We give a complete classification of conformally covariant differential
operators between the spaces of differential $i$-forms on the sphere $S^n$ and
$j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the
restriction of principal series representations of the Lie group $O(n+1,1)$.
  Further, we provide explicit formul\ae{} for these matrix-valued operators in
the flat coordinates and find factorization identities for them.Comment: This note was published in C. R. Acad. Sci. Paris, Ser. I, (2016),
  http://dx.doi.org/10.1016/j.crma.2016.04.01

Kobayashi, Toshiyuki

Kubo, Toshihisa

Pevzner, Michael

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comGroup theory/Differential geometryClassification of differential symmetry breaking operators for differential forms ✩Classification des opérateurs de brisure de symétrie pour les formes différentiellesToshiyuki Kobayashi a, Toshihisa Kubo b, Michael Pevzner ca Kavli IPMU and Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japanb Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japanc Laboratoire de Mathématiques, Université de Reims-Champagne-Ardenne, CNRS FR 3399, 51687 Reims, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 31 March 2016Accepted after revision 28 April 2016Available online 17 May 2016Presented by Michel DufloWe give a complete classification of conformally covariant differential operators between the spaces of differential i-forms on the sphere Sn and j-forms on the totally geodesic hypersphere Sn−1 by analyzing the restriction of principal series representations of the Lie group O (n + 1, 1). Further, we provide explicit formulæ for these matrix-valued operators in the flat coordinates and find factorization identities for them.© 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).r é s u m éNous présentons une classification complète des opérateurs différentiels conformément covariants agissant entre les espaces des i-formes différentielles sur la sphère Sn et ceux des j-formes sur la hypershère totalement géodésique Sn−1 en analysant les restrictions des représentations des séries principales du groupe de Lie O (n + 1, 1). Pour de tels opórateurs à valeurs matricielles, nous donnons des formules explicites dans les coordonnées plates et trouvons des identités de factorisation.© 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).✩ The first named author was partially supported by Institut des Hautes Études Scientifiques, France and Grant-in-Aid for Scientific Research (A) (25247006), Japan Society for the Promotion of Science. All three authors were partially supported by CNRS Grant PICS No. 7270.E-mail addresses: toshi@ms.u-tokyo.ac.jp (T. Kobayashi), toskubo@ms.u-tokyo.ac.jp (T. Kubo), pevzner@univ-reims.fr (M. Pevzner).http://dx.doi.org/10.1016/j.crma.2016.04.0121631-073X/© 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).672 T. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–6761. IntroductionSuppose a Lie group G acts conformally on a Riemannian manifold (X, g). This means that there exists a positive-valued function  ∈ C∞(G × X) (conformal factor) such thatL∗h gh·x = (h, x)2 gx for all h ∈ G and x ∈ X,where Lh : X → X , x → h · x denotes the action of G on X . Since  satisfies a cocycle condition, we can form a family of representations (i)u for u ∈C and 0 ≤ i ≤ dim X on the space E i(X) of differential i-forms on X by(i)u (h)α := (h−1, ·)u L∗h−1α (h ∈ G). (1)The representation (i)u of the conformal group G on E i(X) will be simply denoted by E i(X)u .If Y is a submanifold of X , then we can also define a family of representations ( j)v on E j(Y ) (v ∈ C, 0 ≤ j ≤ dim Y ) of the subgroupG ′ := {h ∈ G : h · Y = Y } ,which acts conformally on the Riemannian submanifold (Y , g |Y ).We study differential operators D : E i(X) −→ E j(Y ) that intertwine the two representations (i)u |G ′ and ( j)v of G ′ . Here (i)u |G ′ stands for the restriction of the G-representation (i)u to the subgroup G ′ . We say that such D is a differential symmetry breaking operator, and denote by DiffG ′ (E i(X)u, E j(Y )v ) the space of all differential symmetry breaking operators. We address the following problems:Problem 1. Determine the dimension of the space DiffG ′(E i(X)u,E j(Y )v). In particular, find a necessary and sufficient condition on a quadruple (i, j, u, v) such that there exist nontrivial differential symmetry breaking operators.Problem 2. Construct explicitly a basis of DiffG ′(E i(X)u,E j(Y )v).In the case where X = Y , G = G ′ , and i = j = 0, a classical prototype of such operators is a second-order differential operator called the Yamabe operator + n − 24(n − 1)κ ∈ DiffG(E0(X) n2 −1,E0(X) n2 +1),where  is the Laplace–Beltrami operator, n is the dimension of X , and κ is the scalar curvature of X . Conformally covari-ant differential operators of higher order are also known: the Paneitz operator (fourth order) [11], which appears in four dimensional supergravity [2], or more generally, the so-called GJMS operators [3] are such examples. Analogous conformally covariant operators on forms (i = j case) were studied by Branson [1]. On the other hand, the insight of representation theory of conformal groups is useful in studying Maxwell’s equations, see [10], for instance.Let us consider the more general case where Y 	= X and G ′ 	= G . An obvious example of symmetry breaking operators is the restriction operator RestY which belongs to DiffG ′(E i(X)u,E i(Y )u)for all u ∈ C. Another elementary example is RestY ◦ ιNY (X) ∈ DiffG ′(E i(X)u,E i−1(Y )v)if v = u + 1 where ιNY (X) denotes the interior multiplication by the normal vector field to Y when Y is of codimension one in X .In the model space where (X, Y ) = (Sn, Sn−1), the pair (G, G ′) of conformal groups amounts to (O (n + 1, 1), O (n, 1))modulo center, and Problems 1 and 2 have been recently solved for i = j = 0 by Juhl [4], see also [5,7] and [9] for different approaches by the F-method and the residue calculus, respectively.Problems 1 and 2 for general i and j for the model space can be reduced to analogous problems for (nonspherical) principal series representations by the isomorphism (3) below. In this note we shall give complete solutions to Problems 1and 2 in those terms (see Theorems 3 and 4).Notation: N = {0, 1, 2, · · · }, N+ = {1, 2, · · · }.2. Principal series representations of G = O (n + 1, 1)We set up notations. Let P = M AN be a Langlands decomposition of a minimal parabolic subgroup of G = O (n +1, 1). For 0 ≤ i ≤ n, δ ∈ Z/2Z, and λ ∈ C, we extend the outer tensor product representation ∧i(Cn) ⊗ (−1)δ ⊗Cλ of M A  (O (n) ×O (1)) ×R to P by letting N act trivially, and form a G-equivariant vector bundle V iλ,δ := G ×P(∧i(Cn) ⊗ (−1)δ ⊗Cλ)over the real flag variety X = G/P  Sn . Then we define an unnormalized principal series representationsI(i, λ)δ := IndGP(i∧(Cn) ⊗ (−1)δ ⊗Cλ)(2)of G on the Fréchet space C∞(X, V i ) of smooth sections.λ,δT. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676 673In our parameterization, I(i, n − 2i)δ and I(i, i)δ have the same infinitesimal character with the trivial one-dimensional representation of G . Then, for all u ∈ C, we have a natural G-isomorphism(i)u  I(i, u + i)i mod 2. (3)Similarly, for 0 ≤ j ≤ n − 1, ε ∈ Z/2Z and ν ∈ C, we define an unnormalized principal series representationJ ( j, ν)ε := IndG ′P ′(∧ j(Cn−1) ⊗ (−1)ε ⊗Cν)of the subgroup G ′ = O (n, 1) on C∞(Y , W jν,ε), where W jν,ε := G ′ ×P ′(∧ j(Cn−1) ⊗ (−1)ε ⊗Cν)is a G ′-equivariant vector bundle over Y = G ′/P ′  Sn−1.3. Existence condition for differential symmetry breaking operatorsA continuous G ′-intertwining operator T : I(i, λ)δ −→ J ( j, ν)ε is said to be a symmetry breaking operator (SBO). We say that T is a differential operator if T satisfies Supp(T f ) ⊂ Supp f for all f ∈ C∞(X, V iλ,δ), and DiffG ′ (I(i, λ)δ, J ( j, ν)ε) denotes the space of differential SBOs. We give a complete solution to Problem 1 for (X, Y ) = (Sn, Sn−1) in terms of principal series representations:Theorem 3. Let n ≥ 3. Suppose 0 ≤ i ≤ n, 0 ≤ j ≤ n − 1, λ, ν ∈ C, and δ, ε ∈ Z/2Z. Then the following three conditions on 6-tuple (i, j, λ, ν, δ, ε) are equivalent:(i) DiffO (n,1)(I(i, λ)δ, J ( j, ν)ε) 	= {0}.(ii) dim DiffO (n,1)(I(i, λ)δ, J ( j, ν)ε) = 1.(iii) The 6-tuple belongs to one of the following six cases:Case 1. j = i, 0 ≤ i ≤ n − 1, ν − λ ∈N, ε − δ ≡ ν − λ mod 2.Case 2. j = i − 1, 1 ≤ i ≤ n, ν − λ ∈N, ε − δ ≡ ν − λ mod 2.Case 3. j = i + 1, 1 ≤ i ≤ n − 2, (λ, ν) = (i, i + 1), ε ≡ δ + 1 mod 2.Case 3′ . (i, j) = (0, 1), −λ ∈ N, ν = 1, ε ≡ δ + λ + 1 mod 2.Case 4. j = i − 2, 2 ≤ i ≤ n − 1, (λ, ν) = (n − i, n − i + 1), ε ≡ δ + 1 mod 2.Case 4′ . (i, j) = (n, n − 2), −λ ∈N, ν = 1, ε ≡ δ + λ + 1 mod 2.We set  := {(i, j, λ, ν): the 6-tuple (i, j, λ, ν, δ, ε) satisfies one of the equivalent conditions of Theorem 3 for some δ, ε ∈Z/2Z}.4. Construction of differential symmetry breaking operatorsIn this section, we describe an explicit generator of the space of differential SBOs if one of the equivalent conditions in Theorem 3 is satisfied. For this, we use the flat picture of the principal series representations I(i, λ)δ of G , which realizes the representation space C∞(X, V iλ,δ) as a subspace of C∞(Rn, ∧i(Cn)) by trivializing the bundle V iλ,δ −→ X on the open Bruhat cellRn ↪→ X, (x1, · · · , xn) → exp⎛⎝ n∑j=1x j N−j⎞⎠ P .Here {N−1 , · · · , N−n } is an orthonormal basis of the nilradical n−(R) of the opposite parabolic subalgebra with respect to an M-invariant inner product. Without loss of generality, we may and do assume that the open Bruhat cell Rn−1 ↪→ Y  G ′/P ′is given by putting xn = 0. Then the flat picture of the principal series representation J ( j, ν)ε of G ′ is defined by realizing C∞(Y , W jν,ε) as a subspace of C∞(Rn−1, ∧ j(Cn−1)). For the construction of explicit generators of matrix-valued SBOs, we begin with a scalar-valued differential operator. For α ∈C and  ∈N, we define a polynomial of two variables (s, t) by(IC̃α)(s, t) := s 2 C̃α(t√s),where C̃α (z) is the renormalized Gegenbauer polynomial given byC̃α (z) :=1(α +[+12])[2]∑k=0(−1)k ( − k + α)k!( − 2k)! (2z)−2k.674 T. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676Then C̃α (z) is a nonzero polynomial for all α ∈ C and  ∈ N, and a (normalized) Juhl’s conformally covariant operator C̃λ,ν :C∞(Rn) −→ C∞(Rn−1) is defined byC̃λ,ν := Restxn=0 ◦(IC̃λ− n−12)(−Rn−1 ,∂∂xn),for λ, ν ∈C with  := ν − λ ∈N. For instance,C̃λ,ν = Restxn=0 ◦⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩id if ν = λ,2∂∂xnif ν = λ + 1,Rn−1 + (2λ − n + 3)∂2∂x2nif ν = λ + 2.For (i, j, λ, ν) ∈ , we introduce a new family of matrix-valued differential operatorsC̃i, jλ,ν : C∞(Rn,i∧(Cn)) −→ C∞(Rn−1,j∧(Cn−1)),by using the identifications E i(Rn)  C∞(Rn) ⊗ ∧i(Cn) and E j(Rn−1)  C∞(Rn−1) ⊗ ∧ j(Cn−1), as follows. Let d∗Rnbe the codifferential, which is the formal adjoint of the differential dRn , and ι ∂∂xnthe inner multiplication by the vector field ∂∂xn. Both operators map E i(Rn) to E i−1(Rn). For α ∈ C and  ∈ N, let γ (α, ) := 1 ( is odd); = α + 2 ( is even). Then we setCi,iλ,ν := C̃λ+1,ν−1dRn d∗Rn − γ (λ −n2, ν − λ)C̃λ,ν−1dRn ι ∂∂xn+ 12(ν − i)C̃λ,ν for 0 ≤ i ≤ n − 1.Ci,i−1λ,ν := −C̃λ+1,ν−1dRn d∗Rn ι ∂∂xn− γ (λ − n − 12, ν − λ)C̃λ+1,νd∗Rn +12(λ + i − n)C̃λ,ν ι ∂∂xnfor 1 ≤ i ≤ n.We note that there exist isolated parameters (λ, ν) for which Ci,iλ,ν = 0 or Ci,i−1λ,ν = 0. For instance, C0,0λ,ν = 12 νC̃λ,ν , and thus C0,0λ,ν = 0 if ν = 0. To be precise, we have the following:Ci,iλ,ν = 0 if and only if λ = ν = i or ν = i = 0; Ci,i−1λ,ν = 0 if and only if λ = ν = n − i or ν = n − i = 0.We renormalize these operators byC̃i,iλ,ν :=⎧⎪⎨⎪⎩Restxn=0 if λ = ν,C̃λ,ν if i = 0,Ci,iλ,ν otherwise,and C̃i,i−1λ,ν :=⎧⎪⎪⎨⎪⎪⎩Restxn=0 ◦ ι ∂∂xnif λ = ν,C̃λ,ν ◦ ι ∂∂xnif i = n,Ci,i−1λ,ν otherwise.Then C̃i,iλ,ν (0 ≤ i ≤ n − 1) and C̃i,i−1λ,ν (1 ≤ i ≤ n) are nonzero differential operators of order ν − λ for any λ, ν ∈ C with ν − λ ∈N.The differential operators C̃i,i+1λ,ν and C̃i,i−2λ,ν are defined only for special parameters (λ, ν) as follows.C̃i,i+1λ,i+1 :={Restxn=0 ◦ dRn for 1 ≤ i ≤ n − 2, λ = i,dRn−1 ◦ C̃λ,0 for i = 0, λ ∈ −N,C̃i,i−2λ,n−i+1 :={Restxn=0 ◦ ι ∂∂xnd∗Rnfor 2 ≤ i ≤ n, λ = n − i,−d∗Rn−1 ◦Cn,n−1λ,0 for i = n, λ ∈ −N.Then we give a complete solution to Problem 2 for the model space (X, Y ) = (Sn, Sn−1) in terms of the flat picture of principal series representations as follows:Theorem 4. Suppose a 6-tuple (i, j, λ, ν, δ, ε) satisfies one of the equivalent conditions in Theorem 3. Then the operators C̃i, jλ,ν :C∞(Rn) ⊗ ∧i(Cn) −→ C∞(Rn−1) ⊗ ∧ j(Cn−1) extend to differential SBOs I(i, λ)δ −→ J ( j, ν)ε , to be denoted by the same letters. Conversely, any differential SBO from I(i, λ)δ to J ( j, ν)ε is proportional to the following differential operators: ̃Ci,iλ,ν in Case 1, ̃Ci,i−1λ,νin Case 2, ̃Ci,i+1i,i+1 in Case 3, ̃C0,1λ,1 in Case 3′ , ̃Ci,i−2n−i,n−i+1 in Case 4, and ̃Cn,n−2λ,1 in Case 4′ .5. Matrix-valued factorization identitiesSuppose that T X : I(i, λ′)δ → I(i, λ)δ or TY : J ( j, ν)ε → J ( j, ν ′)ε are G- or G ′-intertwining operators, respectively. Then the composition T Y ◦ D X→Y or D X→Y ◦ T X of a symmetry breaking operator D X→Y : I(i, λ)δ → J ( j, ν)ε gives another symmetry breaking operator:T. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676 675I(i, λ)δD X→Y J ( j, ν)εT YI(i, λ′)δT XJ ( j, ν ′)εThe multiplicity-free property (see Theorem 3 (ii)) assures the existence of matrix-valued factorization identities for differential SBOs, namely, D X→Y ◦ T X must be a scalar multiple of C̃i, jλ′,ν , and TY ◦ D X→Y must be a scalar multiple of C̃i, jλ,ν ′ .We shall determine these constants explicitly when T X or TY are Branson’s conformally covariant operators [1] defined below. Let 0 ≤ i ≤ n. For  ∈N+ , we setT (i)2 := ((n2− i − )dRn d∗Rn + (n2− i + )d∗Rn dRn )−1Rn= (−2dRn d∗Rn − (n2− i + )Rn)−1Rn .Then the differential operator T (i)2 : E i(Rn) −→ E i(Rn) induces a nonzero O (n +1, 1)-intertwining operator, to be denoted by the same letter T (i)2 , from I(i, n2 − )δto I(i, n2 + )δ, for δ ∈ Z/2Z. Similarly, we define a G ′-intertwining operator T ′( j)2 : J(j, n−12 − )ε−→ J(j, n−12 + )εfor 0 ≤ j ≤ n − 1 and ε ∈ Z/2Z as the lift of the differential operator T ′( j)2 :E j(Rn−1) −→ E j(Rn−1), which is given byT ′( j)2 = ((n − 12− j − )dRn−1d∗Rn−1 + (n − 12− j + )d∗Rn−1dRn−1)−1Rn−1 .Consider the following diagrams for j = i and j = i − 1:I(i, n2 − )δT (i)2C̃i, jn2 −, n2 +a+I(i, n2 + )δC̃i, jn2 +, n2 +a+J(j, n2 + a + )ε,I(i, n−12 − a − )δC̃i, jn−12 −a−, n−12 −C̃i, jn−12 −a−, n−12 +J(j, n−12 − )εT ′( j)2J(j, n−12 + )ε,where parameters δ and ε ∈ Z/2Z are chosen according to Theorem 3 (iii). In what follows, we putp± =⎧⎨⎩i ±  −n2if a 	= 0±2 if a = 0, q =⎧⎪⎪⎪⎨⎪⎪⎪⎩i +  − n − 12if i 	= 0,a 	= 0−2 if i 	= 0,a = 0−( + n − 12)if i = 0, r =⎧⎪⎪⎪⎨⎪⎪⎪⎩i −  − n + 12if i 	= n,a 	= 02 if i 	= n,a = 0−( + n + 12)if i = n,K,a :=∏k=1([a2]+ k).Then the factorization identities for differential SBOs C̃i, jλ,ν for j ∈ {i − 1, i} and Branson’s conformally covariant operators T (i)2 or T ′( j)2 are given as follows.Theorem 5. Suppose 0 ≤ i ≤ n − 1, a ∈N and  ∈N+ . Then(1) C̃i,in2 +,a++ n2 ◦ T(i)2 = p−K,aC̃i,in2 −,a++ n2 .(2) T ′(i)2 ◦ C̃i,in−12 −a−, n−12 −= qK,aC̃i,in−12 −a−, n−12 +.Theorem 6. Suppose 1 ≤ i ≤ n, a ∈N and  ∈N+ . Then(1) C̃i,i−1n2 +,a++ n2 ◦ T(i)2 = p+K,aC̃i,i−1n2 −,a++ n2 .(2) T ′(i−1)2 ◦ C̃i,i−1n−12 −a−, n−12 −= rK,aC̃i,i−1n−12 −a−, n−12 +.In the case where i = 0, C̃i,iλ,ν is a scalar-valued operator, and the corresponding factorization identities in Theorem 5 were studied in [4,8,9].The main results are proved by using the F-method [5,6,9]. Details will appear elsewhere.676 T. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676References[1] T.P. Branson, Conformally covariant equations on differential forms, Commun. Partial Differ. Equ. 7 (1982) 393–431.[2] E.S. Fradkin, A.A. Tseytlin, Asymptotic freedom in extended conformal supergravities, Phys. Lett. B 110 (1982) 117–122.[3] C.R. Graham, R. Jenne, L.J. Mason, G.A.J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. Lond. Math. Soc. (2) 46 (1992) 557–565.[4] A. Juhl, Families of Conformally Covariant Differential Operators, Q -Curvature and Holography, Prog. Math., vol. 275, Birkhäuser, Basel, Switzerland, 2009.[5] T. Kobayashi, F-method for symmetry breaking operators, Differ. Geom. Appl. 33 (2014) 272–289.[6] T. Kobayashi, M. Pevzner, Differential symmetry breaking operators. I. General theory and F-method, Sel. Math. (N.S.) 22 (2016) 801–845.[7] T. Kobayashi, M. Pevzner, Differential symmetry breaking operators. II. Rankin–Cohen operators for symmetric pairs, Sel. Math. (N.S.) 22 (2016) 847–911.[8] T. Kobayashi, B. Speh, Symmetry Breaking for Representations of Rank One Orthogonal Groups, Mem. Amer. Math. Soc., vol. 238, ISBN 978-1-4704-1922-6, 2015, 118 p.[9] T. Kobayashi, B. Ørsted, P. Somberg, V. Souček, Branching laws for Verma modules and applications in parabolic geometry. I, Adv. Math. 285 (2015) 1796–1852.[10] B. Kostant, N.R. Wallach, Action of the conformal group on steady state solutions to Maxwell’s equations and background radiation, in: Symmetry: Representation Theory and Its Applications, in: Prog. Math., vol. 257, Birkhäuser/Springer, New York, 2014, pp. 385–418.[11] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008) paper 036, 3 p.

Classification of differential symmetry breaking operators for differential forms

AbstractWe give a complete classification of conformally covariant differential operators between the spaces of differential i-forms on the sphere Sn and j-forms on the totally geodesic hypersphere Sn−1 by analyzing the restriction of principal series representations of the Lie group O(n+1,1). Further, we provide explicit formulæ for these matrix-valued operators in the flat coordinates and find factorization identities for them

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C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676Contents lists available at ScienceDirect
C. R. Acad. Sci. Paris, Ser. I
www.sciencedirect.com
Group theory/Differential geometry
Classiﬁcation of differential symmetry breaking operators for 
differential forms✩
Classiﬁcation des opérateurs de brisure de symétrie pour les formes 
différentielles
Toshiyuki Kobayashi a, Toshihisa Kubo b, Michael Pevzner c
a Kavli IPMU and Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan
b Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan
c Laboratoire de Mathématiques, Université de Reims-Champagne-Ardenne, CNRS FR 3399, 51687 Reims, France
a r t i c l e i n f o a b s t r a c t
Article history:
Received 31 March 2016
Accepted after revision 28 April 2016
Available online 17 May 2016
Presented by Michel Duﬂo
We give a complete classiﬁcation of conformally covariant differential operators between 
the spaces of differential i-forms on the sphere Sn and j-forms on the totally geodesic 
hypersphere Sn−1 by analyzing the restriction of principal series representations of the Lie 
group O (n + 1, 1). Further, we provide explicit formulæ for these matrix-valued operators 
in the ﬂat coordinates and ﬁnd factorization identities for them.
© 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access 
article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
r é s u m é
Nous présentons une classiﬁcation complète des opérateurs différentiels conformément 
covariants agissant entre les espaces des i-formes différentielles sur la sphère Sn et 
ceux des j-formes sur la hypershère totalement géodésique Sn−1 en analysant les 
restrictions des représentations des séries principales du groupe de Lie O (n + 1, 1). Pour 
de tels opórateurs à valeurs matricielles, nous donnons des formules explicites dans les 
coordonnées plates et trouvons des identités de factorisation.
© 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access 
article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
✩ The ﬁrst named author was partially supported by Institut des Hautes Études Scientiﬁques, France and Grant-in-Aid for Scientiﬁc Research (A) 
(25247006), Japan Society for the Promotion of Science. All three authors were partially supported by CNRS Grant PICS No. 7270.
E-mail addresses: toshi@ms.u-tokyo.ac.jp (T. Kobayashi), toskubo@ms.u-tokyo.ac.jp (T. Kubo), pevzner@univ-reims.fr (M. Pevzner).http://dx.doi.org/10.1016/j.crma.2016.04.012
1631-073X/© 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
672 T. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–6761. Introduction
Suppose a Lie group G acts conformally on a Riemannian manifold (X, g). This means that there exists a positive-valued 
function  ∈ C∞(G × X) (conformal factor) such that
L∗h gh·x = (h, x)2gx for all h ∈ G and x ∈ X,
where Lh : X → X , x → h · x denotes the action of G on X . Since  satisﬁes a cocycle condition, we can form a family of 
representations (i)u for u ∈C and 0 ≤ i ≤ dim X on the space E i(X) of differential i-forms on X by

(i)
u (h)α := (h−1, ·)u L∗h−1α (h ∈ G). (1)
The representation (i)u of the conformal group G on E i(X) will be simply denoted by E i(X)u .
If Y is a submanifold of X , then we can also deﬁne a family of representations ( j)v on E j(Y ) (v ∈C, 0 ≤ j ≤ dim Y ) of 
the subgroup
G ′ := {h ∈ G : h · Y = Y } ,
which acts conformally on the Riemannian submanifold (Y , g |Y ).
We study differential operators D : E i(X) −→ E j(Y ) that intertwine the two representations (i)u |G ′ and ( j)v of G ′ . 
Here (i)u |G ′ stands for the restriction of the G-representation (i)u to the subgroup G ′ . We say that such D is a differential 
symmetry breaking operator, and denote by DiffG ′ (E i(X)u, E j(Y )v ) the space of all differential symmetry breaking operators. 
We address the following problems:
Problem 1. Determine the dimension of the space DiffG ′
(E i(X)u,E j(Y )v). In particular, ﬁnd a necessary and suﬃcient 
condition on a quadruple (i, j, u, v) such that there exist nontrivial differential symmetry breaking operators.
Problem 2. Construct explicitly a basis of DiffG ′
(E i(X)u,E j(Y )v).
In the case where X = Y , G = G ′ , and i = j = 0, a classical prototype of such operators is a second-order differential 
operator called the Yamabe operator
+ n − 2
4(n − 1)κ ∈ DiffG(E
0(X) n
2−1,E
0(X) n
2+1),
where  is the Laplace–Beltrami operator, n is the dimension of X , and κ is the scalar curvature of X . Conformally covari-
ant differential operators of higher order are also known: the Paneitz operator (fourth order) [11], which appears in four 
dimensional supergravity [2], or more generally, the so-called GJMS operators [3] are such examples. Analogous conformally 
covariant operators on forms (i = j case) were studied by Branson [1]. On the other hand, the insight of representation 
theory of conformal groups is useful in studying Maxwell’s equations, see [10], for instance.
Let us consider the more general case where Y 	= X and G ′ 	= G . An obvious example of symmetry breaking operators 
is the restriction operator RestY which belongs to DiffG ′
(E i(X)u,E i(Y )u) for all u ∈ C. Another elementary example is 
RestY ◦ ιNY (X) ∈ DiffG ′
(E i(X)u,E i−1(Y )v) if v = u + 1 where ιNY (X) denotes the interior multiplication by the normal vector 
ﬁeld to Y when Y is of codimension one in X .
In the model space where (X, Y ) = (Sn, Sn−1), the pair (G, G ′) of conformal groups amounts to (O (n + 1, 1), O (n, 1))
modulo center, and Problems 1 and 2 have been recently solved for i = j = 0 by Juhl [4], see also [5,7] and [9] for different 
approaches by the F-method and the residue calculus, respectively.
Problems 1 and 2 for general i and j for the model space can be reduced to analogous problems for (nonspherical) 
principal series representations by the isomorphism (3) below. In this note we shall give complete solutions to Problems 1
and 2 in those terms (see Theorems 3 and 4).
Notation: N = {0, 1, 2, · · · }, N+ = {1, 2, · · · }.
2. Principal series representations of G = O (n + 1, 1)
We set up notations. Let P = MAN be a Langlands decomposition of a minimal parabolic subgroup of G = O (n +1, 1). For 
0 ≤ i ≤ n, δ ∈ Z/2Z, and λ ∈C, we extend the outer tensor product representation ∧i(Cn) ⊗ (−1)δ ⊗Cλ of MA  (O (n) ×
O (1)) ×R to P by letting N act trivially, and form a G-equivariant vector bundle V iλ,δ := G ×P
(∧i
(Cn) ⊗ (−1)δ ⊗Cλ
)
over 
the real ﬂag variety X = G/P  Sn . Then we deﬁne an unnormalized principal series representations
I(i, λ)δ := IndGP
(
i∧
(Cn)⊗ (−1)δ ⊗Cλ
)
(2)
of G on the Fréchet space C∞(X, V i ) of smooth sections.λ,δ
T. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676 673In our parameterization, I(i, n − 2i)δ and I(i, i)δ have the same inﬁnitesimal character with the trivial one-dimensional 
representation of G . Then, for all u ∈C, we have a natural G-isomorphism

(i)
u  I(i,u + i)imod 2. (3)
Similarly, for 0 ≤ j ≤ n − 1, ε ∈ Z/2Z and ν ∈ C, we deﬁne an unnormalized principal series representation
J ( j, ν)ε := IndG ′P ′
(∧ j
(Cn−1) ⊗ (−1)ε ⊗Cν
)
of the subgroup G ′ = O (n, 1) on C∞(Y , W jν,ε), where W jν,ε := G ′ ×P ′(∧ j
(Cn−1) ⊗ (−1)ε ⊗Cν
)
is a G ′-equivariant vector bundle over Y = G ′/P ′  Sn−1.
3. Existence condition for differential symmetry breaking operators
A continuous G ′-intertwining operator T : I(i, λ)δ −→ J ( j, ν)ε is said to be a symmetry breaking operator (SBO). We say 
that T is a differential operator if T satisﬁes Supp(T f ) ⊂ Supp f for all f ∈ C∞(X, V iλ,δ), and DiffG ′ (I(i, λ)δ, J ( j, ν)ε) denotes 
the space of differential SBOs. We give a complete solution to Problem 1 for (X, Y ) = (Sn, Sn−1) in terms of principal series 
representations:
Theorem 3. Let n ≥ 3. Suppose 0 ≤ i ≤ n, 0 ≤ j ≤ n − 1, λ, ν ∈ C, and δ, ε ∈ Z/2Z. Then the following three conditions on 6-tuple 
(i, j, λ, ν, δ, ε) are equivalent:
(i) DiffO (n,1)(I(i, λ)δ, J ( j, ν)ε) 	= {0}.
(ii) dimDiffO (n,1)(I(i, λ)δ, J ( j, ν)ε) = 1.
(iii) The 6-tuple belongs to one of the following six cases:
Case 1. j = i, 0 ≤ i ≤ n − 1, ν − λ ∈N, ε − δ ≡ ν − λ mod2.
Case 2. j = i − 1, 1 ≤ i ≤ n, ν − λ ∈N, ε − δ ≡ ν − λ mod2.
Case 3. j = i + 1, 1 ≤ i ≤ n − 2, (λ, ν) = (i, i + 1), ε ≡ δ + 1 mod2.
Case 3′ . (i, j) = (0, 1), −λ ∈N, ν = 1, ε ≡ δ + λ + 1 mod2.
Case 4. j = i − 2, 2 ≤ i ≤ n − 1, (λ, ν) = (n − i, n − i + 1), ε ≡ δ + 1 mod2.
Case 4′ . (i, j) = (n, n − 2), −λ ∈N, ν = 1, ε ≡ δ + λ + 1 mod2.
We set  := {(i, j, λ, ν): the 6-tuple (i, j, λ, ν, δ, ε) satisﬁes one of the equivalent conditions of Theorem 3 for some δ, ε ∈
Z/2Z}.
4. Construction of differential symmetry breaking operators
In this section, we describe an explicit generator of the space of differential SBOs if one of the equivalent conditions in 
Theorem 3 is satisﬁed. For this, we use the ﬂat picture of the principal series representations I(i, λ)δ of G , which realizes 
the representation space C∞(X, V iλ,δ) as a subspace of C∞(Rn, 
∧i
(Cn)) by trivializing the bundle V iλ,δ −→ X on the open 
Bruhat cell
R
n ↪→ X, (x1, · · · , xn) → exp
⎛⎝ n∑
j=1
x jN
−
j
⎞⎠ P .
Here {N−1 , · · · , N−n } is an orthonormal basis of the nilradical n−(R) of the opposite parabolic subalgebra with respect to an 
M-invariant inner product. Without loss of generality, we may and do assume that the open Bruhat cell Rn−1 ↪→ Y  G ′/P ′
is given by putting xn = 0. Then the ﬂat picture of the principal series representation J ( j, ν)ε of G ′ is deﬁned by realizing 
C∞(Y , W jν,ε) as a subspace of C∞(Rn−1, 
∧ j
(Cn−1)). For the construction of explicit generators of matrix-valued SBOs, we 
begin with a scalar-valued differential operator. For α ∈C and  ∈N, we deﬁne a polynomial of two variables (s, t) by(
IC˜
α

)
(s, t) := s 2 C˜α
(
t√
s
)
,
where C˜α (z) is the renormalized Gegenbauer polynomial given by
C˜α (z) :=
1

(
α +
[
+1
2
])
[

2
]∑
k=0
(−1)k (− k + α)
k!( − 2k)! (2z)
−2k.
674 T. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676Then C˜α (z) is a nonzero polynomial for all α ∈ C and  ∈ N, and a (normalized) Juhl’s conformally covariant operator C˜λ,ν :
C∞(Rn) −→ C∞(Rn−1) is deﬁned by
C˜λ,ν := Restxn=0 ◦
(
IC˜
λ− n−12

)(
−Rn−1 ,
∂
∂xn
)
,
for λ, ν ∈C with  := ν − λ ∈N. For instance,
C˜λ,ν = Restxn=0 ◦
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
id if ν = λ,
2
∂
∂xn
if ν = λ+ 1,
Rn−1 + (2λ− n + 3)
∂2
∂x2n
if ν = λ+ 2.
For (i, j, λ, ν) ∈ , we introduce a new family of matrix-valued differential operators
C˜
i, j
λ,ν : C∞(Rn,
i∧
(Cn)) −→ C∞(Rn−1,
j∧
(Cn−1)),
by using the identiﬁcations E i(Rn)  C∞(Rn) ⊗∧i(Cn) and E j(Rn−1)  C∞(Rn−1) ⊗∧ j(Cn−1), as follows. Let d∗
Rn
be the 
codifferential, which is the formal adjoint of the differential dRn , and ι ∂
∂xn
the inner multiplication by the vector ﬁeld ∂
∂xn
. 
Both operators map E i(Rn) to E i−1(Rn). For α ∈C and  ∈N, let γ (α, ) := 1 ( is odd); = α + 2 ( is even). Then we set
C
i,i
λ,ν := C˜λ+1,ν−1dRnd∗Rn − γ (λ−
n
2
, ν − λ)C˜λ,ν−1dRn ι ∂
∂xn
+ 1
2
(ν − i)C˜λ,ν for 0 ≤ i ≤ n − 1.
C
i,i−1
λ,ν := −C˜λ+1,ν−1dRnd∗Rn ι ∂
∂xn
− γ (λ− n − 1
2
, ν − λ)C˜λ+1,νd∗Rn +
1
2
(λ+ i − n)C˜λ,ν ι ∂
∂xn
for 1 ≤ i ≤ n.
We note that there exist isolated parameters (λ, ν) for which Ci,iλ,ν = 0 or Ci,i−1λ,ν = 0. For instance, C0,0λ,ν = 12νC˜λ,ν , and thus 
C
0,0
λ,ν = 0 if ν = 0. To be precise, we have the following:
C
i,i
λ,ν = 0 if and only if λ = ν = i or ν = i = 0; Ci,i−1λ,ν = 0 if and only if λ = ν = n − i or ν = n − i = 0.
We renormalize these operators by
C˜
i,i
λ,ν :=
⎧⎪⎨⎪⎩
Restxn=0 if λ = ν,
C˜λ,ν if i = 0,
C
i,i
λ,ν otherwise,
and C˜i,i−1λ,ν :=
⎧⎪⎪⎨⎪⎪⎩
Restxn=0 ◦ ι ∂
∂xn
if λ = ν,
C˜λ,ν ◦ ι ∂
∂xn
if i = n,
C
i,i−1
λ,ν otherwise.
Then C˜i,iλ,ν (0 ≤ i ≤ n − 1) and C˜i,i−1λ,ν (1 ≤ i ≤ n) are nonzero differential operators of order ν − λ for any λ, ν ∈ C with 
ν − λ ∈N.
The differential operators C˜i,i+1λ,ν and C˜
i,i−2
λ,ν are deﬁned only for special parameters (λ, ν) as follows.
C˜
i,i+1
λ,i+1 :=
{
Restxn=0 ◦ dRn for 1 ≤ i ≤ n − 2, λ = i,
dRn−1 ◦ C˜λ,0 for i = 0, λ ∈ −N,
C˜
i,i−2
λ,n−i+1 :=
{
Restxn=0 ◦ ι ∂
∂xn
d∗
Rn
for 2 ≤ i ≤ n, λ = n − i,
−d∗
Rn−1 ◦Cn,n−1λ,0 for i = n, λ ∈ −N.
Then we give a complete solution to Problem 2 for the model space (X, Y ) = (Sn, Sn−1) in terms of the ﬂat picture of 
principal series representations as follows:
Theorem 4. Suppose a 6-tuple (i, j, λ, ν, δ, ε) satisﬁes one of the equivalent conditions in Theorem 3. Then the operators C˜i, jλ,ν :
C∞(Rn) ⊗∧i(Cn) −→ C∞(Rn−1) ⊗∧ j(Cn−1) extend to differential SBOs I(i, λ)δ −→ J ( j, ν)ε , to be denoted by the same letters. 
Conversely, any differential SBO from I(i, λ)δ to J ( j, ν)ε is proportional to the following differential operators: ˜C
i,i
λ,ν in Case 1, ˜C
i,i−1
λ,ν
in Case 2, ˜Ci,i+1i,i+1 in Case 3, ˜C
0,1
λ,1 in Case 3
′ , ˜Ci,i−2n−i,n−i+1 in Case 4, and ˜C
n,n−2
λ,1 in Case 4
′ .
5. Matrix-valued factorization identities
Suppose that T X : I(i, λ′)δ → I(i, λ)δ or TY : J ( j, ν)ε → J ( j, ν ′)ε are G- or G ′-intertwining operators, respectively. Then 
the composition TY ◦ DX→Y or DX→Y ◦ T X of a symmetry breaking operator DX→Y : I(i, λ)δ → J ( j, ν)ε gives another 
symmetry breaking operator:
T. Kobayashi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 671–676 675I(i, λ)δ
DX→Y J ( j, ν)ε
TY
I(i, λ′)δ
T X
J ( j, ν ′)ε
The multiplicity-free property (see Theorem 3 (ii)) assures the existence of matrix-valued factorization identities for 
differential SBOs, namely, DX→Y ◦ T X must be a scalar multiple of C˜i, jλ′,ν , and TY ◦ DX→Y must be a scalar multiple of C˜i, jλ,ν ′ .
We shall determine these constants explicitly when T X or TY are Branson’s conformally covariant operators [1] deﬁned 
below. Let 0 ≤ i ≤ n. For  ∈N+ , we set
T (i)2 := ((
n
2
− i − )dRnd∗Rn + (
n
2
− i + )d∗
RndRn )
−1
Rn
= (−2dRnd∗Rn − (
n
2
− i + )Rn)−1Rn .
Then the differential operator T (i)2 : E i(Rn) −→ E i(Rn) induces a nonzero O (n +1, 1)-intertwining operator, to be denoted 
by the same letter T (i)2 , from I
(
i, n2 − 
)
δ
to I
(
i, n2 + 
)
δ
, for δ ∈ Z/2Z. Similarly, we deﬁne a G ′-intertwining operator 
T ′( j)2 : J
(
j, n−12 − 
)
ε
−→ J
(
j, n−12 + 
)
ε
for 0 ≤ j ≤ n − 1 and ε ∈ Z/2Z as the lift of the differential operator T ′( j)2 :
E j(Rn−1) −→ E j(Rn−1), which is given by
T ′( j)2 = ((
n − 1
2
− j − )dRn−1d∗Rn−1 + (
n − 1
2
− j + )d∗
Rn−1dRn−1)
−1
Rn−1 .
Consider the following diagrams for j = i and j = i − 1:
I
(
i, n2 − 
)
δ
T (i)2
C˜
i, j
n
2 −, n2 +a+
I
(
i, n2 + 
)
δ
C˜
i, j
n
2 +, n2 +a+
J
(
j, n2 + a + 
)
ε
,
I
(
i, n−12 − a − 
)
δ
C˜
i, j
n−1
2 −a−, n−12 −
C˜
i, j
n−1
2 −a−, n−12 +
J
(
j, n−12 − 
)
ε
T ′( j)2
J
(
j, n−12 + 
)
ε
,
where parameters δ and ε ∈ Z/2Z are chosen according to Theorem 3 (iii). In what follows, we put
p± =
⎧⎨⎩i ±  −
n
2
if a 	= 0
±2 if a = 0
, q =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
i + − n − 1
2
if i 	= 0,a 	= 0
−2 if i 	= 0,a = 0
−
(
+ n − 1
2
)
if i = 0
, r =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
i − − n + 1
2
if i 	= n,a 	= 0
2 if i 	= n,a = 0
−
(
+ n + 1
2
)
if i = n
,
K,a :=
∏
k=1
([a
2
]
+ k
)
.
Then the factorization identities for differential SBOs C˜i, jλ,ν for j ∈ {i − 1, i} and Branson’s conformally covariant operators 
T (i)2 or T ′( j)2 are given as follows.
Theorem 5. Suppose 0 ≤ i ≤ n − 1, a ∈N and  ∈N+ . Then
(1) C˜i,in
2+,a++ n2 ◦ T
(i)
2 = p−K,aC˜i,in
2−,a++ n2 .
(2) T ′(i)2 ◦ C˜i,in−1
2 −a−, n−12 −
= qK,aC˜i,in−1
2 −a−, n−12 +
.
Theorem 6. Suppose 1 ≤ i ≤ n, a ∈N and  ∈N+ . Then
(1) C˜i,i−1n
2+,a++ n2 ◦ T
(i)
2 = p+K,aC˜i,i−1n
2−,a++ n2 .
(2) T ′(i−1)2 ◦ C˜i,i−1n−1
2 −a−, n−12 −
= rK,aC˜i,i−1n−1
2 −a−, n−12 +
.
In the case where i = 0, C˜i,iλ,ν is a scalar-valued operator, and the corresponding factorization identities in Theorem 5 were 
studied in [4,8,9].
The main results are proved by using the F-method [5,6,9]. Details will appear elsewhere.
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Classification of differential symmetry breaking operators for differential forms 

Toshiyuki Kobayashi

Toshihisa Kubo

Michael Pevzner

Comptes Rendus Mathématique

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2016.04.012.pdf

Classification of differential symmetry breaking operators for differential forms

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