt{a},\pm\sqrt{a})$,
$(0,-\sqrt{|a|},-\sqrt{|a|})$, $(0,0,\pm a)$ and $(0,0,0)$. Hence, we obtain a
bijection between the coadjoints orbits of $\mathrm{SL}(2,\mathbb{R})$ and the
ones in the image of $\mu$, in accordance with [3].

Now, we determine the algebra of regular functions on each coadjoint orbit of
$G$ in the image of $\mu$. We have
${\mathfrak{g}}\simeq\Lambda^{2}\mathbb{R}^{p+1,q+1}$, that we represent by
the Young diagram ${\Yboxdim{5pt}\yng(1,1)}$. Accordingly, elementary
representation theory of the orthogonal Lie algebra leads to

(4.10)
${\mathfrak{g}}\odot{\mathfrak{g}}={\mbox{\tiny$\yng(2,2)\oplus\yng(1,1,1,1)$}}\quad\text{and}\quad{\mbox{\tiny$\yng(2,2)={\yng(2,2)}_{\,0}\oplus{\yng(2)}_{\,0}$}}\oplus\mathbb{R}.$

In the second decomposition, the index $0$ denotes the trace free part, and
the three components correspond to $\mathcal{K}_{2,0}$, $\mathcal{K}_{2,1}$
and the one-dimensional space generated by the Casimir element in
$\mathrm{S}_{2}({\mathfrak{g}})$, still denoted by $C$. The extra term in the
decomposition of ${\mathfrak{g}}\odot{\mathfrak{g}}$ is generated by exterior
products in $\Lambda\mathbb{R}^{p+1,q+1}$ of elements of
${\mathfrak{g}}\simeq\Lambda^{2}\mathbb{R}^{p+1,q+1}$.

######  Lemma 4.5.

The kernel of the pull-back
$\mu^{*}:\mathrm{S}({\mathfrak{g}})\rightarrow{\mathcal{C}^{\infty}}(T^{*}\mathbb{R}^{p+1,q+1})$
by the moment map of ${\mathfrak{g}}$ is the ideal generated by
${\Yboxdim{5pt}\yng(1,1,1,1)}$.

###### Proof.

Since elements of ${\mathfrak{g}}$ are skew-symmetric $2$-tensors $V^{AB}$ on
$\mathbb{R}^{p+1,q+1}$, the map $\mu^{*}$ is explicitly given by $V^{AB\cdots
CD}\mapsto x_{A}\cdots x_{C}V^{AB\cdots CD}p_{B}\cdots p_{D}$, and vanishes
then on tensors $V^{AB\cdots CD}$ which are skew-symmetric in any $3$ indices.
Hence, the module ${\Yboxdim{5pt}\yng(1,1,1,1)}$ is in the kernel of $\mu^{*}$
and $\mu^{*}(\mathrm{S}_{k}({\mathfrak{g}}))$ is contained in the module
$\mathrm{S}_{k}({\mathfrak{g}})/\left(\,{\Yboxdim{5pt}\yng(1,1,1,1)}\,\right)$,
described by the Young diagram with $2$ lines and $k$ columns. But none of the
irreducible components of such a Young diagram is in the kernel of $\mu^{*}$,
as all the traces, $x^{2},xp,p^{2}$, can occur in
${\mathcal{C}^{\infty}}(T^{*}\mathbb{R}^{p+1,q+1})$. In conclusion, the
algebra $\mu^{*}(\mathrm{S}({\mathfrak{g}}))$ is isomorphic to
$\mathrm{S}({\mathfrak{g}})/\left(\,{\Yboxdim{5pt}\yng(1,1,1,1)}\,\right)$. ∎

######  Proposition 4.6.

Let $a\in\mathbb{R}$ and
$\mathcal{I}_{a}=\left([C-a]\mathbb{R}\oplus{\Yboxdim{5pt}\yng(1,1,1,1)}\,\right)$.
The algebras of regular functions are given by
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}$ on $\mathcal{O}_{a^{(\pm)}}$, and
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{00}$ on $\mathcal{O}_{00}$, with
$\mathcal{I}_{00}=\left({\Yboxdim{5pt}{\yng(2)}_{\,0}}\right)+\mathcal{I}_{0}$.
Moreover, the algebras of regular functions on $\mathcal{O}_{0^{\pm}}$ and
$\mathcal{O}_{00}$ are isomorphic to $\mathcal{K}$ and $\mathcal{K}^{1}$
respectively.

###### Proof.

According to (4.8), we get that
$T_{\pm}^{*}(\mathbb{R}^{p+1,q+1}\setminus\\{0\\})//\left\langle
xp,C\right\rangle$ is a $\mathbb{Z}_{2}$-covering of $T^{*}_{\pm}M$. Together
with theorem 4.3, this proves that each coadjoint orbit of $G$ in the image of
$\mu$ is finitely covered by a symplectic reduction of
$T^{*}\mathbb{R}^{p+1,q+1}$. Therefore, its algebra of regular functions is
isomorphic to the corresponding reduction of
$\mu^{*}(\mathrm{S}({\mathfrak{g}}))$. The reduction with respect to $xp$
modifies only the fibers of $\mu$ and the Casimir $C$ Poisson commutes with
all elements in $\mu^{*}(\mathrm{S}({\mathfrak{g}}))$, so that reduction with
respect to $\left\langle xp,C-a\right\rangle$ amounts to modding out by
$(C-a)$. This gives the result for the orbits $\mathcal{O}_{a^{(\pm)}}$ for
every $a\in\mathbb{R}$. By Theorem 4.1, the algebra $\mathcal{K}$ is generated
by ${\mathfrak{g}}$. The isomorphism
$\mathcal{K}\simeq\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{0}$ follows then
from the local diffeomorphism between $T^{*}_{\pm}M$ and
$\mathcal{O}_{0^{\pm}}$.

Resorting to Theorem 4.3, the coadjoint orbit $\mathcal{O}_{00}$ is locally
diffeomorphic to the symplectic quotient $(T^{*}M\setminus M)//\left\langle
R\right\rangle$, i.e. to the quotient of the submanifold $T^{*}_{0}M\setminus
M$ of null covectors by the Hamiltonian flow of $R$. Thus, the algebra of
regular functions on $\mathcal{O}_{00}$ arises as a reduction of
$\mathcal{K}$. Since $\\{R,\mathcal{K}\\}\subset(R)$, this reduced algebra is
$\mathcal{K}/(R)$, which is isomorphic to $\mathcal{K}_{1}$. As
$R\in\mathcal{K}_{2,1}$ is the pull-back of an element in
${\Yboxdim{5pt}{\yng(2)}_{\,0}}$, we finally obtain $\mathcal{I}_{00}$. ∎

###  4.3. The Joseph ideal

We return now to a general conformally flat manifold $(M,[\mathrm{g}])$, and
use notation of Paragraph 4.1, in particular
$\mathcal{K}\simeq\mathrm{S}({\mathfrak{g}})/I$ and
$\mathcal{K}^{1}\simeq\mathrm{S}({\mathfrak{g}})/I^{1}$. The algebraic
description of these symmetry algebras obtained in Proposition 4.6 are of
local nature and thus carry on over $(M,[\mathrm{g}])$. Hence, the ideals
$I,I^{1}$ identify to $\mathcal{I}_{0},\mathcal{I}_{00}$. We compute now the
corresponding ideals $J^{\lambda}$ and $J^{\lambda,1}$ in
$\mathfrak{U}({\mathfrak{g}})$, which define the algebras
$\mathcal{A}^{\lambda}$ and $\mathcal{A}^{\lambda,1}$. Recall that we define
the Killing form by $\frac{1}{2}\mathrm{Tr}(XY)$, for every
$X,Y\in{\mathfrak{g}}$. The corresponding Casimir operator ${\mathcal{C}}$ in
$\mathfrak{U}({\mathfrak{g}})$ is given by the symmetrization of the Casimir
element $C$ in $\mathrm{S}({\mathfrak{g}})$.

######  Proposition 4.7.

For every $\lambda\in\mathbb{R}$, the ideals $J^{\lambda}$ are equal to
$\big{(}\mathrm{Sym}({\Yboxdim{5pt}\yng(1,1,1,1)})\oplus[{\mathcal{C}}-\rho(\lambda)]\mathbb{R}\big{)}$,
where ${\mathcal{C}}$ is the Casimir operator of ${\mathfrak{g}}$ and
$\rho(\lambda)=n^{2}\lambda(1-\lambda)$ its eigenvalue on $\lambda$-densities.

For $\lambda=\frac{n-2}{n}$,
$J^{\lambda,1}=\left(\mathrm{Sym}({\Yboxdim{5pt}\yng(2)}_{\,0})\right)+J^{\lambda}$
is the Joseph ideal.

###### Proof.

Since the graded ideal associated to $J^{\lambda}$ is $I$, we deduce that
$J^{\lambda}$ is also quadratic and resorting to Theorem 4.1, we have
$J^{\lambda}_{2}=\Phi^{\lambda}(I_{2})$ with
$\Phi^{\lambda}=\mathrm{Sym}\circ\phi^{\lambda}$. The map $\phi^{\lambda}$
being ${\mathfrak{g}}$-equivariant, the space $\phi^{\lambda}(I_{2})$ is a
${\mathfrak{g}}$-submodule of
$\mathfrak{U}_{2}({\mathfrak{g}})\simeq\mathbb{R}\oplus{\mathfrak{g}}\oplus\mathrm{S}_{2}({\mathfrak{g}})$.
Hence, $J_{2}^{\lambda}$ is generated by
$\mathrm{Sym}\left({\Yboxdim{5pt}\yng(1,1,1,1)}\right)$ and the Casimir
operator ${\mathcal{C}}$ of $\mathfrak{U}({\mathfrak{g}})$, modified by some
real number. Since this element projects onto $0$ on
$\mathcal{D}^{\lambda,\lambda}$, this real number is necessarily the
eigenvalue of $\ell^{\lambda}({\mathcal{C}})$ on $\lambda$-densities. The
latter has been computed in [17], where the opposite Killing form is used. The
formula giving $J^{\lambda,1}$ is trivially deduced from Corollary 4.2. Thanks
to Theorem 3.5, we have the isomorphism of algebras
${\mathrm{gr}}\,(\mathfrak{U}({\mathfrak{g}})/J^{\lambda,1})\simeq\mathrm{S}({\mathfrak{g}})/I^{1}$.
As $I_{1}$ is prime, we deduce that $J^{\lambda,1}$ is completely prime.
Besides, their common characteristic variety is the closure of the minimal
nilpotent coadjoint orbit of $G$. These two properties characterize the Joseph
ideal [25]. ∎

The identification of the Joseph ideal in the context of the higher symmetries
of the Laplacian was already obtained in different manners [22, 41], but never
from its original definition like here. From the latter proposition, the
ideals $J^{\lambda,\ell}$ associated to symmetries of $\Delta^{\ell}$ are
straightforwardly deduced as they are generated by $J^{\lambda}$ and
$(\ell^{\lambda})^{-1}(\Delta^{\ell})$. The latter corresponds to the Young
diagram  … 0 of length $2\ell$. The determination of those ideals has been
already performed in the context of higher symmetries of $\Delta^{\ell}$ in
[19, 20, 40], but in different terms. Let us make clear the link between the
two approaches. We denote by $\left\langle\cdot,\cdot\right\rangle$ the chosen
Killing form and $C$ the associated Casimir element in
$\mathrm{S}({\mathfrak{g}})$. In the previous works, the projections of
$X\odot Y\in{\mathfrak{g}}\odot{\mathfrak{g}}$ on each irreducible component
are used. Following
${\mathfrak{g}}\odot{\mathfrak{g}}={\Yboxdim{5pt}\yng(2,2)}_{\,0}\oplus{\Yboxdim{5pt}\yng(2)}_{\,0}\oplus\mathbb{R}\oplus{\Yboxdim{5pt}\yng(1,1,1,1)}$,
we have $X\odot Y=X\boxtimes Y+X\bullet Y+\frac{\left\langle
X,Y\right\rangle}{2\dim{\mathfrak{g}}}C+X\wedge Y$. Then, the ideal
$J^{\lambda}$ is clearly generated by $\mathrm{Sym}\big{(}\frac{\left\langle
X,Y\right\rangle}{2\dim{\mathfrak{g}}}(C-\rho(\lambda))+X\wedge Y\big{)}$ for
$X,Y\in{\mathfrak{g}}$ or equivalently by

$\mathrm{Sym}\big{(}X\odot Y-X\boxtimes Y-X\bullet
Y+\frac{\rho(\lambda)}{2\dim{\mathfrak{g}}}\left\langle
X,Y\right\rangle\big{)},$

which is the obtained expression in [19, 20, 40], modulo the extra generator
associated to $R^{\ell}$.

###  4.4. Quantization of a family of coadjoint orbits of $G$

We have described the algebras of regular functions on the coadjoint orbits of
$G$ in the image of $\mu$ as quotients
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}$ for various ideals $\mathcal{I}$. We
perform now their deformation quantization, which should arise from
quantization-like maps
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}\rightarrow\mathfrak{U}({\mathfrak{g}})/\mathcal{J}$,
for ideals $\mathcal{J}$ to be defined. This means to find deformations of the
Poisson algebra $\mathrm{S}({\mathfrak{g}})$ which restrict to coadjoint
orbits, and this is in general not trivial. Thus, for ${\mathfrak{g}}$ a semi-
simple Lie algebra, Cahen, Gutt and Rawnsley have proved the non-existence of
a star product on ${\mathcal{C}^{\infty}}({\mathfrak{g}}^{*})$ tangent to each
coadjoint orbit [11]. Here, for the graded algebras $\mathcal{K}$ and
$\mathcal{K}^{1}$, such a quantization map is provided by the conformally
equivariant quantization, we have proved that it induces a graded
${\mathfrak{g}}$-equivariant star-product on them and determines the
corresponding ideals $\mathcal{J}$. As the remaining algebras
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}$ for $a\neq 0$ are only filtered,
this direct approach fails and we resort to the following Lemma.

######  Lemma 4.8.

Let $a\in\mathbb{R}$. There exists a ${\mathfrak{g}}$-equivariant linear map
$\phi_{a}=\mathrm{Id}+N_{a}$ on $\mathrm{S}({\mathfrak{g}})$, such that
$N_{a}$ lowers the degree and $\phi_{a}(\mathcal{I}_{a})=\mathcal{I}_{0}$,
hence $\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}\simeq\mathcal{K}$.

###### Proof.

We know that $\mathrm{S}({\mathfrak{g}})\simeq I\oplus\mathcal{K}$ and
$I=(C)+\left({\Yboxdim{5pt}\yng(1,1,1,1)}\right)$. Resorting to the semi-
simplicity of ${\mathfrak{g}}$ and the filtration of
$\mathcal{I}_{a}=(C-a)+\left({\Yboxdim{5pt}\yng(1,1,1,1)}\right)$, we get that
$\mathrm{S}({\mathfrak{g}})\simeq\mathcal{I}_{a}+\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}$
and $(C-a)$ admits a ${\mathfrak{g}}$-stable complement in $\mathcal{I}_{a}$.
The map $\phi_{a}$ defined by by $\frac{C}{C-a}\mathrm{Id}$ on $(C-a)$ and the
identity on a ${\mathfrak{g}}$-stable complementary space satisfies the
required properties. ∎

######  Theorem 4.9.

There exists a family of ${\mathfrak{g}}$-equivariant quantizations
$(\Phi_{a}^{\lambda})_{a,\lambda\in\mathbb{R}}$ of
$\mathrm{S}({\mathfrak{g}})$ such that: (i) it lifts
$({\mathcal{Q}}^{\lambda,\lambda})_{\lambda\in\mathbb{R}}$ to
$\mathrm{S}({\mathfrak{g}})$ for $a=0$, (ii) it induces a family of symmetric
${\mathfrak{g}}$-invariant star products on the coadjoint orbits
$\mathcal{O}_{a^{(\pm)}}$ for $a\in\mathbb{R}$, (iii) if $a=0$ and
$\lambda=\frac{n-2}{2n}$, it induces the unique graded
${\mathfrak{g}}$-equivariant star-product on $\mathcal{O}_{00}$, introduced by
Arnal-Benhamor-Cahen [1] and Astashkevich-Brylinski [2].

###### Proof.

The Theorem 4.1 ensures the existence of a ${\mathfrak{g}}$-equivariant
quantization $\Phi_{\lambda}$ of $\mathrm{S}({\mathfrak{g}})$ lifting
${\mathcal{Q}}^{\lambda,\lambda}$ for every $\lambda\in\mathbb{R}$. The lift
property is equivalent to $\Phi^{\lambda}(I)=J^{\lambda}$. We define then the
family of ${\mathfrak{g}}$-equivariant quantizations
$\Phi^{\lambda}_{a}=\Phi^{\lambda}\circ\phi_{a}$, where $\phi_{a}$ is
introduced in Lemma 4.8. It can be chosen such that $\phi_{0}=\mathrm{Id}$, so
(i) is trivially satisfied. The Lemma 4.8 ensures that
$\Phi^{\lambda}_{a}(\mathcal{I}_{a})$ is an ideal and a
${\mathfrak{g}}$-module, hence the ${\mathfrak{g}}$-invariant star product
$\star_{\Phi_{a}^{\lambda}}$ on $\mathrm{S}({\mathfrak{g}})$, induced by
$\Phi_{a}^{\lambda}$, descends on the quotient
$\mathrm{S}({\mathfrak{g}})/\mathcal{I}_{a}$. We recall that
$\star_{\Phi_{a}^{\lambda}}$ is symmetric if $\Phi_{a}^{\lambda}$ satisfies
$\tau\circ\Phi_{a}^{\lambda}=\Phi_{a}^{\lambda}\circ\gamma$. Redefining
$\Phi_{a}^{\lambda}$ by
$\frac{1}{2}(\Phi_{a}^{\lambda}+\tau\circ\Phi_{a}^{\lambda}\circ\gamma)$ this
is trivially the case, and the quantization $\Phi_{0}^{\lambda}$ is still a
lift of ${\mathcal{Q}}^{\lambda,\lambda}$ by uniqueness of the latter. This
proves (ii). The last point follows then from Corollary 4.2, Proposition 4.6
and the uniqueness result in [1, 2]. ∎

######  Remark 4.10.

For two distinct coadjoint orbits, the star products obtained above do not
coincide in general. This is reminiscent to the work of Fioresi and Lledo
[23], dealing with star products tangential to semi-simple coadjoint orbits of
semi-simple Lie groups.

Quantization of a coadjoint orbit of a Lie group is more usually understood as
building an irreducible unitary representation of this group from that orbit.
Since the Hamiltonian $G$-action on
$T^{*}(\mathbb{S}^{p}\times\mathbb{S}^{q})$ preserves its vertical
polarization, the Kirillov’s orbit method [26] applies to the coadjoint orbits
$\mathcal{O}_{0^{\pm}}$. It leads to the unitary representation of $G$ onto
the pre-Hilbert space of compactly supported half-densities on
$\mathbb{S}^{p}\times\mathbb{S}^{q}$, endowed with its canonical Hermitian
product $(\phi,\psi)=\int_{M}\bar{\phi}\psi$. The corresponding infinitesimal
action of the Lie algebra ${\mathfrak{g}}$ by Lie derivative extends to a
deformed representation of the whole Poisson algebra $\mathcal{K}$ of regular
functions on $\mathcal{O}_{0^{\pm}}$, via the conformally equivariant
quantization ${\mathcal{Q}}^{\frac{1}{2},\frac{1}{2}}$.

In contradistinction, the minimal coadjoint orbit cannot be quantize via the
orbit method as it admits no $G$-invariant polarization. This key fact is
proved in [43] and stems from the following result of [8]:
${\mathfrak{g}}=\mathrm{o}(p+1,q+1)$ can be identified to a Lie subalgebra of
polynomial vector fields on $\mathbb{R}^{m}$ iff $m\geq p+q$. The following
proposition shows how conformally equivariant quantization provides a
supplement to the orbit method for the minimal nilpotent orbit.

######  Proposition 4.11.

Let $\lambda=\frac{n-2}{2n}$. The algebra $\mathcal{K}^{1}$ of conformal
Killing tensors on $\mathbb{S}^{p}\times\mathbb{S}^{q}$ identifies to the
algebra of regular functions on the minimal coadjoint orbit
$\mathcal{O}_{00}$. Its conformally equivariant quantization
${\mathcal{Q}}^{\lambda,\lambda}(\mathcal{K}^{1})\simeq\mathfrak{U}({\mathfrak{g}})/J^{\lambda,1}$
is the algebra of higher symmetries of the conformal Laplacian on
$\mathbb{S}^{p}\times\mathbb{S}^{q}$, which acts faithfully on its kernel.
Moreover, $J^{\lambda,1}$ is the Joseph ideal.

###### Proof.

Since the Joseph ideal is a maximal primitive ideal, the action of
${\mathcal{Q}}^{\lambda,\lambda}(\mathcal{K}^{1})\simeq\mathcal{A}^{\lambda,1}$
on the kernel of $\Delta$ is trivial or faithful, hence it is faithful. ∎

In particular, the infinitesimal action of ${\mathfrak{g}}$ on harmonic
functions is given by the Lie derivative of $\frac{n-2}{2n}$-densities and the
kernel of the induced representation of $\mathfrak{U}({\mathfrak{g}})$ is the
Joseph ideal. This result dates back to [5] and was so far the only link
between the minimal coadjoint orbit of $G$ and its minimal representation,
which integrates the one of ${\mathfrak{g}}$ on $\ker\Delta$.

### Acknowledgements

It is a pleasure to acknowledge Christian Duval and Valentin Ovsienko for
their constant interest in this work, and Josef Šilhan for invaluable
discussions.

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