Another Approach to Juhl’s Conformally Covariant Differential Operators

Another Approach to Juhl’s Conformally Covariant
Differential Operators from $\boldsymbol{S^{n}}$ to $\boldsymbol{S^{n-1}}$

Jean-Louis CLERC

J.-L. Clerc

Institut Elie Cartan de Lorraine, Université de Lorraine, France jean-
<EMAIL_ADDRESS>
Received December 07, 2016, in final form April 11, 2017; Published online
April 19, 2017

A family $({\mathbf{D}}_{\lambda})_{\lambda\in\mathbb{C}}$ of differential
operators on the sphere $S^{n}$ is constructed. The operators are conformally
covariant for the action of the subgroup of conformal transformations of
$S^{n}$ which preserve the smaller sphere $S^{n-1}\subset S^{n}$. The family
of conformally covariant differential operators from $S^{n}$ to $S^{n-1}$
introduced by A. Juhl is obtained by composing these operators on $S^{n}$ and
taking restrictions to $S^{n-1}$.

conformally covariant differential operators; Juhl’s covariant differential
operators

58J70; 43A85

##  1 Introduction

Let $S=S^{n}$ be the $n$-dimensional sphere in $\mathbb{R}^{n+1}$ and let
$G={\rm SO}_{0}(1,n+1)$ be (the neutral component of) the group of conformal
transformations of $S$. Let $S^{\prime}\simeq S^{n-1}$ be the subspace of
points of $S$ with vanishing last coordinate ($x_{n}=0$ in our notation) and
let $G^{\prime}\simeq{\rm SO}_{0}(1,n)$ be the conformal group of
$S^{\prime}$, viewed as the subgroup of $G$ which stabilizes $S^{\prime}$. Let
$(\pi_{\lambda})_{\lambda\in\mathbb{C}}$ be the scalar principal series of
representations of $G$ acting on $C^{\infty}(S)$. Denote by
$\pi_{\lambda|G^{\prime}}$ its restriction to $G^{\prime}$. Let
$(\pi^{\prime}_{\mu})_{\mu\in\mathbb{C}}$ be the scalar principal series of
$G^{\prime}$ acting on $C^{\infty}(S^{\prime})$.

In [6] A. Juhl has constructed a family
${\mathcal{D}}_{N}(\lambda)_{\lambda\in\mathbb{C},N\in\mathbb{N}}$ of
differential operators from $C^{\infty}(S)$ into $C^{\infty}(S^{\prime})$,
which are intertwining operators between ${\pi_{\lambda}}_{|G^{\prime}}$ and
$\pi^{\prime}_{\lambda+N}$.111Our $\lambda$ corresponds to $-\lambda$ in
Juhl’s notation. Later, these operators were obtained by T. Kobayashi and B.
Speh in [11] as residues of a meromorphic family of _symmetry breaking
operators_ associated to the restriction problem for the pair
$(G,G^{\prime})$. A third point of view was proposed by T. Kobayashi and M.
Pevzner in [9, 10], based on the $F$-method. Similar operators were recently
constructed for differential forms on spheres [4, 8].

The new approach to Juhl’s operators which I present in this article follows a
method that I used for similar problems, in the context of the restriction
problem for a pair $(G\times G,G^{\prime})$ where $G^{\prime}=G$ embedded
diagonally in $G\times G$. I was influenced by a reminiscence of the _$\Omega$
-process_ which yields both the _transvectants_ and the _Rankin–Cohen
brackets_. These operators may be viewed as covariant bi-differential
operators for the group ${\rm SL}(2,\mathbb{R})$, or symmetry breaking
differential operators from ${\rm SL}(2,\mathbb{R})\times{\rm
SL}(2,\mathbb{R})$ to its diagonal subgroup. For a presentation of these
classical results see Section 5 of [2] for a quick overview or [12] for a
thorough exposition of the transvectants.

The new method was introduced in a collaboration with R. Beckmann for the
conformal group of the sphere (see [1]) and the scalar principal series, then
for $G={\rm SL}(2n,\mathbb{R})$ and the degenerate principal series acting on
the Grasmmannian ${\rm Gr}(n,2n;\mathbb{R})$ (see [2]).

The first step of the method, for the present case, is to introduce the
multiplication by $x_{n}$, viewed as an operator $M$ on $C^{\infty}(S)$. The
operator $M$ is a “universal” $G^{\prime}$-intertwining operator, in the sense
that, for any $\lambda\in\mathbb{C}$, the operator $M$ intertwines
${\pi_{\lambda}}_{|G^{\prime}}$ and ${\pi_{\lambda-1}}_{|G^{\prime}}$. Next
recall the family of Knapp–Stein operators
$(I_{\lambda})_{\lambda\in\mathbb{C}}$ which are $G$-intertwining operators
with respect to $(\pi_{\lambda},\pi_{n-\lambda})$. The operator222For
technical reasons, a normalizing factor is introduced, see (3.4).

$\displaystyle{\mathbf{D}}_{\lambda}=I_{n-\lambda-1}\circ M\circ I_{\lambda}$

obtained by twisting $M$ by the appropriate Knapp–Stein intertwining operators
is clearly an intertwining operator with respect to
$(\pi_{\lambda|G^{\prime}},\pi_{\lambda+1|G^{\prime}})$. Our main result (see
Theorem 3.2) is that ${\mathbf{D}}_{\lambda}$ is a _differential_ operator.
The proof is obtained in the non compact realization of the principal series
(passing from $S^{n}$ to $\mathbb{R}^{n}$ by a conformal map) and uses
Euclidean Fourier transform.

The construction of conformally covariant differential operators from $S^{n}$
to $S^{n-1}$ is now easy. For $N$ a non-negative integer, consider

$\displaystyle{\mathbf{D}}_{N,\lambda}={\mathbf{D}}_{\lambda+N-1}\circ\cdots\circ{\mathbf{D}}_{\lambda+1}\circ{\mathbf{D}}_{\lambda}\qquad\text{or}\qquad{\mathbb{D}}_{N,\lambda}=I_{n-\lambda-N}\circ
M^{N}\circ I_{\lambda}.$

The two families of differential operators on $S$ (which coincide up to a
meromorphic function of $\lambda$) are covariant with respect to
$(\pi_{\lambda|G^{\prime}},\pi_{\lambda+N|G^{\prime}})$. Finally, let

$\displaystyle{\mathbf{D}}_{N}(\lambda)=\operatorname{res}\circ{\mathbf{D}}_{N,\lambda},$

where $\operatorname{res}$ is the restriction map from $C^{\infty}(S)$ to
$C^{\infty}(S^{\prime})$. The operator ${\mathbf{D}}_{N}(\lambda)$ is a
differential operator from $S$ to $S^{\prime}$ which is covariant with respect
to $(\pi_{\lambda|G^{\prime}},\pi^{\prime}_{\lambda+N})$. The family
${\mathbf{D}}_{N}(\lambda)_{\lambda\in\mathbb{C},N\in\mathbb{N}}$ essentially
coincides with Juhl’s family.

The operator $\mathbf{D}_{\lambda}$ has a simple expression in the non compact
picture, see (4.6). It is tempting to find a more direct approach to this
operator. This is achieved in the last section, by using yet another
realization of the principal series, sometimes called the _ambient space_
realization. The way the operator is constructed is much simpler, and it is
then easy to determine its expression in the non compact picture (recovering
the expression of $\mathbf{D}_{\lambda}$ on $\mathbb{R}^{n}$, see Proposition
7.8), but also in the compact realization (see Proposition 7.9), that is to
say as a $G^{\prime}$-conformally covariant differential operator on $S$. Some
generalization of these formulæ in the realm of conformal geometry on a
Riemannian manifold seems plausible.

##  2 The principal series of $\boldsymbol{{\rm SO}_{0}(1,n+1)}$
and the Knapp–Stein intertwining operators

Let $E$ be a Euclidean space of dimension $n+1$, and choose an orthonormal
basis $\\{e_{0},e_{1},\dots,e_{n}\\}$. Let $S=S^{n}$ be the unit sphere of
$E$, i.e.,

$\displaystyle S=\big{\\{}x=(x_{0},x_{1},\dots,x_{n}),\
x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=1\big{\\}}.$

Let $\mathbf{E}$ be the vector space $\mathbb{R}\oplus E$, with the Lorentzian
quadratic form

$\displaystyle
Q(\mathbf{x})=[(t,x),(t,x)]=t^{2}-|x|^{2}\qquad\text{for}\quad\mathbf{x}=(t,x),\quad
t\in\mathbb{R},\quad x\in E.$

For ${\bf x}=(t,x)\in\mathbb{E}$, we let

$\displaystyle t({\bf x})=t,\qquad{\bf x}_{E}=x.$

The space of isotropic lines $\mathcal{S}$ in $\mathbf{E}$ can be identified
with $S$ by the map

$\displaystyle S\ni x\longmapsto
d_{x}=\mathbb{R}(1,x)\in\mathcal{S},\qquad\mathcal{S}\ni d\longmapsto
d\cap\\{t=1\\}.$

Let $G={\rm SO}_{0}(1,n+1)$ be the connected component of the group of
isometries of $\mathbf{E}$. Then $G$ acts on $\mathcal{S}$ and this action can
be transferred to an action on $S$. More explicitly, if
$x=(x_{0},x_{1},\dots,x_{n})\in S$, and $g\in G$, observe that $t(g(1,x))>0$
and define $g(x)\in S$ by

$\displaystyle\big{(}1,g(x)\big{)}=t\big{(}g.(1,x)\big{)}^{-1}g.(1,x).$

Set, for $g\in G$ and $x\in S$

$\displaystyle\kappa(g,x)=t\big{(}g.(1,x)\big{)}^{-1}.$

Clearly $\kappa(g,x)$ is a smooth, strictly positive function on $G\times S$.
Moreover $\kappa(g,x)$ satisfies the _cocycle property_ : for any $g_{1}$,
$g_{2}$ and any $x\in S$,

$\displaystyle\kappa(g_{1}g_{2},x)=\kappa\big{(}g_{1},g_{2}(x)\big{)}\kappa(g_{2},x).$

This action of $G$ on $S$ is known to be _conformal_. For $g\in G$, $x\in S$
and $\xi$ an arbitrary tangent vector to $S$ at $x$

$\displaystyle|Dg(x)\xi|=\kappa(g,x)|\xi|,$

and hence $\kappa(g,x)$ is the _conformal factor_ of $g$ at $x$.

Associated to the action of $G$ on $S$ there is a family of representations on
$C^{\infty}(S)$, which, from the point of view of harmonic analysis is the
_scalar principal series_ of $G$. For $\lambda\in\mathbb{C}$, $g\in G$ and
$f\in C^{\infty}(S)$, let

$\displaystyle\pi_{\lambda}(g)f(x)=\kappa\big{(}g^{-1},x\big{)}^{\lambda}f\big{(}g^{-1}(x)\big{)}.$

The formula defines a (smooth) representation $\pi_{\lambda}$ of $G$ on
$C^{\infty}(S)$.

The Knapp–Stein intertwining operators are a major tool in harmonic analysis
of $G$ (as of any semi-simple Lie group, see, e.g., [7]). For
$\lambda\in\mathbb{C}$ and $f\in C^{\infty}(S)$, let

$\displaystyle
I_{\lambda}f(x)=\frac{1}{\Gamma(\lambda-\frac{n}{2})}\int_{S}|x-y|^{-2n+2\lambda}f(y)dy,$
(2.1)

where $dy$ stands for the Lebesgue measure on $S$ induced by the Euclidean
structure. For $\operatorname{Re}\lambda>\frac{n}{2}$, this formula defines a
continuous operator $I_{\lambda}$ on $C^{\infty}(S)$.

######  Proposition 2.1.

  1. $i)$

The definition (2.1) can be analytically continued in $\lambda$ to all of
$\mathbb{C}$.

  2. $ii)$

The analytic continuation yields a holomorphic family of operators
$I_{\lambda}$ on $C^{\infty}(S)$, which satisfy the intertwining relation

$\displaystyle\forall\,g\in G,\qquad
I_{\lambda}\circ\pi_{\lambda}(g)=\pi_{n-\lambda}(g)\circ I_{\lambda}.$ (2.2)

The following complementary result will be needed later.

######  Proposition 2.2.

For any $\lambda\in\mathbb{C}$

$\displaystyle I_{\lambda}\circ
I_{n-\lambda}=\frac{\pi^{n}}{\Gamma(\lambda)\Gamma(n-\lambda)}\operatorname{id}.$
(2.3)

The next result corresponds to reducibility points for the scalar principal
series. Let $\mathcal{P}(S)$ be the space of restrictions to $S$ of polynomial
functions on $E$, and for $k\in\mathbb{N}$, let $\mathcal{P}_{k}$ be the space
of restrictions to $S$ of polynomials on $E$ of degree $\leq k$. Finally, let
$\mathcal{P}_{k}^{\perp}$ be the subspace of $C^{\infty}(S)$ given by

$\displaystyle\mathcal{P}_{k}^{\perp}=\left\\{f\in
C^{\infty}(S),\,\int_{S}f(x)p(x)dx=0,\ \text{for any}\
p\in\mathcal{P}_{k}\right\\}.$

.

######  Proposition 2.3.

  1. $i)$

Let $\lambda=n+k,k\in\mathbb{N}$. Then

$\displaystyle\operatorname{Im}(I_{n+k})=\mathcal{P}_{k},\qquad\operatorname{Ker}(I_{n+k})=\mathcal{P}_{k}^{\perp}.$
(2.4)

  2. $ii)$

Let $\lambda=-k,k\in\mathbb{N}$. Then

$\displaystyle\operatorname{Ker}(I_{-k})=\mathcal{P}_{k},\qquad\operatorname{Im}(I_{-k})=\mathcal{P}_{k}^{\perp}.$
(2.5)

##  3 Construction of the family
$\boldsymbol{\widetilde{\mathbf{D}}_{\lambda}}$,
$\boldsymbol{\lambda\in\mathbb{C}}$

Now let $E^{\prime}=\\{x\in E,\,x_{n}=0\\}$ and $S^{\prime}=S\cap E^{\prime}$.
Then $S^{\prime}$ is an $(n-1)$-dimensional sphere. Let $G^{\prime}$ be the
subgroup of elements of $G$ of the form

$\displaystyle g=\begin{pmatrix}&&&0\\\ &g^{\prime}&&\vdots\\\ &&&0\\\
0&\dots&0&1\end{pmatrix},\qquad g^{\prime}\in{\rm SO}_{0}(1,n).$

Clearly, $G^{\prime}$ is a subgroup of $G$, isomorphic to ${\rm SO}_{0}(1,n)$.
Elements of $G^{\prime}$ preserve the hyperplane $\\{x_{n}=0\\}$ in
$\mathbf{E}$ and hence the action of $G^{\prime}$ on $S$ preserves
$S^{\prime}$.

For $x\in E$, write $x=(x^{\prime},x_{n})$, with
$x^{\prime}\in\mathbb{R}^{n}$. For $g\in G^{\prime}$,

$\displaystyle
g(1,x)=g(1,x^{\prime},x_{n})=\big{(}g^{\prime}.(1,x^{\prime}),x_{n}\big{)}.$

If $x\in S$, the last equation can be rewritten as

$\displaystyle\kappa(g,x)^{-1}\big{(}1,g(x)\big{)}=\big{(}g^{\prime}.(1,x^{\prime}),x_{n}\big{)},$

so that

$\displaystyle g(x)_{n}=\kappa(g,x)x_{n}.$ (3.1)

In the sequel, the distinction between $g$ and $g^{\prime}$ in the notation is
abandoned, the context providing the correct interpretation.

Let $M$ be the operator defined on $C^{\infty}(S)$ by

$\displaystyle Mf(x)=x_{n}f(x),\qquad f\in C^{\infty}(S).$

######  Proposition 3.1.

The operator $M$ satisfies

$\displaystyle\forall\,g\in G^{\prime}\qquad
M\circ\pi_{\lambda}(g)=\pi_{\lambda-1}(g)\circ M.$ (3.2)

###### Proof.

This is an immediate consequence of (3.1). ∎

Next let ${\mathbf{D}}_{\lambda}$ be the operator on $C^{\infty}(S)$ defined
by

$\displaystyle{\mathbf{D}}_{\lambda}=I_{n-\lambda-1}\circ M\circ I_{\lambda},$

which corresponds to the following diagram

$\displaystyle\begin{CD}C^{\infty}(S)@>{{\mathbf{D}}_{\lambda}}>{}>C^{\infty}(S)\\\
@V{}V{I_{\lambda}}V@A{}A{I_{n-\lambda-1}}A\\\
C^{\infty}(S)@>{M}>{}>C^{\infty}(S).\end{CD}$

As a consequence of the intertwining property of the Knapp–Stein operators
(2.2) and Proposition 3.1, ${\mathbf{D}}_{\lambda}$ satisfies for $g\in
G^{\prime}$

$\displaystyle{\mathbf{D}}_{\lambda}\circ\pi_{\lambda}(g)=\pi_{\lambda+1}(g)\circ{\mathbf{D}}_{\lambda}.$
(3.3)

Otherwise said, the operator ${\mathbf{D}}_{\lambda}$ is covariant with
respect to $({\pi_{\lambda}}_{|G^{\prime}},{\pi_{\lambda+1}}_{|G^{\prime}})$.

######  Theorem 3.2.

The operator ${\mathbf{D}}_{\lambda}$ is a differential operator on $S$.

The proof of Theorem 3.2 will be given at the end the next section.

######  Proposition 3.3.

Let $\lambda\in(n+\mathbb{N})\cup(-1-\mathbb{N})$. Then
${\mathbf{D}}_{\lambda}=0$.

###### Proof.

Let first $\lambda=n+k$ for some $k\in\mathbb{N}$. Then $I_{\lambda}=I_{n+k}$,
and by (2.4) $\operatorname{Im}(I_{\lambda})=\mathcal{P}_{k}$. Next
$\operatorname{Im}(M\circ I_{\lambda})\subset\mathcal{P}_{k+1}$. Now
$I_{n-\lambda-1}=I_{-k-1}$ and using (2.5), $I_{n-\lambda-1}\circ M\circ
I_{\lambda}=0$.

Now let $\lambda=-k$, with $k\geq 1$. Then $I_{\lambda}=I_{-k}$ and by (2.5),
$\operatorname{Im}(I_{\lambda})=\mathcal{P}_{k}^{\perp}$. Next
$\operatorname{Im}(M\circ
I_{\lambda})\subset\mathcal{P}_{1}\mathcal{P}_{k}^{\perp}\subset\mathcal{P}_{k-1}^{\perp}$.
Now $I_{n-\lambda-1}=I_{n+k-1}$ which using (2.4) implies
$I_{n-\lambda-1}\circ M\circ I_{\lambda}=0$. ∎

To compensate for these zeroes of ${\mathbf{D}}_{\lambda}$, introduce

$\displaystyle\widetilde{\mathbf{D}}_{\lambda}=\Gamma(\lambda+1)\Gamma(n-\lambda){\mathbf{D}}_{\lambda}$
(3.4)

for $\lambda\notin(n+\mathbb{N})\cup(-1-\mathbb{N})$ and extend continuously
to all of $\mathbb{C}$ to get a holomorphic family
${(\widetilde{\mathbf{D}}_{\lambda})}_{\lambda\in\mathbb{C}}$ of differential
operators on $S$ covariant with respect to
$(\pi_{\lambda|G^{\prime}},\pi_{\lambda+1|G^{\prime}})$.

##  4 The expression of $\boldsymbol{{\widetilde{\mathbf{D}}}_{\lambda}}$ in
the non-compact picture

Consider the point $-{\mathbf{1}}=(-1,0,\dots,0)\in S$. The stereographic
projection with source at $-{\mathbf{1}}$ provides a diffeomorphism from
$S\setminus\\{-\mathbf{1}\\}$ onto the hyperplane $\\{x_{n}=1\\}$. The inverse
map (up to a scaling by a factor 2) $c\colon\mathbb{R}^{n}\longrightarrow S$
is given by

$\displaystyle
c(\xi)=\begin{pmatrix}\dfrac{1-|\xi|^{2}}{1+|\xi|^{2}}\vspace{1mm}\\\
\dfrac{2\xi_{1}}{1+|\xi|^{2}}\\\ \vdots\\\
\dfrac{2\xi_{n}}{1+|\xi|^{2}}\end{pmatrix}.$ (4.1)

When using this local chart on $S$, we refer to the _non-compact picture_ , as
a reference to semi-simple harmonic analysis.

Geometric considerations (or an elementary computation) show that, for
$\xi,\eta\in\mathbb{R}^{n}$

$\displaystyle|c(\xi)-c(\eta)|^{2}=\kappa(c,\xi)|\xi-\eta|^{2}\kappa(c,\eta),$

where, for $\xi\in\mathbb{R}^{n}$, we set

$\displaystyle\kappa(c,\xi)=2\big{(}1+|\xi|^{2}\big{)}^{-1}.$

There is an infinitesimal version of this result, namely

$\displaystyle|Dc(\xi)\eta|=\kappa(c,\xi)|\eta|$

for $\xi,\eta\in\mathbb{R}^{n}$. This last statement shows that $c$ is
conformal from $\mathbb{R}^{n}$ with its standard Euclidean structure into
$S$.

The action of $g$ on $S$ can be transferred as a (rational) action of $G$ on
$\mathbb{R}^{n}$, namely $c^{-1}\circ g\circ c$. For notational convenience,
we still denote this action on $\mathbb{R}^{n}$ by $(g,\xi)\longmapsto
g(\xi)$, $g\in G$, $\xi\in\mathbb{R}^{n}$. As the map $c$ is conformal, the
transferred action of $G$ on $\mathbb{R}^{n}$ is still conformal. For $g\in G$
defined at $\xi\in\mathbb{R}^{n}$, we let $\kappa(g,\xi)$ be the corresponding
conformal factor of $g$ at $\xi$.

Let $\lambda\in\mathbb{C}$. For $f\in C^{\infty}(S)$ let $C_{\lambda}(f)$ be
defined by

$\displaystyle
C_{\lambda}(f)(\xi)=\kappa(c,\xi)^{\lambda}f(c(\xi)),\qquad\xi\in\mathbb{R}^{n}$

and let $\mathcal{H}_{\lambda}$ be the image of $C_{\lambda}$. It is easily
proved that

$\displaystyle\mathcal{S}\big{(}\mathbb{R}^{n}\big{)}\subset\mathcal{H}_{\lambda}\subset\mathcal{S}^{\prime}\big{(}\mathbb{R}^{n}\big{)},$

where $\mathcal{S}(\mathbb{R}^{n})$ stands for the Schwartz space on
$\mathbb{R}^{n}$ and $\mathcal{S}^{\prime}(\mathbb{R}^{n})$ for its dual, the
space of tempered distributions.

The representation $\pi_{\lambda}$ can be transferred in the non-compact
model, using $C_{\lambda}$ as intertwining map, i.e., set

$\displaystyle\rho_{\lambda}(g)=C_{\lambda}\circ\pi_{\lambda}(g)\circ
C_{\lambda}^{-1}.$

Using the cocycle property of $\kappa$, $\rho_{\lambda}$ can be realized as

$\displaystyle\rho_{\lambda}(g)f(\xi)=\kappa\big{(}g^{-1},\xi\big{)}^{\lambda}f\big{(}g^{-1}(\xi)\big{)},$

where $f\in\mathcal{H}_{\lambda}$ and $g\in G$.

Similarly, the Knapp–Stein operators can be transferred to the non-compact
picture. For $s\in\mathbb{C}$, consider the expression

$\displaystyle
h_{s}(\xi)=\frac{1}{\Gamma\big{(}\frac{n}{2}+\frac{s}{2}\big{)}}|\xi|^{s},\qquad\xi\in\mathbb{R}^{n}.$

For $\operatorname{Re}(s)>-n$, $h_{s}$ is locally summable with moderate
growth at infinity, hence defines a tempered distribution. The
($\mathcal{S}^{\prime}(\mathbb{R}^{n})$-valued) function $s\mapsto h_{s}$ can
be extended by analytic continuation to $\mathbb{C}$ and the $\Gamma$ factor
in the definition of $h_{s}$ is so chosen that it extends as an _entire_
function with values in $\mathcal{S}^{\prime}(\mathbb{R}^{n})$ (for more
details see, e.g., [5]).

For $\lambda\in\mathbb{C}$, the Knapp–Stein operator $J_{\lambda}$ is given by

$\displaystyle J_{\lambda}f=h_{-2n+2\lambda}\star f,$

or more concretely

$\displaystyle
J_{\lambda}f(\xi)=\frac{1}{\Gamma\big{(}\lambda-\frac{n}{2}\big{)}}\int_{\mathbb{R}^{n}}|\xi-\eta|^{-2n+2\lambda}f(\eta)d\eta.$

As for any $s\in\mathbb{C}$ $h_{s}$ is a tempered distribution, $J_{\lambda}$
maps $\mathcal{S}(\mathbb{R}^{n})$ into
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$.

######  Proposition 4.1.

Let $\lambda\in\mathbb{C}$. Then for $f\in\mathcal{S}(\mathbb{R}^{n})$

$\displaystyle J_{\lambda}f=\big{(}C_{n-\lambda}\circ I_{\lambda}\circ
C_{\lambda}^{-1}\big{)}f.$

###### Proof.

As
$\mathcal{S}(\mathbb{R}^{n})\subset\mathcal{H}_{\lambda}\subset\mathcal{S}^{\prime}(\mathbb{R}^{n})$,
both sides are well-defined and belong to
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$. For
$\operatorname{Re}\lambda>\frac{n}{2}$, both sides are given by convergent
integrals, and the equality is proved by a change of variable. The general
case follows by analytic continuation. The intertwining property of the
Knapp–Stein operators can be formulated in the following way. ∎

######  Proposition 4.2.

Let $f\in C_{c}^{\infty}(S)$ and let $g\in G$ such that $g^{-1}$ is defined on
$\operatorname{Supp}(f)$. Then

$\displaystyle
J_{\lambda}\big{(}\rho_{\lambda}(g)f\big{)}=\rho_{n-\lambda}(g)\big{(}J_{\lambda}f\big{)},$

where the two sides of the equation are viewed as tempered distributions on
$\mathbb{R}^{n}$.

###### Proof.

The condition implies that both $f$ and $\rho_{\lambda}(g)f$ are contained in
$\mathcal{S}(\mathbb{R}^{n})$. Hence

$\displaystyle
J_{\lambda}\big{(}\rho_{\lambda}(g)f\big{)}=\big{(}C_{n-\lambda}\circ
I_{\lambda}\circ C_{\lambda}^{-1}\big{)}\big{(}\rho_{\lambda}(g)f\big{)}$
$\displaystyle\hphantom{J_{\lambda}\big{(}\rho_{\lambda}(g)f\big{)}}{}=(C_{n-\lambda}\circ
I_{\lambda})\big{(}\pi_{\lambda}(g)C_{\lambda}^{-1}f\big{)}=\big{(}C_{n-\lambda}\circ\pi_{n-\lambda}(g)\big{)}\circ\big{(}I_{\lambda}\circ
C_{\lambda}^{-1}\big{)}f$
$\displaystyle\hphantom{J_{\lambda}\big{(}\rho_{\lambda}(g)f\big{)}}{}=\rho_{n-\lambda}(g)\circ\big{(}C_{n-\lambda}\circ
I_{\lambda}\circ C_{\lambda}^{-1}\big{)}f=\rho_{n-\lambda}(g)(J_{\lambda}f).$
∎

The following formulæ will be needed in the sequel

$\displaystyle|\xi|^{2}h_{s}(\xi)=\frac{n+s}{2}h_{s+2}(\xi),$ (4.2)
$\displaystyle\frac{\partial}{\partial\xi_{n}}h_{s}(\xi)=\frac{2s}{n+s-2}\xi_{n}h_{s-2}(\xi),$
(4.3)

where at $s=-n+2$, the last formula has to be understood by analytic
continuation.

As the pole of the stereographic projection has been chosen in $S^{\prime}$,
the map $c$ maps the hyperplane $\\{\xi_{n}=0\\}$ into $S^{\prime}$. It allows
to transfer the map $M$ to the non-compact picture.

######  Lemma 4.3.

Let $g\in G^{\prime}$ be defined at $\xi\in\mathbb{R}^{n}$. Then

$\displaystyle g(\xi)_{n}=\kappa(g,\xi)\xi_{n}.$ (4.4)

###### Proof.

Let $\xi\in\mathbb{R}^{n}$ and let $x=c(\xi)\in S\setminus\\{-\mathbf{1}\\}$.
Then

$\displaystyle c(\xi)_{n}=\kappa(c,\xi)\xi_{n},\qquad
g(x)_{n}=\kappa(g,x)x_{n},\qquad c^{-1}(x)=\kappa\big{(}c^{-1},x\big{)}x_{n}$

the first equality by (4.1), the second by (3.1), and the third also by (4.1).
As $\kappa$ satisfies a cocycle relation, we get

$\displaystyle\big{(}\big{(}c^{-1}\circ g\circ
c\big{)}(\xi)\big{)}_{n}=\kappa\big{(}c^{-1}\circ g\circ c,\xi\big{)}\xi_{n},$

which gives (4.4). ∎

######  Lemma 4.4.

Let $\lambda\in\mathbb{C}$ and $f\in C^{\infty}(S)$. Then

$\displaystyle
C_{\lambda-1}(Mf)(\xi)=\xi_{n}C_{\lambda}(f)(\xi),\qquad\xi\in\mathbb{R}^{n}.$

###### Proof.

Let $\xi=(\xi^{\prime},\xi_{n})$. By (4.1), $c(\xi)_{n}=\kappa(c,\xi)\xi_{n}$,
so that

$\displaystyle
C_{\lambda-1}(Mf)(\xi)=\kappa(c,\xi)^{\lambda-1}Mf\big{(}c(\xi)\big{)}=\kappa(c,\xi)^{\lambda}\xi_{n}f\big{(}c(\xi)\big{)}=\xi_{n}C_{\lambda}(f)(\xi).$
∎

Abusing notation, $M$ will be used for the operator (on
$C^{\infty}(\mathbb{R}^{n})$ say) of multiplication by $\xi_{n}$. The operator
$M$ maps $\mathcal{S}(\mathbb{R}^{n})$ (resp.
$\mathcal{S}^{\prime}(\mathbb{R}^{n}))$ into $\mathcal{S}(\mathbb{R}^{n})$
(resp. $\mathcal{S}^{\prime}(\mathbb{R}^{n}))$, and for any
$\lambda\in\mathbb{C}$, the operator $M$ maps $\mathcal{H}_{\lambda}$ into
$\mathcal{H}_{\lambda-1}$ (Lemma 4.4).

######  Proposition 4.5.

Let $\lambda\in\mathbb{C}$. The operator
$M\colon\mathcal{H}_{\lambda}\longrightarrow\mathcal{H}_{\lambda-1}$ satisfies

$\displaystyle\forall\,g\in G^{\prime},\qquad
M\circ\rho_{\lambda}(g)=\rho_{\lambda-1}(g)\circ M.$

Otherwise said, the operator $M$ intertwines the representations
${\pi_{\lambda}}_{|G^{\prime}}$ and ${\pi_{\lambda-1}}_{|G^{\prime}}$.

###### Proof.

Let $f\in\mathcal{H}_{\lambda}$. Then

$\displaystyle\big{(}M\circ\rho_{\lambda}(g)\big{)}f(\xi)=\xi_{n}\kappa\big{(}g^{-1},\xi\big{)}^{\lambda}f\big{(}g^{-1}(\xi)\big{)}=\kappa\big{(}g^{-1},\xi\big{)}^{\lambda-1}\big{(}g^{-1}(\xi)\big{)}_{n}f\big{(}g^{-1}(\xi)\big{)}$
$\displaystyle\hphantom{\big{(}M\circ\rho_{\lambda}(g)\big{)}f(\xi)}{}=\rho_{\lambda-1}(g)(Mf)\big{(}g^{-1}(\xi)\big{)}$

and the statement follows. ∎

Having introduced the non-compact version of the main ingredients, we observe
that the Knapp–Stein operators are convolution operators, whereas $M$ is the
multiplication by an elementary polynomial. So the Fourier transform is well-
fitted for computations in this context. Define the Fourier transform on
$\mathbb{R}^{n}$ as usual by

$\displaystyle\widehat{f}(\eta)=\int_{\mathbb{R}^{n}}e^{i\langle\eta,\xi\rangle}f(\xi)d\xi$

initially for functions in $\mathcal{S}(\mathbb{R}^{n})$ and extend by duality
to $\mathcal{S}^{\prime}(\mathbb{R}^{n})$.

The Fourier transform of $h_{s}$ is given by

$\displaystyle\widehat{h}_{s}=2^{n+s}\pi^{\frac{n}{2}}h_{-n-s}.$

For this result see, e.g., [5].

Thanks to the above observations, it is possible to define the composition
$M\circ J_{\lambda}$ as an operator from $\mathcal{S}(\mathbb{R}^{n})$ into
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$.

######  Lemma 4.6.

For $f\in\mathcal{S}(\mathbb{R}^{n})$,

$\displaystyle\big{(}(M\circ J_{\lambda})f\big{)}^{\widehat{\
}}(\eta)=-i\pi^{\frac{n}{2}}2^{-n+2\lambda}\left(h_{n-2\lambda}(\eta)\frac{\partial\widehat{f}}{\partial\eta_{n}}(\eta)+\frac{n-2\lambda}{n-\lambda-1}\eta_{n}h_{n-2-2\lambda}(\eta)\widehat{f}(\eta)\right).\\!\\!\\!\\!$
(4.5)

###### Proof.

As observed earlier, the Knapp–Stein operator $J_{\lambda}$ is a convolution
operator on $\mathbb{R}^{n}$, so that

$\displaystyle(J_{\lambda}f)^{\widehat{\
}}(\eta)=\widehat{h}_{-2n+2\lambda}(\eta)\widehat{f}(\eta)=2^{-n+2\lambda}\pi^{\frac{n}{2}}h_{n-2\lambda}(\eta)\widehat{f}(\eta).$

Next, for any distribution $\varphi\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$

$\displaystyle\widehat{M\varphi}=-i\frac{\partial}{\partial\eta_{n}}\widehat{\varphi}$

and (4.5) follows, using (4.3). ∎

The composition $J_{n-\lambda-1}\circ M\circ J_{\lambda}$ is not well-defined
on $\mathcal{S}(\mathbb{R}^{n})$. However, a formal computation (using again
Fourier transforms) can be made and leads to a differential operator, which is
at the origin of the definition (4.6) below. In order to give a rigorous
argument, it is necessary to follow an indirect route.

For $\lambda\in\mathbb{C}$, let $E_{\lambda}$ be the differential operator on
$\mathbb{R}^{n}$ defined by

$\displaystyle
E_{\lambda}=(2\lambda-n+2)\frac{\partial}{\partial\xi_{n}}+\xi_{n}\Delta,$
(4.6)

where $\Delta=\sum\limits_{j=1}^{n}\frac{\partial^{2}}{\partial\xi_{j}^{2}}$
is the usual Laplacian on $\mathbb{R}^{n}$. Notice that the operator
$E_{\lambda}$ maps $\mathcal{S}(\mathbb{R}^{n})$ (resp.
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$) into $\mathcal{S}(\mathbb{R}^{n})$
(resp. $\mathcal{S}^{\prime}(\mathbb{R}^{n})$), so that we may consider the
composition $J_{\lambda+1}\circ E_{\lambda}$.

######  Lemma 4.7.

For $f\in\mathcal{S}(\mathbb{R}^{n})$,

$\displaystyle\big{(}(J_{\lambda+1}\circ E_{\lambda})f\big{)}^{\widehat{\
}}(\eta)$
$\displaystyle{}=-i2^{-n+2+2\lambda}\pi^{\frac{n}{2}}\left((\lambda-n+1)h_{n-2\lambda}(\eta)\frac{\partial\widehat{f}}{\partial\eta_{n}}(\eta)+(2\lambda-n)\eta_{n}h_{-2\lambda+n-2}(\eta)\widehat{f}(\eta)\right).$
(4.7)

###### Proof.

Using (4.2) and (4.3),

$\displaystyle\big{(}E_{\lambda}f\big{)}^{\widehat{\
}}(\eta)=(-i)(2\lambda-n+2)\eta_{n}\widehat{f}(\eta)+(-i)\frac{\partial}{\partial\eta_{n}}\big{(}{-}|\eta|^{2}\widehat{f}(\eta)\big{)}$
$\displaystyle\hphantom{\big{(}E_{\lambda}f\big{)}^{\widehat{\
}}(\eta)}{}=(-i)\left((2\lambda-n)\eta_{n}\widehat{f}(\eta)-|\eta|^{2}\frac{\partial\widehat{f}}{\partial\eta_{n}}(\eta)\right).$

Next

$\displaystyle\big{(}(J_{\lambda+1}\circ E_{\lambda})f\big{)}^{\widehat{\
}}(\eta)=\widehat{h}_{-2n+2\lambda+2}(\eta)(E_{\lambda}f)^{\widehat{\
}}(\eta)$
$\displaystyle{}=2^{-n+2+2\lambda}\pi^{\frac{n}{2}}(-i)\left((2\lambda-n)\eta_{n}h_{n-2-2\lambda}(\eta)\widehat{f}(\eta)-(-\lambda+n-1)h_{n-2\lambda}(\eta)\frac{\partial\widehat{f}}{\partial\eta_{n}}(\eta)\right).$
∎

Comparison of (4.5) and (4.7) yields the next result.

######  Proposition 4.8.

$\displaystyle M\circ J_{\lambda}=\frac{1}{4(\lambda-n+1)}J_{\lambda+1}\circ
E_{\lambda}.$ (4.8)

######  Remark 4.9.

This equality has to be understood as an equality of operators from
$\mathcal{S}(\mathbb{R}^{n})$ into $\mathcal{S}^{\prime}(\mathbb{R}^{n})$. For
$\lambda=n-1$, $J_{\lambda+1}=J_{n}$ is equal (up to a constant $\neq 0$) to
the operator $f\longmapsto\left(\int_{\mathbb{R}^{n}}f(\xi)d\xi\right)1$. Now
for $f\in\mathcal{S}(\mathbb{R}^{n})$,
$\int_{\mathbb{R}^{n}}E_{\lambda}f(\xi)d\xi=0$ as is easily seen by
integration by parts. Hence, $J_{\lambda+1}\circ E_{\lambda}$ vanishes for
$\lambda=n-1$, so that (4.8) has to be interpreted as a residue formula.

######  Proposition 4.10.

Let $f\in C^{\infty}_{c}(\mathbb{R}^{n})$ and assume that $g\in G^{\prime}$ is
such that $g^{-1}$ is defined on $\operatorname{Supp}(f)$. Then

$\displaystyle\big{(}E_{\lambda}\circ\rho_{\lambda}(g)\big{)}f=\big{(}\rho_{\lambda+1}(g)\circ
E_{\lambda}\big{)}f.$

###### Proof.

As a consequence of the intertwining property of $J_{\lambda}$ (Proposition
4.2) and of $M$ (Proposition 3.1),

$\displaystyle(M\circ
J_{\lambda})\big{(}\rho_{\lambda}(g)f\big{)}=\rho_{n-\lambda-1}(g)(M\circ
J_{\lambda})f.$

Hence, by (4.8) (assuming for a while that $\lambda\neq n-1$)

$\displaystyle\big{(}J_{\lambda+1}\circ
E_{\lambda}\big{)}\rho_{\lambda}(g)f=\rho_{n-\lambda-1}(g)\big{(}(J_{\lambda+1}\circ
E_{\lambda})f\big{)}.$

Now $\operatorname{Supp}(E_{\lambda}f)\subset\operatorname{Supp}(f)$, so that
$g^{-1}$ is defined on $\operatorname{Supp}(E_{\lambda}f)$. Hence, by
Proposition 4.2

$\displaystyle\big{(}\rho_{n-\lambda-1}(g)\circ
J_{\lambda+1}\big{)}E_{\lambda}f=\big{(}J_{\lambda+1}\circ\rho_{\lambda+1}(g)\big{)}E_{\lambda}f,$

so that

$\displaystyle
J_{\lambda+1}\big{(}(E_{\lambda}\circ\rho_{\lambda}(g))f\big{)}=J_{\lambda+1}\big{(}(\rho_{\lambda+1}(g)\circ
E_{\lambda})f\big{)}.$

Now, for $\lambda$ generic, the operator $J_{\lambda+1}$ is injective on
$\mathcal{S}(\mathbb{R}^{n})$, hence

$\displaystyle E_{\lambda}\circ\rho_{\lambda}(g)f=\rho_{\lambda+1}(g)\circ
E_{\lambda}f.$

The general result follows by continuity, as the family $E_{\lambda}$ depends
holomorphically on $\lambda$. ∎

###### Proof of Theorem 3.2.

The covariance property of the differential operator $E_{\lambda}$ allows to
construct a _global_ differential operator on $S$ which is expressed in the
non-compact picture to $E_{\lambda}$. In fact to fully cover the sphere $S$,
we only need another chart, which can be chosen as the analog of the map $c$
but constructed from the stereographic projection corresponding to the pole
$\mathbf{1}=(1,0,\dots,0)$ instead of $-{\mathbf{1}}$. Consider the element
$s$ of $G$ given by

$\displaystyle s=\begin{pmatrix}1&0&0&0&\dots&0\\\ 0&-1&0&0&\dots&0\\\
0&0&-1&0&\dots&0\\\ 0&0&0&1&&0\\\ \vdots&\vdots&\vdots&&\ddots&\vdots\\\
0&0&0&0&\dots&1\end{pmatrix}.$

Then $s$ acts on $S$ by

$\displaystyle s(x)=s\left(\begin{matrix}x_{0}\\\ x_{1}\\\ x_{2}\\\ \vdots\\\
x_{n}\end{matrix}\right)=\left(\begin{matrix}-x_{0}\\\ -x_{1}\\\ x_{2}\\\
\vdots\\\ x_{n}\end{matrix}\right).$

In particular, $s$ maps $-\mathbf{1}$ to $\mathbf{1}$ and preserves
$S^{\prime}$. In the non-compact picture, the map $s$ is defined for $\xi\neq
0$ and is given by

$\displaystyle(\xi_{1},\xi_{2},\dots,\xi_{n})\longmapsto\left(-\frac{\xi_{1}}{|\xi|^{2}},\frac{\xi_{2}}{|\xi|^{2}},\dots,\frac{\xi_{n}}{|\xi|^{2}}\right).$
(4.9)

The two charts $\xi\longmapsto c(\xi)$ and $\xi\longmapsto
s\big{(}c(\xi)\big{)}$ cover $S$. Their common domain corresponds to $\xi\neq
0$, the change of chart being given by (4.9), which is the local expression in
the non-compact picture of the transform $s$. So Proposition 4.10, when
applied to $g=s$ is exactly what is needed to prove that there is a global
differential operator $\mathbf{E}_{\lambda}$ on $S$ which is expressed by
$E_{\lambda}$ in the non-compact model. Clearly $\mathbf{E}_{\lambda}$
satisfies

$\displaystyle\forall\,g\in
G^{\prime},\qquad\mathbf{E}_{\lambda}\circ\pi_{\lambda}(g)=\pi_{\lambda+1}(g)\circ\mathbf{E}_{\lambda}.$

By (4.8),

$\displaystyle M\circ
I_{\lambda}=\frac{1}{4(\lambda-n+1)}I_{\lambda+1}(g)\circ\mathbf{E}_{\lambda}.$

Compose both sides with $I_{n-\lambda-1}$ and use (2.3) to get

$\displaystyle{\mathbf{D}}_{\lambda}=\frac{\pi^{-n}}{4(\lambda-n+1)\Gamma(n-\lambda-1)\Gamma(\lambda+1)}\mathbf{E}_{\lambda}$

or equivalently

$\displaystyle\widetilde{\mathbf{D}}_{\lambda}=-\frac{1}{4\pi^{n}}\mathbf{E}_{\lambda}.$

This relation implies in particular that $\widetilde{\mathbf{D}}_{\lambda}$ is
a differential operator on $S$. ∎

##  5 The families $\boldsymbol{{\mathbf{D}}_{\lambda,N}}$,
$\boldsymbol{\widetilde{\mathbf{D}}_{\lambda,N}}$ and
$\boldsymbol{{\mathbf{D}}_{N}(\lambda)}$

For $N\geq 1$, set

$\displaystyle\widetilde{\mathbf{D}}_{\lambda,N}=\widetilde{\mathbf{D}}_{\lambda+N-1}\circ\cdots\circ\widetilde{\mathbf{D}}_{\lambda}.$

Let $M^{N}$ be the operator on $C^{\infty}(S)$ given by multiplication by
$x_{n}^{N}$. Set

$\displaystyle{\mathbb{D}}_{\lambda,N}=I_{n-N-\lambda}\circ M^{N}\circ
I_{\lambda}.$

######  Proposition 5.1.

  1. $i)$

$\widetilde{\mathbf{D}}_{\lambda,N}$ and ${\mathbb{D}}_{\lambda,N}$ are
differential operators on $S$ which intertwine $\pi_{\lambda|G^{\prime}}$ and
$\pi_{\lambda+N|G^{\prime}}$.

  2. $ii)$

$\displaystyle\widetilde{\mathbf{D}}_{\lambda,N}=\pi^{n(N-1)}\Gamma(\lambda+N)\Gamma(n-\lambda-N){\mathbb{D}}_{\lambda,N}.$
(5.1)

###### Proof.

Repeated uses of (3.2) show that, for any $\mu\in\mathbb{C}$, $M^{N}$
intertwines ${\pi_{\mu}}_{|G^{\prime}}$ and ${\pi_{\mu-N}}_{|G^{\prime}}$.
Hence ${\mathbb{D}}_{\lambda,N}$ intertwines ${\pi_{\lambda}}_{|G^{\prime}}$
and ${\pi_{\lambda+N}}_{|G^{\prime}}$. On the other hand, repeated uses of
(3.3) proves that $\widetilde{\mathbf{D}}_{\lambda,N}$ also intertwines
${\pi_{\lambda}}_{|G^{\prime}}$ and ${\pi_{\lambda+N}}_{|G^{\prime}}$.

Next, $\widetilde{\mathbf{D}}_{\lambda,N}$ as a composition of differential
operators on $S$ is a differential operator. So it remains to prove (5.1).

Substitute ${\mathbf{D}}_{\lambda+j}=I_{n-\lambda-j-1}\circ M\circ
I_{\lambda+j}$ for $0\leq j\leq N-1$ to get

$\displaystyle{\mathbf{D}}_{\lambda+N-1}\circ{\mathbf{D}}_{\lambda+N-2}\circ\dots\circ{\mathbf{D}}_{\lambda}$
$\displaystyle\qquad{}=I_{-\lambda+n-N}\circ\dots\circ I_{-\lambda+n-j-1}\circ
M\circ I_{\lambda+j}\circ I_{-\lambda+n-j}\circ M\circ
I_{\lambda+j-1}\circ\dots\circ I_{\lambda},$

and use (2.3) repeatedly for $\lambda+j$ to obtain

$\displaystyle{\mathbf{D}}_{\lambda+N-1}\circ{\mathbf{D}}_{\lambda+N-2}\circ\dots\circ{\mathbf{D}}_{\lambda}$
$\displaystyle\qquad{}=\pi^{n(N-1)}\left(\prod_{j=1}^{N-1}\Gamma(\lambda+j)\Gamma(n-\lambda-j)\right)^{-1}I_{-\lambda+n-N}\circ
M^{N}\circ I_{\lambda}.$

Multiply by the appropriate $\Gamma$ factors coming from (3.4) to get the
formula. ∎

The group $G^{\prime}$ acts conformally on $S^{\prime}$. The scalar principal
series for $G^{\prime}\simeq{\rm SO}_{0}(1,n)$ is defined as follows: for
$\mu\in\mathbb{C}$, for $g\in G^{\prime}$ and $f\in C^{\infty}(S^{\prime})$,

$\displaystyle\pi^{\prime}_{\mu}(g)f(x)=\kappa\big{(}g^{-1},x\big{)}^{\mu}f\big{(}g^{-1}(x)\big{)},\qquad
x\in S^{\prime}.$ (5.2)

Let $\operatorname{res}\colon C^{\infty}(S)\longrightarrow
C^{\infty}(S^{\prime})$ be the restriction map from $S$ to $S^{\prime}$,
defined for $f\in C^{\infty}(S)$ by $(\operatorname{res}f)(x)=f(x),x\in
S^{\prime}$. The last remark makes clear that for $\lambda\in\mathbb{C}$ and
for $g\in G^{\prime}$,

$\displaystyle\operatorname{res}\circ\pi_{\lambda}(g)=\pi^{\prime}_{\lambda}(g)\circ\operatorname{res}.$
(5.3)

Define the differential operator ${\mathbf{D}}_{N}(\lambda)\colon
C^{\infty}(S)\longrightarrow C^{\infty}(S^{\prime})$ by

$\displaystyle{\mathbf{D}}_{N}(\lambda)=\operatorname{res}\circ\widetilde{\mathbf{D}}_{\lambda,N}.$

######  Theorem 5.2.

${\mathbf{D}}_{N}(\lambda)$ satisfies

$\displaystyle\forall\,g\in
G^{\prime}\qquad{\mathbf{D}}_{N}(\lambda)\circ\pi_{\lambda}(g)=\pi^{\prime}_{\lambda+N}(g)\circ{\mathbf{D}}_{N}(\lambda).$

The proof follows immediately from the covariance property of
$\widetilde{\mathbf{D}}_{\lambda,N}$ and of the restriction map (5.3).

##  6 The family $\boldsymbol{E_{N}(\lambda)}$

The previous constructions of differential operators made for $S$ and
$S^{\prime}$ can be made in a similar manner in the non compact picture, i.e.,
for $\mathbb{R}^{n}$ and $\mathbb{R}^{n-1}$. For $N\in\mathbb{N}$, let
$E_{\lambda,N}$ be defined by

$\displaystyle E_{\lambda,N}=E_{\lambda+N-1}\circ\dots\circ E_{\lambda}$

and

$\displaystyle E_{N}(\lambda)=\operatorname{res}\circ E_{\lambda,N},$

where $\operatorname{res}$ is the restriction from $\mathbb{R}^{n}$ to
$\mathbb{R}^{n-1}$. Then $E_{\lambda,N}$ is a differential operator on
$\mathbb{R}^{n}$ which is covariant with respect to
$(\rho_{\lambda|G^{\prime}},\rho_{\lambda+N|G^{\prime}})$ and $E_{N}(\lambda)$
is a differential operator from $\mathbb{R}^{n}$ to $\mathbb{R}^{n-1}$ which
is covariant with respect to
$(\rho_{\lambda|G^{\prime}},\rho^{\prime}_{\lambda+N})$.333The representation
$\rho^{\prime}$ is the principal series for $G^{\prime}$ realized in the
$\mathbb{R}^{n-1}$, defined in analogy with (5.2).

In this section, for the sake of completeness, we compare $E_{N}(\lambda)$
with Juhl’s operator for the non compact model. For $\xi\in\mathbb{R}^{n}$,
introduce the notation $\xi=(\xi^{\prime},\xi_{n})$ where
$\xi^{\prime}\in\mathbb{R}^{n-1}$. Let
$\Delta^{\prime}=\sum\limits_{j=1}^{n-1}\frac{\partial^{2}}{\partial\xi_{j}^{2}}$.

######  Proposition 6.1.

Let $E\colon C^{\infty}(\mathbb{R}^{n})\longrightarrow
C^{\infty}(\mathbb{R}^{n-1})$ be a differential operator and assume that $E$
is covariant with respect to
$(\rho_{\lambda|G^{\prime}},\rho^{\prime}_{\lambda+N})$ for some
$N\in\mathbb{N}$. Then there exits a family of complex constants $a_{j}$,
$0\leq j\leq[\frac{N}{2}]$ such that

$\displaystyle
E=\operatorname{res}\circ\sum_{j=0}^{[\frac{N}{2}]}a_{j}\left(\frac{\partial}{\partial\xi_{n}}\right)^{N-2j}\Delta^{\prime
j}.$

###### Proof.

By the definition of a differential operator from $\mathbb{R}^{n}$ to
$\mathbb{R}^{n-1}$, $E$ can be written as a locally finite sum

$\displaystyle\sum_{i,J}a_{i,J}(\xi^{\prime})\operatorname{res}\circ\left(\frac{\partial}{\partial\xi_{n}}\right)^{i}\partial^{J},$

where $J=(j_{1},j_{2},\dots,j_{n-1})$ is a $(n-1)$-tuple,
$\partial^{J}=\prod\limits_{k=1}^{n-1}\big{(}\frac{\partial}{\partial\xi_{k}}\big{)}^{j_{k}}$
and $a_{i,J}$ is a smooth function of $\xi^{\prime}\in\mathbb{R}^{n-1}$.

The invariance by translations forces the $a_{i,J}$ to be constants (and also
the sum to be finite). The invariance by ${\rm SO}(n-1)$ forces the expression
to be of the form

$\displaystyle\sum_{i,j}a_{i,j}\left(\frac{\partial}{\partial\xi_{n}}\right)^{i}(\Delta^{\prime})^{j}$

and finally the covariance under the action of the dilations forces $i+2j=N$.
The statement follows. ∎

Notice that the proof uses only the covariance property for the parabolic
subgroup of affine conformal diffeomorphisms of $\mathbb{R}^{n-1}$. The full
covariance condition implies further conditions on the coefficients $a_{i,j}$,
explicitly written by A. Juhl (see [6], condition (5.1.2) for $N$ even and
(5.1.22) for $N$ odd), proving in particular that there exists (up to a
constant) a unique covariant differential operator. Now let

$\displaystyle
E_{N}(\lambda)=\sum_{j=0}^{[\frac{N}{2}]}a_{j}(\lambda,N)\left(\frac{\partial}{\partial\xi_{n}}\right)^{N-2j}\Delta^{\prime
j},$

where $a_{j}(\lambda,N)$ are complex numbers.

To find the ratio between $E_{N}(\lambda)$ and the corresponding Juhl’s
operator, it is enough to know some coefficient of $E_{N}(\lambda)$ and to
compare it to the corresponding coefficient of Juhl’s operator. It turns out
that the coefficient $a_{0}(\lambda,N)$ is rather easy to compute.

######  Lemma 6.2.

  1. $i)$

For $k\in\mathbb{N}$ and $\mu\in\mathbb{C}$,

$\displaystyle E_{\mu}\xi_{n}^{k}=k(2\mu-n+1+k)\xi_{n}^{k-1}.$

  2. $ii)$

For $N\in\mathbb{N}$ and for $\lambda\in\mathbb{C}$,

$\displaystyle
E_{\lambda,N}\big{(}\xi_{n}^{N}\big{)}=N!(2\lambda-n+N+1)(2\lambda-n+N+2)\cdots(N+2\lambda-n+2N).$

  3. $iii)$

The constant $a_{0}(\lambda,N)$ is given by

$\displaystyle
a_{0}(\lambda,N)=(2\lambda-n+N+1)(2\lambda-n+N+2)\cdots(2\lambda-n+2N).$

###### Proof.

Let $f$ be a function on $\mathbb{R}^{n}$ which depends only on $\xi_{n}$.
Then $\Delta^{\prime}f=0$, and

$\displaystyle
E_{\mu}f=\left((2\mu-n+2)\frac{\partial}{\partial\xi_{n}}+\xi_{n}\frac{\partial^{2}}{\partial\xi_{n}^{2}}\right)f,$

so that $i)$ and $ii)$ are reduced to elementary one variable computations.
For $iii)$ observe that

$\displaystyle
E_{\lambda,N}\big{(}\xi_{n}^{N}\big{)}=a_{0}(\lambda,N)\left(\frac{\partial}{\partial\xi_{n}}\right)^{N}\big{(}\xi_{n}^{N}\big{)}+0+\dots+0=N!a_{0}(\lambda,N),$

hence $E_{N}(\lambda)(\xi_{n}^{N})=N!a_{0}(\lambda,N)$ and $iii)$ follows. ∎

The comparison with Juhl’s operator is then easy. As his normalization depends
on the parity of $N$, one has to examine two cases.

  * •

In the even case, $E_{N}(\lambda)$ is obtained by multiplying Juhl’s operator
by

$\displaystyle\frac{N!}{\big{(}\frac{N}{2}\big{)}!}2^{\frac{N}{2}-1}\prod_{j=1}^{\frac{N}{2}}(2\lambda-n+N+2j).$

  * •

In the odd case, $E_{N}(\lambda)$ is obtained by multiplying Juhl’s operator
by

$\displaystyle\frac{N!}{\big{(}\frac{N-1}{2}\big{)}!}2^{\frac{N+1}{2}}\prod_{j=0}^{\frac{N-1}{2}}(2\lambda-n+N+1+2j).$

##  7 The operator $\boldsymbol{{\mathbf{D}}_{\lambda}}$ in the ambient space
model

This last section is devoted to another (simpler) construction of (a multiple
of) the operator ${\mathbf{D}}_{\lambda}$, using the _ambient space_
realization of the principal series.

Let $\Xi^{+}$ be the positive light cone,

$\displaystyle\Xi^{+}=\big{\\{}\mathbf{x}\in\mathbf{E},\,Q(\mathbf{x})=[\mathbf{x},\mathbf{x}]=0,\,t(\mathbf{x})>0\big{\\}}.$

For $\lambda\in\mathbb{C}$, let

$\displaystyle\mathcal{H}_{\lambda}=\big{\\{}F\in
C^{\infty}(\Xi^{+}),\,F(t\mathbf{x})=t^{-\lambda}F(\mathbf{x}),\text{ for
}t\in\mathbb{R}^{+}\big{\\}}.$

The space $\mathcal{H}_{\lambda}$ is in one-to-one correspondence with the
space $C^{\infty}(S)$ through the map $R_{\lambda}$

$\displaystyle\mathcal{H}_{\lambda}\ni F\longmapsto R_{\lambda}F\in
C^{\infty}(S),\qquad R_{\lambda}F(x)=F\big{(}(1,x)\big{)}.$

The space $\mathcal{H}_{\lambda}$ inherits the corresponding topology. For
$g\in G$, and $F\in\mathcal{H}_{\lambda}$, let

$\displaystyle\Pi_{\lambda}(g)F=F\circ g^{-1}.$

Then $\Pi_{\lambda}$ defines a representation of $G$ on
$\mathcal{H}_{\lambda}$ and it is easily verified that

$\displaystyle R_{\lambda}\circ\Pi_{\lambda}(g)=\pi_{\lambda}(g)\circ
R_{\lambda},$ (7.1)

so that $\Pi_{\lambda}$ is yet another model for the representation
$\pi_{\lambda}$ of $G$.

Let $\square=\frac{\partial^{2}}{\partial
t^{2}}-\sum\limits_{j=0}^{n}\frac{\partial^{2}}{\partial x_{j}^{2}}$ be the
d’Alembertian on $\mathbf{E}$. It satisfies, for any $g\in G$ and $F$ a smooth
function on $\mathbf{E}$

$\displaystyle\square(F\circ g)=(\square F)\circ g.$ (7.2)

The following lemma, which I learnt from [3] is a key result for what follows.

######  Lemma 7.1.

Let $F_{1}$, $F_{2}$ be two smooth functions defined in a neighborhood of
$\Xi^{+}$, positively homogeneous of degree $-\frac{n}{2}+1$ and which
coincide on $\Xi^{+}$. Then $\square F_{1}$ and $\square F_{2}$ coincide on
$\Xi^{+}$.

###### Proof.

The function $F_{1}-F_{2}$ vanishes on $\Xi^{+}$. Notice that
$dQ(\mathbf{x})\neq 0$ for any $\mathbf{x}\in\Xi^{+}$. Hence, there exists a
smooth function $G$ defined on a neighborhood of $\Xi^{+}$ such that

$\displaystyle
F_{1}(\mathbf{x})-F_{2}(\mathbf{x})=Q(\mathbf{x})G(\mathbf{x}).$

Moreover, $G$ is positively homogeneous of degree $-\frac{n}{2}-1$. Now, for
any smooth function $H$ on $\mathbf{E}$

$\displaystyle\square(QH)=2(n+2)H+4EH+Q\square H,$

where $E=t\frac{\partial}{\partial
t}+\sum\limits_{j=0}^{n}x_{j}\frac{\partial}{\partial x_{j}}$ is the Euler
operator. As $G$ is homogeneous of degree $-\frac{n}{2}-1$,

$\displaystyle EG(\mathbf{x})=\left(-\frac{n}{2}-1\right)G(\mathbf{x}),$

and hence $\square(QG)(x)=0$ for $x\in\Xi^{+}$. The lemma follows. ∎

The next result is a reformulation of the previous lemma.

######  Lemma 7.2.

Let $F\in\mathcal{H}_{\frac{n}{2}-1}$. Extend $F$ smoothly to a positively
homogeneous function of degree $-\frac{n}{2}+1$ to neighborhood of $\Xi^{+}$.
Then the restriction to $\Xi^{+}$ of $\square F$ does not depend on the
extension.

The operator $\square$ induces a map from $\mathcal{H}_{\frac{n}{2}-1}$ to
$\mathcal{H}_{\frac{n}{2}+1}$ and intertwines the action of $G$. Let
$\Delta_{S}$ be the operator defined on $C^{\infty}(S)$ by

$\displaystyle\Delta_{S}=R_{\frac{n}{2}+1}\circ\square\circ
R^{-1}_{\frac{n}{2}-1}.$

The invariance of $\square$ (see (7.2)) and the covariance of $R_{\lambda}$
(see (7.1)) imply the following proposition.

######  Proposition 7.3.

The operator $\Delta_{S}$ $($conformal Laplacian or Yamabe operator on $S)$ is
a differential operator on $S$ which is covariant with respect to
$(\pi_{\frac{n}{2}-1},\pi_{\frac{n}{2}+1})$.

Let $\mathbf{B}_{\mu}$ be the differential operator on $\mathbf{E}$ defined by

$\displaystyle\mathbf{B}_{\mu}F(\mathbf{x})=x_{n}\square
F(\mathbf{x})-2\mu\frac{\partial F}{\partial x_{n}}.$

######  Lemma 7.4.

Let $\mu\in\mathbb{C}$. Let $F$ be a smooth function on $\mathbf{E}$. Then, on
$\\{x_{n}\neq 0\\}$,

$\displaystyle\mathbf{B}_{\mu}F(\mathbf{x})=x_{n}|x_{n}|^{-\mu}\square\big{(}|x_{n}|^{\mu}F\big{)}(\mathbf{x})+\mu(\mu-1)\frac{1}{x_{n}}F(\mathbf{x}).$
(7.3)

###### Proof.

By an elementary calculation,

$\displaystyle\square\big{(}|x_{n}|^{\mu}F\big{)}(\mathbf{x})=|x_{n}|^{\mu}\square
F(\mathbf{x})-2\mu\operatorname{sgn}(x_{n})|x_{n}|^{\mu-1}\frac{\partial
F}{\partial x_{n}}(\mathbf{x})-\mu(\mu-1)|x_{n}|^{\mu-2}F(\mathbf{x}),$

so that

$\displaystyle\square\big{(}|x_{n}|^{\mu}F\big{)}+\mu(\mu-1)|x_{n}|^{\mu-2}F=\operatorname{sgn}(x_{n})|x_{n}|^{\mu-1}\mathbf{B}_{\mu}F.$

The conclusion follows, by noticing that
$x_{n}=\operatorname{sgn}(x_{n})|x_{n}|$. ∎

######  Proposition 7.5.

Let $g\in G^{\prime}$. Then for $F$ a smooth function on $\mathbf{E}$,

$\displaystyle\mathbf{B}_{\mu}(F\circ g)=(\mathbf{B}_{\mu}F)\circ g.$

###### Proof.

As $g\in G^{\prime}$, the coordinate $x_{n}$ is unchanged by the action of
$g$, and the action of $g$ commutes with $\frac{\partial}{\partial x_{n}}$ and
with $\square$. The result follows. ∎

######  Proposition 7.6.

Let $F\in\mathcal{H}_{\lambda}$. Extend $F$ smoothly to a neighborhood of
$\Xi^{+}$ as a positively homogeneous function of degree $-\lambda$. Then the
restriction to $\Xi^{+}$ of $\mathbf{B}_{\lambda-\frac{n}{2}+1}F$ does not
depend on the extension.

###### Proof.

The function $|x_{n}|^{\lambda-\frac{n}{2}+1}F(\mathbf{x})$ is homogenous of
degree $-\frac{n}{2}+1$, and hence, by Lemma 7.2, for $x\in\Xi^{+}$,
$\square\big{(}|x_{n}|^{\lambda-\frac{n}{2}+1}F\big{)}(\mathbf{x})$ only
depend on the values of $F$ on $\Xi^{+}$. Hence, by (7.3), for $\mathbf{x}$ in
$\Xi^{+}$, $x_{n}\neq 0$, $\mathbf{B}_{\lambda-\frac{n}{2}+1}F(\mathbf{x})$
does not depend on the extension of $F$. The result follows by continuity. ∎

######  Proposition 7.7.

The differential operator $\mathbf{B}_{\lambda-\frac{n}{2}+1}$ induces a map
from $\mathcal{H}_{\lambda}$ into $\mathcal{H}_{\lambda+1}$, which commutes
with the action of $G^{\prime}$.

###### Proof.

The invariance follows from Proposition 7.5. ∎

Having constructed a covariant operator in the ambient space model, it is
possible to express it both in the non-compact and in the compact picture.

######  Proposition 7.8.

The local expression of the operator $\mathbf{B}_{\lambda-\frac{n}{2}+1}$ in
the non compact picture is equal to $-E_{\lambda}$.

###### Proof.

Let $f$ be a smooth function on $\mathbb{R}^{n}$. Recall the map $c$ (cf.
(4.1)) which realizes the passage from $\mathbb{R}^{n}$ to $S$. Its inverse is
given by

$\displaystyle
S\setminus\\{-\mathbf{1}\\}\ni(x_{0},x_{1},\dots,x_{n})\longmapsto\left(\frac{x_{1}}{1+x_{0}},\dots,\frac{x_{n}}{1+x_{0}}\right).$

So map $f$ to a function on $S$ by

$\displaystyle
C_{\lambda}^{-1}f(x)=(1+x_{0})^{-\lambda}f\left(\frac{x_{1}}{1+x_{0}},\dots,\frac{x_{n}}{1+x_{0}}\right).$

Consider the function $F$ on $\mathbf{E}$ defined by

$\displaystyle
F(\mathbf{x})=(t+x_{0})^{-\lambda}f\left(\frac{x_{1}}{t+x_{0}},\dots,\frac{x_{n}}{t+x_{0}}\right).$

Then $F$ is homogenous of degree $-\lambda$ and coincide on $S$ with
$C_{\lambda}^{-1}f$. To compute $\mathbf{B}_{\lambda-\frac{n}{2}+1}F$, first
observe that

$\displaystyle\frac{\partial F}{\partial t}=\frac{\partial F}{\partial
x_{0}},\qquad\frac{\partial^{2}F}{\partial
t^{2}}=\frac{\partial^{2}F}{\partial x_{0}^{2}},$

so that

$\displaystyle\square F=-\sum_{j=1}^{n}\frac{\partial^{2}F}{\partial
x_{j}^{2}}.$

Hence

$\displaystyle\mathbf{B}_{\lambda-\frac{n}{2}+1}F(\mathbf{x})=-(t+x_{0})^{-\lambda-2}x_{n}(\Delta
f)\left(\frac{x_{1}}{t+x_{0}},\dots,\frac{x_{n}}{t+x_{0}}\right)$
$\displaystyle\hphantom{\mathbf{B}_{\lambda-\frac{n}{2}+1}F(\mathbf{x})=}{}-2\left(\lambda-\frac{n}{2}+1\right)(t+x_{0})^{-\lambda}(t+x_{0})^{-1}\frac{\partial
f}{\partial\xi_{n}}\left(\frac{x_{1}}{t+x_{0}},\dots,\frac{x_{n}}{t+x_{0}}\right).$

Now letting $\mathbf{x}=(1,c(\xi))$,

$\displaystyle\mathbf{B}_{\lambda-\frac{n}{2}+1}F(1,c(\xi))=-\xi_{n}\Delta
f(\xi)-(2\lambda-n+2)\frac{\partial f}{\partial\xi_{n}}(\xi).$

A comparison with (4.6) implies the result. ∎

######  Proposition 7.9.

The expression of the operator $\mathbf{B}_{\lambda-\frac{n}{2}+1}$ on $S$ is
given by

$\displaystyle
x_{n}|x_{n}|^{-\lambda+\frac{n}{2}-1}\Delta_{S}\circ|x_{n}|^{\lambda-\frac{n}{2}+1}+\left(\lambda-\frac{n}{2}+1\right)\left(\lambda-\frac{n}{2}\right)\frac{1}{x_{n}}.$

The expression, a priori defined on $x_{n}\neq 0$ can be continued
continuously to all of $S$.

###### Proof.

Let $f\in C^{\infty}(S)$. Then

$\displaystyle
F(\mathbf{x})=\big{(}x_{0}^{2}+\dots+x_{n}^{2}\big{)}^{-\lambda}f\left(\frac{x_{0}}{\sqrt{x_{0}^{2}+\dots+x_{n}^{2}}},\dots,\frac{x_{n}}{\sqrt{x_{0}^{2}+\dots+x_{n}^{2}}}\right)$

is a function defined on $\mathbf{E}\setminus\\{0\\}$ which is positively
homogeneous of degree $-\lambda$ and such that for $x\in S$,

$\displaystyle F(1,x)=f(x).$

By (7.3) with $\mu=\lambda-\frac{n}{2}+1$ and for $\mathbf{x}\neq 0$,
$x_{n}\neq 0$,

$\displaystyle\mathbf{B}_{\lambda-\frac{n}{2}+1}F(\mathbf{x})=x_{n}|x_{n}|^{-\lambda+\frac{n}{2}+1}\square\big{(}|x_{n}|^{\lambda-\frac{n}{2}+1}F\big{)}(\mathbf{x})+\left(\lambda-\frac{n}{2}+1\right)\left(\lambda-\frac{n}{2}\right)\frac{1}{x_{n}}F(\mathbf{x}).$

The function $|x_{n}|^{\lambda-\frac{n}{2}+1}F(\mathbf{x})$ is positively
homogeneous of degree $-\frac{n}{2}+1$. Thus, by Lemma 7.2 and the definition
of the Yamabe operator $\Delta_{S}$, for $x\in S$,

$\displaystyle\mathbf{B}_{\lambda-\frac{n}{2}+1}F(1,x)=x_{n}|x_{n}|^{-\lambda+\frac{n}{2}-1}\Delta_{S}\big{(}|x_{n}|^{\lambda-\frac{n}{2}+1}f\big{)}(x)+\left(\lambda-\frac{n}{2}+1\right)\left(\lambda-\frac{n}{2}\right)\frac{1}{x_{n}}f(x),$

from which the statement follows, at least for $x_{n}\neq 0$. As
$\mathbf{B}_{\lambda-\frac{n}{2}+1}$ induces a smooth differential operator on
$S$, the formula determines the operator on all of $S$ by continuity. ∎

### Acknowledgements

It is a pleasure to thank the anonymous referees for their contributions which
helped to improve and reshape the initial version of this article.

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