Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹
A family (Dλ)λ∈C of differential operators on the sphere Sⁿ is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of Sⁿ which preserve the smaller sphere Sⁿ⁻¹ ⊂Sⁿ. The family of conformally covariant differential operators from Sⁿ to Sⁿ⁻¹...
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| Cite this: | Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ / Jean-Louis Clerc // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 12 назв. — англ. |
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| citation_txt | Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ / Jean-Louis Clerc // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 12 назв. — англ. |
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| description | A family (Dλ)λ∈C of differential operators on the sphere Sⁿ is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of Sⁿ which preserve the smaller sphere Sⁿ⁻¹ ⊂Sⁿ. The family of conformally covariant differential operators from Sⁿ to Sⁿ⁻¹ introduced by A. Juhl is obtained by composing these operators on Sⁿ and taking restrictions to Sⁿ⁻¹ .
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 026, 18 pages
Another Approach to Juhl’s Conformally Covariant
Differential Operators from Sn to Sn−1
Jean-Louis CLERC
Institut Elie Cartan de Lorraine, Université de Lorraine, France
E-mail: jean-louis.clerc@univ-lorraine.fr
Received December 07, 2016, in final form April 11, 2017; Published online April 19, 2017
https://doi.org/10.3842/SIGMA.2017.026
Abstract. A family (Dλ)λ∈C of differential operators on the sphere Sn is constructed. The
operators are conformally covariant for the action of the subgroup of conformal transfor-
mations of Sn which preserve the smaller sphere Sn−1 ⊂ Sn. The family of conformally
covariant differential operators from Sn to Sn−1 introduced by A. Juhl is obtained by com-
posing these operators on Sn and taking restrictions to Sn−1.
Key words: conformally covariant differential operators; Juhl’s covariant differential opera-
tors
2010 Mathematics Subject Classification: 58J70; 43A85
1 Introduction
Let S = Sn be the n-dimensional sphere in Rn+1 and let G = SO0(1, n + 1) be (the neutral
component of) the group of conformal transformations of S. Let S′ ' Sn−1 be the subspace of
points of S with vanishing last coordinate (xn = 0 in our notation) and let G′ ' SO0(1, n) be
the conformal group of S′, viewed as the subgroup of G which stabilizes S′. Let (πλ)λ∈C be the
scalar principal series of representations of G acting on C∞(S). Denote by πλ|G′ its restriction
to G′. Let (π′µ)µ∈C be the scalar principal series of G′ acting on C∞(S′).
In [6] A. Juhl has constructed a family DN (λ)λ∈C,N∈N of differential operators from C∞(S)
into C∞(S′), which are intertwining operators between πλ|G′ and π′λ+N .1 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′). 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×G,G′) where
G′ = G embedded diagonally in G × G. I was influenced by a reminiscence of the Ω-process
which yields both the transvectants and the Rankin–Cohen brackets. These operators may
be viewed as covariant bi-differential operators for the group SL(2,R), or symmetry breaking
differential operators from SL(2,R) × SL(2,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 = SL(2n,R) and the
degenerate principal series acting on the Grasmmannian Gr(n, 2n;R) (see [2]).
The first step of the method, for the present case, is to introduce the multiplication by xn,
viewed as an operator M on C∞(S). The operator M is a “universal” G′-intertwining operator,
1Our λ corresponds to −λ in Juhl’s notation.
mailto:jean-louis.clerc@univ-lorraine.fr
https://doi.org/10.3842/SIGMA.2017.026
2 J.-L. Clerc
in the sense that, for any λ ∈ C, the operator M intertwines πλ|G′ and πλ−1|G′ . Next recall
the family of Knapp–Stein operators (Iλ)λ∈C which are G-intertwining operators with respect
to (πλ, πn−λ). The operator2
Dλ = In−λ−1 ◦M ◦ Iλ
obtained by twisting M by the appropriate Knapp–Stein intertwining operators is clearly an
intertwining operator with respect to (πλ|G′ , πλ+1|G′). Our main result (see Theorem 3.2) is
that Dλ is a differential operator. The proof is obtained in the non compact realization of
the principal series (passing from Sn to Rn by a conformal map) and uses Euclidean Fourier
transform.
The construction of conformally covariant differential operators from Sn to Sn−1 is now easy.
For N a non-negative integer, consider
DN,λ = Dλ+N−1 ◦ · · · ◦Dλ+1 ◦Dλ or DN,λ = In−λ−N ◦MN ◦ Iλ.
The two families of differential operators on S (which coincide up to a meromorphic function
of λ) are covariant with respect to (πλ|G′ , πλ+N |G′). Finally, let
DN (λ) = res ◦DN,λ,
where res is the restriction map from C∞(S) to C∞(S′). The operator DN (λ) is a differential
operator from S to S′ which is covariant with respect to (πλ|G′ , π
′
λ+N ). The family DN (λ)λ∈C,N∈N
essentially coincides with Juhl’s family.
The operator Dλ 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 Dλ on Rn, see Proposition 7.8), but also
in the compact realization (see Proposition 7.9), that is to say as a G′-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 SO0(1, n+ 1)
and the Knapp–Stein intertwining operators
Let E be a Euclidean space of dimension n+1, and choose an orthonormal basis {e0, e1, . . . , en}.
Let S = Sn be the unit sphere of E, i.e.,
S =
{
x = (x0, x1, . . . , xn), x20 + x21 + · · ·+ x2n = 1
}
.
Let E be the vector space R⊕ E, with the Lorentzian quadratic form
Q(x) = [(t, x), (t, x)] = t2 − |x|2 for x = (t, x), t ∈ R, x ∈ E.
For x = (t, x) ∈ E, we let
t(x) = t, xE = x.
The space of isotropic lines S in E can be identified with S by the map
S 3 x 7−→ dx = R(1, x) ∈ S, S 3 d 7−→ d ∩ {t = 1}.
2For technical reasons, a normalizing factor is introduced, see (3.4).
Another Approach to Juhl’s Conformally Covariant Differential Operators 3
Let G = SO0(1, n+ 1) be the connected component of the group of isometries of E. Then G
acts on S and this action can be transferred to an action on S. More explicitly, if x =
(x0, x1, . . . , xn) ∈ S, and g ∈ G, observe that t(g(1, x)) > 0 and define g(x) ∈ S by(
1, g(x)
)
= t
(
g.(1, x)
)−1
g.(1, x).
Set, for g ∈ G and x ∈ S
κ(g, x) = t
(
g.(1, x)
)−1
.
Clearly κ(g, x) is a smooth, strictly positive function on G × S. Moreover κ(g, x) satisfies the
cocycle property : for any g1, g2 and any x ∈ S,
κ(g1g2, x) = κ
(
g1, g2(x)
)
κ(g2, x).
This action of G on S is known to be conformal. For g ∈ G, x ∈ S and ξ an arbitrary tangent
vector to S at x
|Dg(x)ξ| = κ(g, x)|ξ|,
and hence κ(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∞(S), which,
from the point of view of harmonic analysis is the scalar principal series of G. For λ ∈ C, g ∈ G
and f ∈ C∞(S), let
πλ(g)f(x) = κ
(
g−1, x
)λ
f
(
g−1(x)
)
.
The formula defines a (smooth) representation πλ of G on C∞(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 λ ∈ C and f ∈ C∞(S), let
Iλf(x) =
1
Γ(λ− n
2 )
∫
S
|x− y|−2n+2λf(y)dy, (2.1)
where dy stands for the Lebesgue measure on S induced by the Euclidean structure. For
Reλ > n
2 , this formula defines a continuous operator Iλ on C∞(S).
Proposition 2.1.
i) The definition (2.1) can be analytically continued in λ to all of C.
ii) The analytic continuation yields a holomorphic family of operators Iλ on C∞(S), which
satisfy the intertwining relation
∀ g ∈ G, Iλ ◦ πλ(g) = πn−λ(g) ◦ Iλ. (2.2)
The following complementary result will be needed later.
Proposition 2.2. For any λ ∈ C
Iλ ◦ In−λ =
πn
Γ(λ)Γ(n− λ)
id . (2.3)
4 J.-L. Clerc
The next result corresponds to reducibility points for the scalar principal series. Let P(S) be
the space of restrictions to S of polynomial functions on E, and for k ∈ N, let Pk be the space of
restrictions to S of polynomials on E of degree ≤ k. Finally, let P⊥k be the subspace of C∞(S)
given by
P⊥k =
{
f ∈ C∞(S),
∫
S
f(x)p(x)dx = 0, for any p ∈ Pk
}
.
.
Proposition 2.3.
i) Let λ = n+ k, k ∈ N. Then
Im(In+k) = Pk, Ker(In+k) = P⊥k . (2.4)
ii) Let λ = −k, k ∈ N. Then
Ker(I−k) = Pk, Im(I−k) = P⊥k . (2.5)
3 Construction of the family D̃λ, λ ∈ C
Now let E′ = {x ∈ E, xn = 0} and S′ = S ∩ E′. Then S′ is an (n − 1)-dimensional sphere.
Let G′ be the subgroup of elements of G of the form
g =
0
g′
...
0
0 . . . 0 1
, g′ ∈ SO0(1, n).
Clearly, G′ is a subgroup of G, isomorphic to SO0(1, n). Elements of G′ preserve the hyperplane
{xn = 0} in E and hence the action of G′ on S preserves S′.
For x ∈ E, write x = (x′, xn), with x′ ∈ Rn. For g ∈ G′,
g(1, x) = g(1, x′, xn) =
(
g′.(1, x′), xn
)
.
If x ∈ S, the last equation can be rewritten as
κ(g, x)−1
(
1, g(x)
)
=
(
g′.(1, x′), xn
)
,
so that
g(x)n = κ(g, x)xn. (3.1)
In the sequel, the distinction between g and g′ in the notation is abandoned, the context providing
the correct interpretation.
Let M be the operator defined on C∞(S) by
Mf(x) = xnf(x), f ∈ C∞(S).
Proposition 3.1. The operator M satisfies
∀ g ∈ G′ M ◦ πλ(g) = πλ−1(g) ◦M. (3.2)
Proof. This is an immediate consequence of (3.1). �
Another Approach to Juhl’s Conformally Covariant Differential Operators 5
Next let Dλ be the operator on C∞(S) defined by
Dλ = In−λ−1 ◦M ◦ Iλ,
which corresponds to the following diagram
C∞(S)
Dλ−−−−→ C∞(S)yIλ xIn−λ−1
C∞(S)
M−−−−→ C∞(S).
As a consequence of the intertwining property of the Knapp–Stein operators (2.2) and Proposi-
tion 3.1, Dλ satisfies for g ∈ G′
Dλ ◦ πλ(g) = πλ+1(g) ◦Dλ. (3.3)
Otherwise said, the operator Dλ is covariant with respect to (πλ|G′ , πλ+1|G′).
Theorem 3.2. The operator Dλ 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 λ ∈ (n+ N) ∪ (−1− N). Then Dλ = 0.
Proof. Let first λ = n + k for some k ∈ N. Then Iλ = In+k, and by (2.4) Im(Iλ) = Pk. Next
Im(M ◦ Iλ) ⊂ Pk+1. Now In−λ−1 = I−k−1 and using (2.5), In−λ−1 ◦M ◦ Iλ = 0.
Now let λ = −k, with k ≥ 1. Then Iλ = I−k and by (2.5), Im(Iλ) = P⊥k . Next Im(M ◦ Iλ) ⊂
P1P⊥k ⊂ P⊥k−1. Now In−λ−1 = In+k−1 which using (2.4) implies In−λ−1 ◦M ◦ Iλ = 0. �
To compensate for these zeroes of Dλ, introduce
D̃λ = Γ(λ+ 1)Γ(n− λ)Dλ (3.4)
for λ /∈ (n + N) ∪ (−1 − N) and extend continuously to all of C to get a holomorphic family
(D̃λ)λ∈C of differential operators on S covariant with respect to (πλ|G′ , πλ+1|G′).
4 The expression of D̃λ in the non-compact picture
Consider the point −1 = (−1, 0, . . . , 0) ∈ S. The stereographic projection with source at −1
provides a diffeomorphism from S \ {−1} onto the hyperplane {xn = 1}. The inverse map (up
to a scaling by a factor 2) c : Rn −→ S is given by
c(ξ) =
1− |ξ|2
1 + |ξ|2
2ξ1
1 + |ξ|2
...
2ξn
1 + |ξ|2
. (4.1)
When using this local chart on S, we refer to the non-compact picture, as a reference to semi-
simple harmonic analysis.
6 J.-L. Clerc
Geometric considerations (or an elementary computation) show that, for ξ, η ∈ Rn
|c(ξ)− c(η)|2 = κ(c, ξ)|ξ − η|2κ(c, η),
where, for ξ ∈ Rn, we set
κ(c, ξ) = 2
(
1 + |ξ|2
)−1
.
There is an infinitesimal version of this result, namely
|Dc(ξ)η| = κ(c, ξ)|η|
for ξ, η ∈ Rn. This last statement shows that c is conformal from Rn with its standard Euclidean
structure into S.
The action of g on S can be transferred as a (rational) action of G on Rn, namely c−1 ◦ g ◦ c.
For notational convenience, we still denote this action on Rn by (g, ξ) 7−→ g(ξ), g ∈ G, ξ ∈ Rn.
As the map c is conformal, the transferred action of G on Rn is still conformal. For g ∈ G
defined at ξ ∈ Rn, we let κ(g, ξ) be the corresponding conformal factor of g at ξ.
Let λ ∈ C. For f ∈ C∞(S) let Cλ(f) be defined by
Cλ(f)(ξ) = κ(c, ξ)λf(c(ξ)), ξ ∈ Rn
and let Hλ be the image of Cλ. It is easily proved that
S
(
Rn
)
⊂ Hλ ⊂ S ′
(
Rn
)
,
where S(Rn) stands for the Schwartz space on Rn and S ′(Rn) for its dual, the space of tempered
distributions.
The representation πλ can be transferred in the non-compact model, using Cλ as intertwining
map, i.e., set
ρλ(g) = Cλ ◦ πλ(g) ◦ C−1λ .
Using the cocycle property of κ, ρλ can be realized as
ρλ(g)f(ξ) = κ
(
g−1, ξ
)λ
f
(
g−1(ξ)
)
,
where f ∈ Hλ and g ∈ G.
Similarly, the Knapp–Stein operators can be transferred to the non-compact picture. For
s ∈ C, consider the expression
hs(ξ) =
1
Γ
(
n
2 + s
2
) |ξ|s, ξ ∈ Rn.
For Re(s) > −n, hs is locally summable with moderate growth at infinity, hence defines a tem-
pered distribution. The (S ′(Rn)-valued) function s 7→ hs can be extended by analytic conti-
nuation to C and the Γ factor in the definition of hs is so chosen that it extends as an entire
function with values in S ′(Rn) (for more details see, e.g., [5]).
For λ ∈ C, the Knapp–Stein operator Jλ is given by
Jλf = h−2n+2λ ? f,
or more concretely
Jλf(ξ) =
1
Γ
(
λ− n
2
) ∫
Rn
|ξ − η|−2n+2λf(η)dη.
As for any s ∈ C hs is a tempered distribution, Jλ maps S(Rn) into S ′(Rn).
Another Approach to Juhl’s Conformally Covariant Differential Operators 7
Proposition 4.1. Let λ ∈ C. Then for f ∈ S(Rn)
Jλf =
(
Cn−λ ◦ Iλ ◦ C−1λ
)
f.
Proof. As S(Rn) ⊂ Hλ ⊂ S ′(Rn), both sides are well-defined and belong to S ′(Rn). For
Reλ > 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 ∈ C∞c (S) and let g ∈ G such that g−1 is defined on Supp(f). Then
Jλ
(
ρλ(g)f
)
= ρn−λ(g)
(
Jλf
)
,
where the two sides of the equation are viewed as tempered distributions on Rn.
Proof. The condition implies that both f and ρλ(g)f are contained in S(Rn). Hence
Jλ
(
ρλ(g)f
)
=
(
Cn−λ ◦ Iλ ◦ C−1λ
)(
ρλ(g)f
)
= (Cn−λ ◦ Iλ)
(
πλ(g)C−1λ f
)
=
(
Cn−λ ◦ πn−λ(g)
)
◦
(
Iλ ◦ C−1λ
)
f
= ρn−λ(g) ◦
(
Cn−λ ◦ Iλ ◦ C−1λ
)
f = ρn−λ(g)(Jλf). �
The following formulæ will be needed in the sequel
|ξ|2hs(ξ) =
n+ s
2
hs+2(ξ), (4.2)
∂
∂ξn
hs(ξ) =
2s
n+ s− 2
ξnhs−2(ξ), (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′, the map c maps the
hyperplane {ξn = 0} into S′. It allows to transfer the map M to the non-compact picture.
Lemma 4.3. Let g ∈ G′ be defined at ξ ∈ Rn. Then
g(ξ)n = κ(g, ξ)ξn. (4.4)
Proof. Let ξ ∈ Rn and let x = c(ξ) ∈ S \ {−1}. Then
c(ξ)n = κ(c, ξ)ξn, g(x)n = κ(g, x)xn, c−1(x) = κ
(
c−1, x
)
xn
the first equality by (4.1), the second by (3.1), and the third also by (4.1). As κ satisfies
a cocycle relation, we get((
c−1 ◦ g ◦ c
)
(ξ)
)
n
= κ
(
c−1 ◦ g ◦ c, ξ
)
ξn,
which gives (4.4). �
Lemma 4.4. Let λ ∈ C and f ∈ C∞(S). Then
Cλ−1(Mf)(ξ) = ξnCλ(f)(ξ), ξ ∈ Rn.
Proof. Let ξ = (ξ′, ξn). By (4.1), c(ξ)n = κ(c, ξ)ξn, so that
Cλ−1(Mf)(ξ) = κ(c, ξ)λ−1Mf
(
c(ξ)
)
= κ(c, ξ)λξnf
(
c(ξ)
)
= ξnCλ(f)(ξ). �
8 J.-L. Clerc
Abusing notation, M will be used for the operator (on C∞(Rn) say) of multiplication by ξn.
The operator M maps S(Rn) (resp. S ′(Rn)) into S(Rn) (resp. S ′(Rn)), and for any λ ∈ C, the
operator M maps Hλ into Hλ−1 (Lemma 4.4).
Proposition 4.5. Let λ ∈ C. The operator M : Hλ −→ Hλ−1 satisfies
∀ g ∈ G′, M ◦ ρλ(g) = ρλ−1(g) ◦M.
Otherwise said, the operator M intertwines the representations πλ|G′ and πλ−1|G′.
Proof. Let f ∈ Hλ. Then(
M ◦ ρλ(g)
)
f(ξ) = ξnκ
(
g−1, ξ
)λ
f
(
g−1(ξ)
)
= κ
(
g−1, ξ
)λ−1(
g−1(ξ)
)
n
f
(
g−1(ξ)
)
= ρλ−1(g)(Mf)
(
g−1(ξ)
)
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 ele-
mentary polynomial. So the Fourier transform is well-fitted for computations in this context.
Define the Fourier transform on Rn as usual by
f̂(η) =
∫
Rn
ei〈η,ξ〉f(ξ)dξ
initially for functions in S(Rn) and extend by duality to S ′(Rn).
The Fourier transform of hs is given by
ĥs = 2n+sπ
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 ◦ Jλ as an
operator from S(Rn) into S ′(Rn).
Lemma 4.6. For f ∈ S(Rn),
(
(M ◦ Jλ)f
)̂
(η) = −iπ
n
2 2−n+2λ
(
hn−2λ(η)
∂f̂
∂ηn
(η) +
n− 2λ
n− λ− 1
ηnhn−2−2λ(η)f̂(η)
)
. (4.5)
Proof. As observed earlier, the Knapp–Stein operator Jλ is a convolution operator on Rn, so
that
(Jλf)̂(η) = ĥ−2n+2λ(η)f̂(η) = 2−n+2λπ
n
2 hn−2λ(η)f̂(η).
Next, for any distribution ϕ ∈ S ′(Rn)
M̂ϕ = −i ∂
∂ηn
ϕ̂
and (4.5) follows, using (4.3). �
The composition Jn−λ−1◦M◦Jλ is not well-defined on S(Rn). 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.
Another Approach to Juhl’s Conformally Covariant Differential Operators 9
For λ ∈ C, let Eλ be the differential operator on Rn defined by
Eλ = (2λ− n+ 2)
∂
∂ξn
+ ξn∆, (4.6)
where ∆ =
n∑
j=1
∂2
∂ξ2j
is the usual Laplacian on Rn. Notice that the operator Eλ maps S(Rn)
(resp. S ′(Rn)) into S(Rn) (resp. S ′(Rn)), so that we may consider the composition Jλ+1 ◦ Eλ.
Lemma 4.7. For f ∈ S(Rn),(
(Jλ+1 ◦ Eλ)f
)̂
(η)
= −i2−n+2+2λπ
n
2
(
(λ− n+ 1)hn−2λ(η)
∂f̂
∂ηn
(η) + (2λ− n)ηnh−2λ+n−2(η)f̂(η)
)
. (4.7)
Proof. Using (4.2) and (4.3),(
Eλf
)̂
(η) = (−i)(2λ− n+ 2)ηnf̂(η) + (−i) ∂
∂ηn
(
−|η|2f̂(η)
)
= (−i)
(
(2λ− n)ηnf̂(η)− |η|2 ∂f̂
∂ηn
(η)
)
.
Next (
(Jλ+1 ◦ Eλ)f
)̂
(η) = ĥ−2n+2λ+2(η)(Eλf)̂(η)
= 2−n+2+2λπ
n
2 (−i)
(
(2λ− n)ηnhn−2−2λ(η)f̂(η)− (−λ+ n− 1)hn−2λ(η)
∂f̂
∂ηn
(η)
)
. �
Comparison of (4.5) and (4.7) yields the next result.
Proposition 4.8.
M ◦ Jλ =
1
4(λ− n+ 1)
Jλ+1 ◦ Eλ. (4.8)
Remark 4.9. This equality has to be understood as an equality of operators from S(Rn)
into S ′(Rn). For λ = n − 1, Jλ+1 = Jn is equal (up to a constant 6= 0) to the operator
f 7−→
(∫
Rn f(ξ)dξ
)
1. Now for f ∈ S(Rn),
∫
Rn Eλf(ξ)dξ = 0 as is easily seen by integration by
parts. Hence, Jλ+1 ◦ Eλ vanishes for λ = n− 1, so that (4.8) has to be interpreted as a residue
formula.
Proposition 4.10. Let f ∈ C∞c (Rn) and assume that g ∈ G′ is such that g−1 is defined on
Supp(f). Then(
Eλ ◦ ρλ(g)
)
f =
(
ρλ+1(g) ◦ Eλ
)
f.
Proof. As a consequence of the intertwining property of Jλ (Proposition 4.2) and of M (Propo-
sition 3.1),
(M ◦ Jλ)
(
ρλ(g)f
)
= ρn−λ−1(g)(M ◦ Jλ)f.
Hence, by (4.8) (assuming for a while that λ 6= n− 1)(
Jλ+1 ◦ Eλ
)
ρλ(g)f = ρn−λ−1(g)
(
(Jλ+1 ◦ Eλ)f
)
.
10 J.-L. Clerc
Now Supp(Eλf) ⊂ Supp(f), so that g−1 is defined on Supp(Eλf). Hence, by Proposition 4.2(
ρn−λ−1(g) ◦ Jλ+1
)
Eλf =
(
Jλ+1 ◦ ρλ+1(g)
)
Eλf,
so that
Jλ+1
(
(Eλ ◦ ρλ(g))f
)
= Jλ+1
(
(ρλ+1(g) ◦ Eλ)f
)
.
Now, for λ generic, the operator Jλ+1 is injective on S(Rn), hence
Eλ ◦ ρλ(g)f = ρλ+1(g) ◦ Eλf.
The general result follows by continuity, as the family Eλ depends holomorphically on λ. �
Proof of Theorem 3.2. The covariance property of the differential operator Eλ allows to
construct a global differential operator on S which is expressed in the non-compact picture
to Eλ. 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 1 = (1, 0, . . . , 0) instead of −1. Consider the element s of G given by
s =
1 0 0 0 . . . 0
0 −1 0 0 . . . 0
0 0 −1 0 . . . 0
0 0 0 1 0
...
...
...
. . .
...
0 0 0 0 . . . 1
.
Then s acts on S by
s(x) = s
x0
x1
x2
...
xn
=
−x0
−x1
x2
...
xn
.
In particular, s maps −1 to 1 and preserves S′. In the non-compact picture, the map s is defined
for ξ 6= 0 and is given by
(ξ1, ξ2, . . . , ξn) 7−→
(
− ξ1
|ξ|2
,
ξ2
|ξ|2
, . . . ,
ξn
|ξ|2
)
. (4.9)
The two charts ξ 7−→ c(ξ) and ξ 7−→ s
(
c(ξ)
)
cover S. Their common domain corresponds to
ξ 6= 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 Eλ on S which is expressed by Eλ in the
non-compact model. Clearly Eλ satisfies
∀ g ∈ G′, Eλ ◦ πλ(g) = πλ+1(g) ◦Eλ.
By (4.8),
M ◦ Iλ =
1
4(λ− n+ 1)
Iλ+1(g) ◦Eλ.
Another Approach to Juhl’s Conformally Covariant Differential Operators 11
Compose both sides with In−λ−1 and use (2.3) to get
Dλ =
π−n
4(λ− n+ 1)Γ(n− λ− 1)Γ(λ+ 1)
Eλ
or equivalently
D̃λ = − 1
4πn
Eλ.
This relation implies in particular that D̃λ is a differential operator on S. �
5 The families Dλ,N , D̃λ,N and DN(λ)
For N ≥ 1, set
D̃λ,N = D̃λ+N−1 ◦ · · · ◦ D̃λ.
Let MN be the operator on C∞(S) given by multiplication by xNn . Set
Dλ,N = In−N−λ ◦MN ◦ Iλ.
Proposition 5.1.
i) D̃λ,N and Dλ,N are differential operators on S which intertwine πλ|G′ and πλ+N |G′.
ii)
D̃λ,N = πn(N−1)Γ(λ+N)Γ(n− λ−N)Dλ,N . (5.1)
Proof. Repeated uses of (3.2) show that, for any µ ∈ C, MN intertwines πµ|G′ and πµ−N |G′ .
Hence Dλ,N intertwines πλ|G′ and πλ+N |G′ . On the other hand, repeated uses of (3.3) proves
that D̃λ,N also intertwines πλ|G′ and πλ+N |G′ .
Next, D̃λ,N as a composition of differential operators on S is a differential operator. So it
remains to prove (5.1).
Substitute Dλ+j = In−λ−j−1 ◦M ◦ Iλ+j for 0 ≤ j ≤ N − 1 to get
Dλ+N−1 ◦Dλ+N−2 ◦ · · · ◦Dλ
= I−λ+n−N ◦ · · · ◦ I−λ+n−j−1 ◦M ◦ Iλ+j ◦ I−λ+n−j ◦M ◦ Iλ+j−1 ◦ · · · ◦ Iλ,
and use (2.3) repeatedly for λ+ j to obtain
Dλ+N−1 ◦Dλ+N−2 ◦ · · · ◦Dλ
= πn(N−1)
N−1∏
j=1
Γ(λ+ j)Γ(n− λ− j)
−1 I−λ+n−N ◦MN ◦ Iλ.
Multiply by the appropriate Γ factors coming from (3.4) to get the formula. �
The group G′ acts conformally on S′. The scalar principal series for G′ ' SO0(1, n) is defined
as follows: for µ ∈ C, for g ∈ G′ and f ∈ C∞(S′),
π′µ(g)f(x) = κ
(
g−1, x
)µ
f
(
g−1(x)
)
, x ∈ S′. (5.2)
12 J.-L. Clerc
Let res : C∞(S) −→ C∞(S′) be the restriction map from S to S′, defined for f ∈ C∞(S) by
(res f)(x) = f(x), x ∈ S′. The last remark makes clear that for λ ∈ C and for g ∈ G′,
res ◦πλ(g) = π′λ(g) ◦ res . (5.3)
Define the differential operator DN (λ) : C∞(S) −→ C∞(S′) by
DN (λ) = res ◦D̃λ,N .
Theorem 5.2. DN (λ) satisfies
∀ g ∈ G′ DN (λ) ◦ πλ(g) = π′λ+N (g) ◦DN (λ).
The proof follows immediately from the covariance property of D̃λ,N and of the restriction
map (5.3).
6 The family EN(λ)
The previous constructions of differential operators made for S and S′ can be made in a similar
manner in the non compact picture, i.e., for Rn and Rn−1. For N ∈ N, let Eλ,N be defined by
Eλ,N = Eλ+N−1 ◦ · · · ◦ Eλ
and
EN (λ) = res ◦Eλ,N ,
where res is the restriction from Rn to Rn−1. Then Eλ,N is a differential operator on Rn which is
covariant with respect to (ρλ|G′ , ρλ+N |G′) and EN (λ) is a differential operator from Rn to Rn−1
which is covariant with respect to (ρλ|G′ , ρ
′
λ+N ).3
In this section, for the sake of completeness, we compare EN (λ) with Juhl’s operator for the
non compact model. For ξ ∈ Rn, introduce the notation ξ = (ξ′, ξn) where ξ′ ∈ Rn−1. Let
∆′ =
n−1∑
j=1
∂2
∂ξ2j
.
Proposition 6.1. Let E : C∞(Rn) −→ C∞(Rn−1) be a differential operator and assume that E
is covariant with respect to (ρλ|G′ , ρ
′
λ+N ) for some N ∈ N. Then there exits a family of complex
constants aj, 0 ≤ j ≤ [N2 ] such that
E = res ◦
[N
2
]∑
j=0
aj
(
∂
∂ξn
)N−2j
∆′j .
Proof. By the definition of a differential operator from Rn to Rn−1, E can be written as
a locally finite sum
∑
i,J
ai,J(ξ′) res ◦
(
∂
∂ξn
)i
∂J ,
where J = (j1, j2, . . . , jn−1) is a (n− 1)-tuple, ∂J =
n−1∏
k=1
(
∂
∂ξk
)jk and ai,J is a smooth function of
ξ′ ∈ Rn−1.
3The representation ρ′ is the principal series for G′ realized in the Rn−1, defined in analogy with (5.2).
Another Approach to Juhl’s Conformally Covariant Differential Operators 13
The invariance by translations forces the ai,J to be constants (and also the sum to be finite).
The invariance by SO(n− 1) forces the expression to be of the form
∑
i,j
ai,j
(
∂
∂ξn
)i
(∆′)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 Rn−1. The full covariance condition implies further conditions
on the coefficients ai,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
EN (λ) =
[N
2
]∑
j=0
aj(λ,N)
(
∂
∂ξn
)N−2j
∆′j ,
where aj(λ,N) are complex numbers.
To find the ratio between EN (λ) and the corresponding Juhl’s operator, it is enough to know
some coefficient of EN (λ) and to compare it to the corresponding coefficient of Juhl’s operator.
It turns out that the coefficient a0(λ,N) is rather easy to compute.
Lemma 6.2.
i) For k ∈ N and µ ∈ C,
Eµξ
k
n = k(2µ− n+ 1 + k)ξk−1n .
ii) For N ∈ N and for λ ∈ C,
Eλ,N
(
ξNn
)
= N !(2λ− n+N + 1)(2λ− n+N + 2) · · · (N + 2λ− n+ 2N).
iii) The constant a0(λ,N) is given by
a0(λ,N) = (2λ− n+N + 1)(2λ− n+N + 2) · · · (2λ− n+ 2N).
Proof. Let f be a function on Rn which depends only on ξn. Then ∆′f = 0, and
Eµf =
(
(2µ− n+ 2)
∂
∂ξn
+ ξn
∂2
∂ξ2n
)
f,
so that i) and ii) are reduced to elementary one variable computations. For iii) observe that
Eλ,N
(
ξNn
)
= a0(λ,N)
(
∂
∂ξn
)N (
ξNn
)
+ 0 + · · ·+ 0 = N !a0(λ,N),
hence EN (λ)(ξNn ) = N !a0(λ,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.
14 J.-L. Clerc
• In the even case, EN (λ) is obtained by multiplying Juhl’s operator by
N !(
N
2
)
!
2
N
2
−1
N
2∏
j=1
(2λ− n+N + 2j).
• In the odd case, EN (λ) is obtained by multiplying Juhl’s operator by
N !(
N−1
2
)
!
2
N+1
2
N−1
2∏
j=0
(2λ− n+N + 1 + 2j).
7 The operator Dλ in the ambient space model
This last section is devoted to another (simpler) construction of (a multiple of) the operator Dλ,
using the ambient space realization of the principal series.
Let Ξ+ be the positive light cone,
Ξ+ =
{
x ∈ E, Q(x) = [x,x] = 0, t(x) > 0
}
.
For λ ∈ C, let
Hλ =
{
F ∈ C∞(Ξ+), F (tx) = t−λF (x), for t ∈ R+
}
.
The space Hλ is in one-to-one correspondence with the space C∞(S) through the map Rλ
Hλ 3 F 7−→ RλF ∈ C∞(S), RλF (x) = F
(
(1, x)
)
.
The space Hλ inherits the corresponding topology. For g ∈ G, and F ∈ Hλ, let
Πλ(g)F = F ◦ g−1.
Then Πλ defines a representation of G on Hλ and it is easily verified that
Rλ ◦Πλ(g) = πλ(g) ◦Rλ, (7.1)
so that Πλ is yet another model for the representation πλ of G.
Let � = ∂2
∂t2
−
n∑
j=0
∂2
∂x2j
be the d’Alembertian on E. It satisfies, for any g ∈ G and F a smooth
function on E
�(F ◦ g) = (�F ) ◦ g. (7.2)
The following lemma, which I learnt from [3] is a key result for what follows.
Lemma 7.1. Let F1, F2 be two smooth functions defined in a neighborhood of Ξ+, posi-
tively homogeneous of degree −n
2 + 1 and which coincide on Ξ+. Then �F1 and �F2 coincide
on Ξ+.
Proof. The function F1 − F2 vanishes on Ξ+. Notice that dQ(x) 6= 0 for any x ∈ Ξ+. Hence,
there exists a smooth function G defined on a neighborhood of Ξ+ such that
F1(x)− F2(x) = Q(x)G(x).
Another Approach to Juhl’s Conformally Covariant Differential Operators 15
Moreover, G is positively homogeneous of degree −n
2 − 1. Now, for any smooth function H
on E
�(QH) = 2(n+ 2)H + 4EH +Q�H,
where E = t ∂∂t +
n∑
j=0
xj
∂
∂xj
is the Euler operator. As G is homogeneous of degree −n
2 − 1,
EG(x) =
(
−n
2
− 1
)
G(x),
and hence �(QG)(x) = 0 for x ∈ Ξ+. The lemma follows. �
The next result is a reformulation of the previous lemma.
Lemma 7.2. Let F ∈ Hn
2
−1. Extend F smoothly to a positively homogeneous function of degree
−n
2 + 1 to neighborhood of Ξ+. Then the restriction to Ξ+ of �F does not depend on the
extension.
The operator � induces a map from Hn
2
−1 to Hn
2
+1 and intertwines the action of G. Let ∆S
be the operator defined on C∞(S) by
∆S = Rn
2
+1 ◦� ◦R−1n
2
−1.
The invariance of � (see (7.2)) and the covariance of Rλ (see (7.1)) imply the following propo-
sition.
Proposition 7.3. The operator ∆S (conformal Laplacian or Yamabe operator on S) is a dif-
ferential operator on S which is covariant with respect to (πn
2
−1, πn
2
+1).
Let Bµ be the differential operator on E defined by
BµF (x) = xn�F (x)− 2µ
∂F
∂xn
.
Lemma 7.4. Let µ ∈ C. Let F be a smooth function on E. Then, on {xn 6= 0},
BµF (x) = xn|xn|−µ�
(
|xn|µF
)
(x) + µ(µ− 1)
1
xn
F (x). (7.3)
Proof. By an elementary calculation,
�
(
|xn|µF
)
(x) = |xn|µ�F (x)− 2µ sgn(xn)|xn|µ−1
∂F
∂xn
(x)− µ(µ− 1)|xn|µ−2F (x),
so that
�
(
|xn|µF
)
+ µ(µ− 1)|xn|µ−2F = sgn(xn)|xn|µ−1BµF.
The conclusion follows, by noticing that xn = sgn(xn)|xn|. �
Proposition 7.5. Let g ∈ G′. Then for F a smooth function on E,
Bµ(F ◦ g) = (BµF ) ◦ g.
Proof. As g ∈ G′, the coordinate xn is unchanged by the action of g, and the action of g
commutes with ∂
∂xn
and with �. The result follows. �
16 J.-L. Clerc
Proposition 7.6. Let F ∈ Hλ. Extend F smoothly to a neighborhood of Ξ+ as a positively
homogeneous function of degree −λ. Then the restriction to Ξ+ of Bλ−n
2
+1F does not depend
on the extension.
Proof. The function |xn|λ−
n
2
+1F (x) is homogenous of degree −n
2 +1, and hence, by Lemma 7.2,
for x ∈ Ξ+, �
(
|xn|λ−
n
2
+1F
)
(x) only depend on the values of F on Ξ+. Hence, by (7.3), for x
in Ξ+, xn 6= 0, Bλ−n
2
+1F (x) does not depend on the extension of F . The result follows by
continuity. �
Proposition 7.7. The differential operator Bλ−n
2
+1 induces a map from Hλ into Hλ+1, which
commutes with the action of G′.
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 Bλ−n
2
+1 in the non compact picture is
equal to −Eλ.
Proof. Let f be a smooth function on Rn. Recall the map c (cf. (4.1)) which realizes the
passage from Rn to S. Its inverse is given by
S \ {−1} 3 (x0, x1, . . . , xn) 7−→
(
x1
1 + x0
, . . . ,
xn
1 + x0
)
.
So map f to a function on S by
C−1λ f(x) = (1 + x0)
−λf
(
x1
1 + x0
, . . . ,
xn
1 + x0
)
.
Consider the function F on E defined by
F (x) = (t+ x0)
−λf
(
x1
t+ x0
, . . . ,
xn
t+ x0
)
.
Then F is homogenous of degree −λ and coincide on S with C−1λ f . To compute Bλ−n
2
+1F , first
observe that
∂F
∂t
=
∂F
∂x0
,
∂2F
∂t2
=
∂2F
∂x20
,
so that
�F = −
n∑
j=1
∂2F
∂x2j
.
Hence
Bλ−n
2
+1F (x) = −(t+ x0)
−λ−2xn(∆f)
(
x1
t+ x0
, . . . ,
xn
t+ x0
)
− 2
(
λ− n
2
+ 1
)
(t+ x0)
−λ(t+ x0)
−1 ∂f
∂ξn
(
x1
t+ x0
, . . . ,
xn
t+ x0
)
.
Now letting x = (1, c(ξ)),
Bλ−n
2
+1F (1, c(ξ)) = −ξn∆f(ξ)− (2λ− n+ 2)
∂f
∂ξn
(ξ).
A comparison with (4.6) implies the result. �
Another Approach to Juhl’s Conformally Covariant Differential Operators 17
Proposition 7.9. The expression of the operator Bλ−n
2
+1 on S is given by
xn|xn|−λ+
n
2
−1∆S ◦ |xn|λ−
n
2
+1 +
(
λ− n
2
+ 1
)(
λ− n
2
) 1
xn
.
The expression, a priori defined on xn 6= 0 can be continued continuously to all of S.
Proof. Let f ∈ C∞(S). Then
F (x) =
(
x20 + · · ·+ x2n
)−λ
f
(
x0√
x20 + · · ·+ x2n
, . . . ,
xn√
x20 + · · ·+ x2n
)
is a function defined on E \ {0} which is positively homogeneous of degree −λ and such that for
x ∈ S,
F (1, x) = f(x).
By (7.3) with µ = λ− n
2 + 1 and for x 6= 0, xn 6= 0,
Bλ−n
2
+1F (x) = xn|xn|−λ+
n
2
+1�
(
|xn|λ−
n
2
+1F
)
(x) +
(
λ− n
2
+ 1
)(
λ− n
2
) 1
xn
F (x).
The function |xn|λ−
n
2
+1F (x) is positively homogeneous of degree −n
2 + 1. Thus, by Lemma 7.2
and the definition of the Yamabe operator ∆S , for x ∈ S,
Bλ−n
2
+1F (1, x) = xn|xn|−λ+
n
2
−1∆S
(
|xn|λ−
n
2
+1f
)
(x) +
(
λ− n
2
+ 1
)(
λ− n
2
) 1
xn
f(x),
from which the statement follows, at least for xn 6= 0. As Bλ−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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http://arxiv.org/abs/1301.2111
https://doi.org/10.1007/s00029-015-0208-8
http://arxiv.org/abs/1301.2111
https://doi.org/10.1090/memo/1126
https://doi.org/10.1090/memo/1126
http://arxiv.org/abs/1310.3213
https://doi.org/10.1017/CBO9780511623660
1 Introduction
2 The principal series of SO0(1,n+1) and the Knapp–Stein intertwining operators
3 Construction of the family D"0365D, C
4 The expression of D"0365D in the non-compact picture
5 The families D,N, D"0365D,N and DN()
6 The family EN()
7 The operator D in the ambient space model
References
|
| id | nasplib_isofts_kiev_ua-123456789-148574 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-29T13:26:35Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Clerc, Jean-Louis 2019-02-18T16:02:02Z 2019-02-18T16:02:02Z 2017 Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ / Jean-Louis Clerc // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58J70; 43A85 DOI:10.3842/SIGMA.2017.026 https://nasplib.isofts.kiev.ua/handle/123456789/148574 A family (Dλ)λ∈C of differential operators on the sphere Sⁿ is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of Sⁿ which preserve the smaller sphere Sⁿ⁻¹ ⊂Sⁿ. The family of conformally covariant differential operators from Sⁿ to Sⁿ⁻¹ introduced by A. Juhl is obtained by composing these operators on Sⁿ and taking restrictions to Sⁿ⁻¹ . It is a pleasure to thank the anonymous referees for their contributions which helped to improve and reshape the initial version of this article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ Article published earlier |
| spellingShingle | Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ Clerc, Jean-Louis |
| title | Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ |
| title_full | Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ |
| title_fullStr | Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ |
| title_full_unstemmed | Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ |
| title_short | Another Approach to Juhl's Conformally Covariant Differential Operators from Sⁿ to Sⁿ⁻¹ |
| title_sort | another approach to juhl's conformally covariant differential operators from sⁿ to sⁿ⁻¹ |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148574 |
| work_keys_str_mv | AT clercjeanlouis anotherapproachtojuhlsconformallycovariantdifferentialoperatorsfromsntosn1 |