Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations
SSymmetry breaking differential operators fortensor products of spinorial representations
Jean-Louis Clerc and Khalid Koufany
Abstract
Let S be a Clifford module for the complexified Clifford algebra C (cid:96) p R n q , S its dual, ρ and ρ be the corresponding representations ofthe spin group Spin p R n q . The group G “ Spin p R ,n ` q is the (twofoldcovering) of the conformal group of R n . For λ, µ P C , let π ρ,λ (resp. π ρ ,µ ) be the spinorial representation of G on S -valued λ -densities(resp. S -valued µ -densities) on R n . For 0 ď k ď n and m P N ,we construct a symmetry breaking differential operator B p m q k ; λ,µ from C p R n ˆ R n , S b S q into C p R n , Λ ˚ k p R n qq which intertwines the repre-sentations π ρ,λ b π ρ ,µ and π τ ˚ k ,λ ` µ ` m , where τ ˚ k is the representationof Spin p R n q on Λ ˚ k p R n q . Introduction
In the last years there had been a lot of work on symmetry breaking dif-ferential operators (SBDO for short), initiated by a program designed byT. Kobayashi (see [11] and [12] for more information on the subject). Thepresent authors have already contributed to the construction of some SBDO(see [1–3, 5]). In the present paper, we construct such operators in the con-text of tensor product of two spinorial principal series representations ofthe conformal spin group of R n (a two-fold covering of the Lorentz groupSO p , n ` qq .The method we follow has been named source operator method by thefirst author (see [5] for a systematic presentation). An essential ingredient isthe Knapp-Stein operator for the spinorial series, which is presented alongnew lines in the ambient space approach (subsection 2.3). The constructionof the source operator requires some Fourier analysis on R n and in this paper Keywords : Clifford algebra, spinors, tensor product, conformal analysis, symmetrybreaking differential operators2010 AMS Classification : Primary 43A85. Secondary 58J70, 33J45 a r X i v : . [ m a t h . R T ] D ec e develop an approach through an ad hoc symbolic calculus, which easesthe computations (subsection 3.3). As a result, an explicit expression for thesource operator (of degree 4 with polynomial coefficients) is obtained (see(3.11)).Once the source operator is computed, the sequel is standard and yieldsa family of constant coefficients bi-differential operators which are covariantfor the action of the conformal spinor group. In complement, a recurrenceformula on these operators is obtained. An explicit formula is obtained forthe ”simplest” SBDO (scalar-valued and of degree 2).When the dimension n is equal to 1, the operators thus constructedcoincide with the classical Rankin-Cohen brackets of even degree, see (4.3).At the very beginning of the present work, the first author benefitedfrom a discussion with Bent Ørsted during a visit at Aarhus University andwishes to thank him and its institution for the invitation.
Contents S b S . . . . . . . . . . 51.4 Spinors and irreducible representations of Spin p E q . . . . . . 7 R n . . . . . . . . . . . . . . . . 82.2 The Gelfand Naimark decomposition . . . . . . . . . . . . . 102.3 The representation induced from a Clifford module and theassociated Knapp-Stein operators . . . . . . . . . . . . . . . 11 r Ψ p k q . . . . . . . . . . . . . . . . . . . . . . 264.2 Definition of the SBDO . . . . . . . . . . . . . . . . . . . . . 264.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 The dimension n “ Clifford modules
This section contains what is necessary to know about Clifford algebras andtheir modules in order to read this article without any claim to originality.We use [7] and [4] as main references.
Let p E, x¨ , ¨yq be a Euclidean vector space of dimension n . The Cliffordalgebra C (cid:96) p E q is the algebra over R generated by the vector space E andthe relations xy ` yx “ ´ x x, y y for x and y P E , where ´ x x, y y is identified with ´ x x, y y and being the algebra identityelement.There is a natural action of the Clifford algebra on the exterior algebraΛ p E q . For x P E and ω P Λ p E q , let ε p x q ω be the exterior product of x with ω , and let ι p x q ω be the contraction of ω with the covector x x, . y . TheClifford action of a vector x P E on Λ p E q is given by c p x q ω “ ε p x q ω ´ ι p x q ω. The classical formula ε p x q ι p y q ` ι p x q ε p y q “ x x, y y implies that c p x q c p y q ` c p y q c p x q “ ´ x x, y y and by the universal property of the Clifford algebra, the action c can beextended to C (cid:96) p E q .Associated to this action is the symbol map σ : C (cid:96) p E q ÝÑ Λ p E q given by σ p a q “ c p a q for a P C (cid:96) p E q . The symbol map can be shown to be an isomorphism, and its inverse γ :Λ p E q ÝÑ C (cid:96) p E q is called the quantization map , see [4] for more information.To give an explicit realization of the quantization map, let e , e , . . . , e n bean orthonormal basis of E . Then γ p e i ^ e i ^ ¨ ¨ ¨ ^ e i k q “ e i e i . . . e i k , (1.1)where 1 ď i ă i ă ¨ ¨ ¨ ă i k ď n . Moreover, σ (and hence γ ) is anisomorphism of O p E q -modules. 3he conjugation α is the unique anti-involution of C (cid:96) p E q such that for x P E , α p x q “ ´ x . Notice that x P E, | x | “ α p x q “ x ´ The pin group Pin p E q is defined as the multiplicative subset of C (cid:96) p E q given byPin p E q “ t g P C (cid:96) p E q , g “ x x . . . x k , | x j | “ ď j ď k u , the inverse of the element g “ x x . . . x k being the element g ´ “ α p g q . The spin group Spin p E q is defined similarly asSpin p E q “ t g P C (cid:96) p E q , g “ x x . . . x k , | x j | “ ď j ď k u . Let x P E such that | x | “
1. Then for any y P E , xyx ´ belongs to E and xyx ´ “ s x y, where s x is the orthogonal symmetry with respect to the hyperplane perpen-dicular to x . As a consequence, if g P Pin p E q , then for any x P E , gxg ´ P E and the map τ g : x ÞÝÑ gxg ´ belongs to O p E q . If moreover g P Spin p E q ,then τ g belongs to SO p E q . Proposition 1.1.
The map g ÝÑ τ g induces homomorphisms τ : Pin p E q ÝÑ O p E q , τ : Spin p E q ÝÑ SO p E q , which are twofold coverings. Let E be the complexification of E and extend the inner product on E toa symmetric C -bilinear form. Denote by C (cid:96) p E q the complex Clifford of E ,which can be identified with C (cid:96) p E q b C .A Clifford module p S , ρ q is a complex vector space together with a (left)action ρ of C (cid:96) p E q on S . By restriction, the action ρ yields representationsof the groups Pin p E q or Spin p E q , also denoted by ρ . As Pin p E q is compact,there exists an inner product x ¨ , ¨y on S for which the action of the groupPin p E q is unitary. Now for any v P E and for any s, t P S x ρ p v q s, t y “ ´x s, ρ p v q t y .
4n fact, it suffices to prove the formula for v P E such that | v | “
1. But then ρ p v q “ ´
1, so that x ρ p v q s, t y “ ´x ρ p v q s, ρ p v q ρ p v q t y “ ´x s, ρ p v q t y using the unitarity of ρ p v q for v P Pin p E q .The dual space S is also a Clifford module with the action ρ given by x P E, t P S ρ p x q t “ ´ t ˝ ρ p x q , and then extended to a representation of C (cid:96) p E q . The restriction of ρ toSpin p E q (still denoted by ρ ) coincides with the contragredient representationof ρ .Denote the duality between S and S by p s, t q , for s P S and t P S . Then @ x P E, s P S , t P S , p ρ p x q t, t q “ ´p t, ρ p x q t q (1.2) @ g P Spin p E q , s P S , t P S , p ρ p g q s, ρ p g q t q “ p s, t q . (1.3) S b S Define Ψ : S b S ÝÑ Λ ˚ p E q b C by the following formula, for s P S , t P S and ω P Λ p E q Ψ p s b t qp ω q “ p ρ ` γ p ω q ˘ s, t q or more explicitly (see (1.1)),for ω “ e i ^ ¨ ¨ ¨ ^ e i k , where 1 ď i ă i ă ¨ ¨ ¨ ă i k ď n ,Ψ p s b t q ` e i ^ ¨ ¨ ¨ ^ e i k ˘ “ ` ρ p e i q ¨ ¨ ¨ ρ p e i k q s, t ˘ . Proposition 1.2.
The map
Ψ : S b S ÝÑ Λ ˚ p E q b C intertwines therepresentation ρ b ρ and the natural representation τ ˚ of Spin p E q on thedual exterior algebra Λ ˚ p E q b C .Proof. Let g P Spin p E q . Recall that for any x P E , τ p g q x “ gxg ´ , so that ρ p x q ρ p g q “ ρ p g q ρ p g ´ xg q “ ρ p g q ρ ` τ p g ´ q x q ˘ and hence, for 1 ď i ă i ă ¨ ¨ ¨ ă i k ď nρ p e i q ¨ ¨ ¨ ρ p e i k q ρ p g q “ ρ p g q ρ ` τ p g ´ q e i ˘ ¨ ¨ ¨ ρ ` τ p g ´ q e i k ˘ . so that for s P S , t P S Ψ p ρ p g q s, ρ p g q t q ` e i ^ ¨ ¨ ¨ ^ e i k ˘ “ ` ρ p e i q . . . ρ p e i k q ρ p g q s, ρ p g q t ˘ “ ` ρ p g q ρ ` τ p g ´ q e i ˘ ¨ ¨ ¨ ρ ` τ p g ´ q e i k ˘ s, ρ p g q t ˘ “ ` ρ ` τ p g ´ q e i ˘ ¨ ¨ ¨ ρ ` τ p g ´ q e i k ˘ s, t ˘ “ τ p g q ˚ Ψ p s, t qp e i ^ ¨ ¨ ¨ ^ e i k q
5e thus get Ψ p ρ p g q s, ρ p g q t q “ ` τ p g q ˚ Ψ ˘ p s, t q . The space Λ ˚ p E q b C decomposes further under the action of the groupSpin p E q (which reduces to an action of SO p E q ) and in factΛ p E q ˚ b C “ n à k “ Λ ˚ k p E q b C , where Λ ˚ k p E q is the space of alternating k -forms on E . For 0 ď k ď n letΨ p k q : S b S ÝÑ Λ ˚ k p E q b C Ψ p k q “ proj k ˝ Ψ , the operator proj k being the projector from Λ ˚ p E q b C on Λ ˚ k p E q b C .The following lemma will be needed in the proof of the next proposition. Lemma 1.3.
Let J “ t ď j , ă j ă ¨ ¨ ¨ ă j k ď n u and let e J “ e j e j ¨ ¨ ¨ e j k . Then n ÿ i “ e i e J e i “ p´ q k ´ p n ´ k q e J . (1.4) Proof.
Let 1 ď i ď n . Assume first that i R J . Then e i e J e i “ e i p e j e j ¨ ¨ ¨ e j k q e i “ p´ q k p e j e j ¨ ¨ ¨ e j k q e i e i “ p´ q k ´ e J . Assume on the contrary that i “ j (cid:96) for some (cid:96), ď (cid:96) ď k . Then e i p e j e j ¨ ¨ ¨ e j (cid:96) ´ q e j (cid:96) p e j (cid:96) ` ¨ ¨ ¨ e j k q e i “ p´ q (cid:96) ´ p´ q k ´ (cid:96) p e j e j . . . e j (cid:96) ´ q e i e j (cid:96) e i p e j (cid:96) ` . . . e j k q “ p´ q k e J . Hence n ÿ i “ e i e J e i “ p n ´ k qp´ q k ´ ` k p´ q k “ p n ´ k qp´ q k ´ . Let L be the operator on S b S given by L p v b w q “ n ÿ i “ ρ p e i q v b ρ p e i q w . roposition 1.4. Let ď k ď n . Then Ψ p k q ˝ L “ p´ q k p n ´ k q Ψ p k q . (1.5) Proof.
Fix v P S and w P S . Let J “ t ď j ă j ă ¨ ¨ ¨ ă j k ď n u and let e J be the corresponding k -vector. ThenΨ ` L p v b w q ˘ p e J q “ n ÿ i “ p ρ p e j q ρ p e j q ¨ ¨ ¨ ρ p e j k q ρ p e i q v, ρ p e i q w q which by (1.2) is equal to ´ n ÿ i “ p ρ p e i q ρ p e j q ρ p e j q . . . ρ p e j k q ρ p e i q v, w q“ ´ ˜ ρ ˜ n ÿ i “ e i e j e j . . . e j k e i ¸ v, w ¸ , and according to (1.4) we haveΨ ` L p v b w q ˘ p e J q “ p´ q k p n ´ k qp ρ p e j e j . . . e j k q v, w q“ p´ q k p n ´ k q Ψ p k q p v b w qp e J q and the conclusion follows. Spin p E q For sake of completeness, we now discuss the irreducible Clifford modulesand the corresponding representations of the spin group, known as spinorspaces .When n is even, say n “ m , there exists, up to equivalence a uniqueirreducible Clifford module S m of dimension 2 m . As a representation ofSpin p R m q , S m splits into two irreducible non equivalent representations,the half spinors spaces S ` m and S ´ m , each of dimension 2 m ´ .When n is odd, say n “ m `
1, there exist two non-equivalent irreducibleClifford modules, of dimension 2 m . As representations of the spin groupSpin p R m ` q , they are irreducible and equivalent, thus leading to a uniquespinor space S m ` .Wether n is even or odd, the dual of the Clifford module S n is isomorphicto itself as a representation of the spin group Spin p R n q . In the even case,the half spinor space is either self dual or isomorphic to its opposite halfspinor space, depending on m , but in any case, S m b S m “ p S ` m b S ` m q ‘p S ` m b S ´ m q ‘ p S ´ m b S ` m q ‘ p S ´ m b S ´ m q .7he representation of the spin group Spin p R n q on Λ ˚ p R n q goes downto a representation of SO p n q and decomposes as À nk “ Λ ˚ k p R n q . The Hodgeoperator yields an isomorphism Λ ˚ k p R n q » Λ ˚ n ´ k p R n q . In the odd case Λ ˚ k p R n q is irreducible for any k , whereas for n “ m , Λ ˚ k p R n q is irreducible exceptfor k “ m , and in fact, Λ ˚ m p R m q splits in two irreducible non equivalentrepresentations.In the present article we chose to work with Clifford modules. The latterconsiderations show that it is clearly possible to deduce results for spinorspaces, just by refining the decomposition under the action of the spin group. In this Section, we present the construction of the conformal spin group G of the space E , its conformal action on E and the representations of G associated by induction of the Clifford modules. For convenience we identify E and R n . R n Let E “ R ,n ` be the real vector space of dimension n ` Q p x , y q “ x y ´ x y ´ ¨ ¨ ¨ ´ x n ` y n ` . Denote by C (cid:96) p E q the corresponding Clifford algebra, generated by E andsubject to the relation xy ` yx “ Q p x , y q Let α be the conjugation of the Clifford algebra, i.e. the unique anti-involution of C (cid:96) p E q such that α p x q “ ´ x . Let G “ Spin p , n ` q bedefined by ! v v . . . v k , k P N , v j P E , Q p v j q “ ˘ , t j, Q p v j q “ ´ u even ) . Then G is a connected Lie group, the inverse of an element g is equal to g ´ “ α p g q . For x P E and g P G , the element gx α p g q belongs to E and the map τ g : x ÝÑ gx α p g q defines an isometry of p E , Q q . Moreover,the map g ÝÑ τ g is a Lie group homomporphism from G onto SO p E q » SO p , n ` q which turns out to be a twofold covering (see [7] for moredetails). 8he Lie algebra g of G can be realized as the subspace C (cid:96) p E q of bivectorsin C (cid:96) p E q generated by t e i e j , ď i ă j ď n ` u . The Lie algebra g isisomorphic to o p , n ` q . The isomorphism of E given by e ÞÝÑ e , e j ÞÝÑ ´ e j , ď j ď n ` (cid:96) p E q which, when restricted to C (cid:96) p E q yields a Cartan involution θ of g . The corresponding decomposition of g isgiven by g “ k ‘ s , where k “ à ď i ă j ď n ` R e i e j , s “ n ` à j “ R e e j . A Cartan subspace a of s is given by a “ R H, where H “ e e n ` . Now let m “ ÿ ď i ă j ď n R e i e j , n “ n à j “ R e j p e ´ e n ` q , n “ n à j “ R e j p e ` e n ` q , and notice that r H, m s “ , ad H | n “ ` , ad H | n “ ´ . Then g “ n ‘ m ‘ a ‘ n is a Gelfand-Naimark decomposition of g . By elementary calculation, for t P R a t : “ exp p te e n ` q “ cosh t ` sinh t e e n ` “ e ` cosh t e ` sinh t e n ` ˘ and for y P R n n y : “ exp y p e ´ e n ` q “ ` y p e ´ e n ` q z P R n n z : “ exp z p e ` e n ` q “ ` z p e ` e n ` q . The analytic Lie subgroups of G associated to a , n and n are isomorphic totheir counterparts in SO p , n ` q and hence are denoted respectively by A, N and N . 9he Cartan involution of g can be lifted to a Cartan involution of G .The fixed points set of this involution is a maximal compact subgroup K “ t v v . . . , v k , v j P n ` à i “ R e i , Q p v j q “ ´ , ď j ď k u , isomorphic to Spin p n ` q and a twofold covering of K » SO p n ` q . Let M be the centralizer of A in K which is isomorphic to Spin p n q and is a twofoldcovering of M . Let M be the normalizer of A in K . Then the Weyl group M { M has two elements. As a representative of the non trivial Weyl groupelement choose w “ e e n ` and observe in fact that wHw ´ “ e e n ` e e n ` e n ` e “ ´ e e n ` “ ´ H .
To the decomposition of g is associated a (partial) decomposition of thegroup G , often called the Gelfand Naimrak decomposition.More precisely, the map N ˆ M ˆ A ˆ N Q p n, m, a, n q ÞÝÑ nman P G is injective and its image is a dense open subset of full measure in G . Con-versely, let g P G and assume that g belongs to the image. Then there areunique elements n p g q P N , m p g q P M , a p g q P A, n p g q P N such that g “ n p g q m p g q a p g q n p g q . The following result will be needed in the sequel.
Proposition 2.1.
Let x P R n , and assume that x ‰ . Let x “ e xe .Thenthe following identity holds true : w ´ n x “ n x x | ˆ ´ e x | x | ˙ a ln | x | n x | x | . (2.1) In particular, m p w ´ n x q “ ´ e x | x | , ln a p w ´ n x q “ ln | x | . roof. First w ´ n x “ e n ` e ` ` x p e ` e n ` q ˘ “ ´ e x ´ e xe e n ` ´ e e n ` . The right side of the identity (2.1) is equal to ˆ ` x p e ` e n ` | x | ˙ˆ ´ e x | x | ˙ ˆ ˆ | x | ` | x | ˙ ` ˆ | x | ´ | x | ˙ e e n ` ˙ˆ ` x p e ´ e n ` q | x | ˙ whereas the left handside is obtained by a standard computation, using inparticular the fact that e x commutes to e e n ` and the relation x e x “| x | e .Through the covering map G ÝÑ SO p , n ` q , G acts (rationally) on R n » À nj “ R e j . In particular, the action of M on R n is given by m P M , x P R n , v ÞÝÑ mxm ´ , and the action of A is given by a t P A, x P R n n x ÞÝÑ n e t x . Let p S , ρ q be a Clifford module for the Clifford algebra C (cid:96) p E q . The restrictionof ρ to the spin group M yields a representation of M , still denoted by ρ .For λ P C , let χ λ be the character of A given by χ λ p a t q “ e tλ for t P R . Now consider the representation of P “ M AN given by ρ b χ λ b . and let π ρ,λ “ Ind GP ρ b χ λ b P to G . Let S ρ,λ be theassociated bundle G ˆ ρ,λ S over G { P and let H ρ,λ be the space of smoothsections of S ρ,λ . The natural action of G on S ρ,λ gives a realization of π ρ,λ on H ρ,λ .Another realization of the representation π ρ,λ , more fitted for calcula-tions is the noncompact picture , see [10, chapter VII]. In this model, therepresentation is given by π ρ,λ p g q F p n q “ χ λ ` a p g ´ n q ˘ ´ ρ ` m p g ´ n q ˘ ´ F p g ´ n q , where F is a smooth S -valued function on N .Consider now the representation wρ of M defined by @ m P M , p wρ qp m q “ ρ p w ´ mw q . Proposition 2.2.
The representation wρ is equivalent to ρ . More precisely,for all m P M ρ p´ e q ˝ wρ p m q “ ρ p m q ˝ ρ p´ e q . (2.2) Proof.
Recall that w “ e e n ` so that for any m P M wρ p m q “ ρ p e n ` e me e n ` q . As e n ` anticommutes to e and commutes to m , this implies wρ p m q “ ρ p e m p´ e qq “ ρ p e q ρ p m q ρ p´ e q from which (2.2) follows by left multiplication by ρ p´ e q “ ρ p e q ´ .Now form the induced representation π wρ,n ´ λ “ Ind GP p wρ b χ n ´ λ b q . The Knapp-Stein operators J ρ,λ are intertwining operators between π ρ,λ and π wρ,n ´ λ , see again [10] for general information on these operators. Using theequivalence between wρ and ρ , introduce the operators I ρ,λ “ ρ p´ e q ˝ J ρ,λ . Proposition 2.3.
For any g P G , I ρ,λ ˝ π ρ,λ p g q “ π ρ,n ´ λ p g q ˝ I ρ,λ . roof. First, by induction Proposition 2.2 implies for any µ P C ρ p´ e q ˝ π wρ,µ p g q “ π ρ,µ p g q ˝ ρ p´ e q . Hence I ρ,λ ˝ π ρ,λ p g q “ ρ p´ e q ˝ J ρ,λ ˝ π ρ,λ p g q “ ρ p´ e q ˝ π wρ,n ´ λ p g q ˝ J ρ,λ “ π ρ,n ´ λ p g q ˝ ρ p´ e q ˝ J ρ,λ “ π ρ,n ´ λ p g q ˝ I ρ,λ . The expression of the corresponding Knapp-Stein operator in the non-compact picture is given by J ρ,λ F p n x q “ ż R n e ´p n ´ λ q ln a p w ´ n y q ρ p m p w ´ n y qq F p n x ` y q dy, which, using Proposition 2.1, can be rewritten more explicitly as J ρ,λ F p n x q “ ż R n | y | ´ n ` λ ρ ˆ ´ e y | y | ˙ F p n x ` y q dy. In turn, the operator I ρ,λ “ ρ p´ e q ˝ J λ is given by I ρ,λ F p n x q “ ż R n | y | ´ n ` λ ρ ˆ y | y | ˙ F p n x ´ y q dy , after the change of variables y ÞÑ y “ ´ y . The Knapp-Stein operator I ρ,λ is thus shown to be a convolution operator on N , or otherwise said over R n .Notice the these operators were already introduced and studied in [6].Now consider simultaneously S and its dual S . For λ, µ P C the corre-sponding induced representations are π λ “ Ind GP ρ b χ λ b , π µ “ Ind GP ρ b χ µ b . Simplifying the notation, the corresponding intertwining operators are I λ f p x q “ ż R n | y | ´ n ` λ ρ ˆ y | y | ˙ f p x ´ y q dy,I µ f p x q “ ż R n | y | ´ n ` µ ρ ˆ y | y | ˙ f p x ´ y q dy . (2.3)Finally, consider the ”outer” tensor product ρ b ρ as a representation of M ˆ M and, for λ, µ P C form the tensor product representation π λ b π µ “ Ind G ˆ GP ˆ P p ρ b χ λ b q b p ρ b χ µ b q . Proposition 2.4.
The operator I λ b I µ intertwines the representations p π λ b π µ q and p π n ´ λ b π n ´ µ q of G ˆ G . The diagonal subgroup of G ˆ G will be denoted simply by G , andviewed as acting diagonally on R n ˆ R n . Needless to say, the previousproposition implies that I λ b I µ is an intertwining operator for the action of G on C p R n ˆ R n q by the ”inner” tensor product π λ b π µ .13 The source operator
Let M be the operator on C p R n ˆ R n , S b S q defined for F a smoothfunction on R n ˆ R n with values in S b S by M F p x, y q “ | x ´ y | F p x, y q . Proposition 3.1.
Let λ, µ P C . Then for any g P G , M ˝ ` π λ p g q b π µ p g q ˘ “ ` π λ ´ p g q b π µ ´ p g q ˘ ˝ M . Proof.
The result is a consequence of the following covariance property ofthe function | x ´ y | under a conformal transformation g P G | g p x q ´ g p y q| “ e ´ a p g, x q | x ´ y | e ´ a p g, y q , (3.1)where a p g, x q “ a p g ¯ n x q .This is equivalent to the more classical formula | g p x q ´ g p y q| “ κ p g, x q | x ´ y | κ p g, y q (3.2)where κ p g, x q stands the conformal factor of g at x . The equivalence of (3.1)and (3.2) comes from the relation κ p g, x q “ χ ` a p g, x q ˘ ´ “ e ´ a p g,x q , which can be checked easily for g in N , M and A and can be deduced for g “ w ´ from the Gelfand Naimark decomposition of w ´ n x obtained inProposition 2.1.The proof of the intertwining property is then straightforward. In fact,let g P G and F P C p R n ˆ R n q . Then M ˝ ` π λ p g q b π µ p g q ˘ F p x, y q ““ | x ´ y | e ´ λ ln a p g ´ , x q e ´ µ ln a p g ´ , y q ˆˆ ` ρ p m p g ´ n x qq b ρ p m p g ´ n y q ˘ F ` g ´ p x q , g ´ p y q ˘ and by using (3.1) this can be transformed as | g ´ p x q ´ g ´ p y q| e ´ p λ ´ q ln a p g ´ , x q e ´ p µ ´ q ln a p g ´ , y q ˆˆ ` ρ p m p g ´ n x qq b ρ p m p g ´ n y q ˘ F ` g ´ p x q , g ´ p y q ˘ “ p π λ ´ p g q b π µ ´ p g qq M F p x, y q . p λ, µ q there exists an operator M λ,µ : H λ b H µ ÝÑ H λ ´ b H µ ´ intertwining π λ b π µ and π λ ´ b π µ ´ which isexpressed in the local chart N ÝÑ G { P by the operator M .Now let consider the operator E λ,µ defined by the following diagram H λ,µ E λ,µ ÝÝÝÑ H λ ` ,µ ` I λ b I µ §§đ ݧ§ I n ´ λ ´ b I n ´ µ ´ H n ´ λ,n ´ µ M λ,µ ÝÝÝÑ H n ´ λ ´ ,n ´ µ ´ Theorem 3.2.
The operator E λ,µ is a differential operator on G { P ˆ G { P which satisfies, for any g P G E λ,µ ˝ ` π λ p g q b π µ p g q ˘ “ ` π λ ` p g q b π µ ` p g q ˘ ˝ E λ,µ . This is the main theorem of the article. The operator is named the source operator as it is the key to the construction of the SBDO as wewill show later. The fact that E λ,µ is G -intertwining is a consequence of thedefinition. The fact that it is a differential operator is much more subtle andwill be shown by working in the noncompact picture. There is however somedifficulty when using the noncompat picture, due to the fact that the spaceof C vectors in the noncompact picture is not very manageable, speciallywhen using the Fourier transform on R n . Hence we have to use a slightlydifferent path to construct and explicitly calculate the local expression ofthe operator E λ,µ . Coming bak to the compact picture, for generic λ (resp. µ ) the Knapp-Stein operator I λ is invertible, and up to a constant ( ‰ I n ´ λ . So that the operator E λ,µ (up to a the constant)satisfies the relation ` I n ´ λ ´ b I n ´ µ ´ ˘ ˝ M “ E λ,µ ˝ ` I n ´ λ b I n ´ µ ˘ . This is the way we will introduce and calculate the expression of the sourceoperator in the noncompact picture (cf Subsection 3.4).
Up to that point, the intertwining operators are only formally defined andwe need to look more carefully to the convolution kernels of the Knapp-Steinoperators. 15irst recall the classical
Riesz distributions . For s P C , the Riesz distri-bution r s on R n is given by r s p x q “ | x | s . More precisely, for (cid:60) p s q ą ´ n , the function r s is locally integrable and hasmoderate growth at infinity, so that r s is a well-defined tempered distribu-tion. The family of distributions thus defined can be extended analyticallyin the parameter s P C , with poles at ´ n ´ k, k P N .Let p S , ρ q be a Clifford module and for s P C define the associated CliffordRiesz distribution by { r s p x q “ | x | s ρ ˆ x | x | ˙ “ | x | s ´ ρ p x q . (3.3)Let E j “ ρ p e j q , ď j ď n . Then, for x “ ř nj “ x j e j , ρ p x q “ ř nj “ x j E j . Usethe identity x j | x | s ´ “ s ` BB x j p| x | s ` q , to conclude that { r s p x q “ s ` n ÿ j “ B r s ` B x j p x q E j . From this expression it is easy to deduce the next statement.
Proposition 3.3.
The family { r s defined by (3.3) is a meromorphic familyof End p S q -valued tempered distributions with poles at s “ ´ n ´ ´ k, k P Z . Further properties of these distributions will be needed in the sequel.Parts of the present results were already obtained in [6] and in [8].
Proposition 3.4. B j { r s p x q “ ` p s ´ q x j ´ ρ p e j x q ˘ { r s ´ p x q (3.4)∆ { r s p x q “ p s ´ qp s ` n ´ q { r s ´ p x q (3.5) Proof.
First B j ` | x | s ´ ρ p x q ˘ “ ` p s ´ q x j | x | s ´ ˘ ρ p x q ` | x | s ´ ρ p e j q“ ` p s ´ q x j | x | s ´ ˘ ρ p x q ´ | x | s ´ ρ p e j q ρ p x q ρ p x q“ ` p s ´ q x j ´ ρ p e j x qq| x | s ´ ρ p x q , B j { r s p x q “ ` p s ´ q ´ ρ p e j e j q ˘ | x | s ´ ρ p x q` ` p s ´ q x j ´ ρ p e j x q ˘` p s ´ q x j ´ ρ p e j x q ˘ | x | s ´ ρ p x q“ s | x | s ´ ρ p x q ` ` p s ´ qp s ´ q x j ´ p s ´ q x j ρ p e j x q ` ρ p e j xe j x q ˘ | x | s ´ ρ p x q . Now sum over j from j “ j “ n and use that n ÿ j “ x j “ | x | , n ÿ j “ x j e j “ x, n ÿ j “ e j xe j “ p n ´ q x to get∆ { r s p x q “ ` ns | x | ` p s ´ qp s ´ q| x | ` p s ´ q| x | ´ p n ´ q| x | ˘ | x | s ´ ρ p x q“ ` s ` p n ´ q s ´ n ` ˘ | x | s ´ ρ p x q , and (3.5) follows.We will also need the Fourier transform of the Riesz distributions. TheFourier transform of a function f on E is defined by the formula F f p ξ q “ p f p ξ q “ ż E e ´ i x x,ξ y f p x q dx . The Fourier transform is an isomorphism of S p E q onto S p E q , the definition ofthe Fourier transform can be extended by duality to the space of tempereddistribution S p E q . For V a finite dimensional real vector space, denoteby S p E, V q (resp. S p V q ) the space of V -valued Schwartz functions (resp.tempered distributions). The Fourier transform can also be extended tothese spaces.Recall the following classical result for the usual Riesz distributions, seee.g. [9]. Proposition 3.5.
The Fourier transform of the Riesz distribution r s is givenby F p r s qp ξ q “ c s r ´ s ´ n p ξ q , where c s “ s ` n π n Γ p s ` n q Γ p´ s q . Proposition 3.6.
The Fourier transform of { r s is given by F { r s “ { c s { r ´ s ´ n (3.6) where { c s “ ´ i s ` n π n Γ p s ` n ` q Γ p´ s ´ q . roof. For any j, ď j ď n , a basic formula for the Fourier transform yields F p x j | x | s ´ qp ξ q “ i BB ξ j p F | x | s ´ qp ξ q “ ic s ´ BB ξ j p| ξ | ´ s ` ´ n q“ i p´ s ` ´ n q c s ´ ξ j | ξ | ´ s ´ ´ n “ { c s ξ j | ξ | ´ s ´ n ´ . and (3.6) follows easily by using the linearity of x ÞÑ ρ p x q . This subsection describes in a general context a symbolic calculus, inspiredby the calculus for the Weyl algebra or of the pseudo-differential calculus E ,but designed for our specific problems to be treated in the next subsection.Let E a Euclidean vector space of dimension n , and let V a finite dimen-sional vector space.Let k P S p E, End p V qq . Then, for f P S p E, V q , the formula Kf p x q “ ż E k p x ´ y q f p y q dy defines a convolution operator K which maps S p E, V q in S p E, V q .As in the scalar case, these operators have a nice version through theFourier transform, namely y Kf p ξ q “ p k p ξ q p f p ξ q , ξ P E , where p k P S p E , V q is the Fourier transform of the distribution k .Let p p x q be a End p V q -valued polynomial function on E . Then, for f P S p E, V q the formula f ÞÝÑ ` x ÞÑ p p x q f p x q ˘ defines an operator on S p E, V q denoted by f ÞÝÑ pf and referred to as the multiplication operator by p . A multiplication operator can be extended to S p E, V q .We will have to deal with operators from S p E, V q into S p E, V q whichare obtained by composing a convolution operator (say K ) followed by amultiplication operator by an End p V q -valued polynomial function (say p )on E . Such an operator will be denoted by p K . By definition, its symbol isgiven by symb p p K qp x, ξ q “ p p x q ˝ p k p ξ q x P E, ξ P E . viewed as the polynomial function on E with values in S p End p V qq x ÞÝÑ p p x q ˝ p k p ξ q .
18e let Op p E, V q be the family of finite linear combinations of such operators.In other words, an element of Op p E, V q can be written in a unique way as ÿ α x α A α K α where α “ p α , α , . . . , α n q denotes a n -multiindex, A α P End p V q and K α is a convolution operator by a tempered End p V q -valued distribution on E ,with the tacit convention that only a finite number of terms in the sum arenon zero. Then the symbol of such an operator is given by ÿ α x α A α p k α p ξ q . A constant coefficients End p V q -valued differential operators on E is an exam-ple of convolution operator with a tempered distribution, namely a combina-tion of derivatives of the Dirac distribution at 0 P E . In particular Op p E, V q contains the End p V q -valued Weyl algebra on E , denoted by W p E, V q , con-sisting of the differential operators on E with End p V q -valued polynomialcoefficients. Notice that these operators map S p E, V q into S p E, V q and S p E, V q into S p E, V q . Recall the usual definition of the symbol of a dif-ferential operator, namely the End p V q -valued polynomial function σ D on E ˆ E given De i x x,ξ y “ σ D p x, ξ q e i x x,ξ y . Then an elementary computation shows that σ D coincides with symb p D q (see [5] for the scalar case). More explicitly, let D “ ř α p α p x qB αx be in W p E, V q , where p α is a End p V q -valued polynomial. Then its symbol isgiven by symb p D qp x, ξ q “ ÿ α p α p x qp iξ q α Although Op p E q is not an algebra of operators, some compositions are pos-sible. Although a more general result could be stated, we consider only twocases, which will be enough for the present paper. Proposition 3.7.
Let D be a End p V q -valued differential operator on E , andlet K be a convolution operator with a tempered distribution. Then D ˝ K belongs to Op p E q , and its symbol is given by symb p D ˝ K qp x, ξ q “ symb p D qp x, ξ q ˝ symb p K qp ξ q (3.7) Proof.
Let α be a n -multiindex and let k be the kernel of the convolutionoperator K . Then B αx ˝ K is the convolution operator with kernel B αx k p x q .Hence symb pB αx ˝ K qp ξ q “ p iξ q α p k p ξ q , which coincides with formula (3.7) for this particular case. The generalformula follows easily. 19 roposition 3.8. Let p be a scalar-valued polynomial on E , and let L be in Op p E q . Then L ˝ p belongs to Op p E q and its symbol is given by symb p L ˝ p qp x, ξ q “ ÿ α α ! B αx p p x q ˆ i B ξ ˙ α symb p L qp x, ξ q (3.8) Proof.
Assume first that L “ K is a convolution operator by a tempereddistribution k . Then the symbol of the composition K ˝ p can be computedexactly as in the scalar case (see [5, Proposition 1.2]) and the result coin-cides with (3.8), due to the fact that we assume that p is a scalar-valuedpolynomial. The general case follows easily. We now apply the symbolic calculus developed in the previous subsectionto the construction of the source operator in the noncompact picture. Inparticular, we come back to the context and notation of subsection 3.2.For s, t P C , consider the normalized Clifford-Riesz convolution operators { R s : S p R n , S q ÝÑ S p R n , S q { R s f “ { c ´ s { r s ‹ f , { R t : S p R n , S q ÝÑ S p R n , S q { R t f “ { c ´ t { r s ‹ f . (3.9)The technical reason for this normalization is that symb p { R s qp ξ q “ { r ´ n ´ s p ξ q , symb p { R t qp ζ q “ { r n ´ t p ζ q , see (3.6).The family { R s depends meromorphically on the parameter s . Poles andresidues were studied in [6]. An example of residue is the Dirac operators ,which in our context comes in two versions, given by { D “ ÿ j “ ρ p e j q BB x j , { D “ ÿ j “ ρ p e j q BB x j . As a consequence, they satisfy an intertwining property under the action ofthe conformal group, and similar results are valid for their powers, see [6].Further, consider the operator { M s,t : S p R n ˆ R n , S b S q ÝÑ S p R n ˆ R n , S b S q , { M s,t “ ` { R s b { R t ˘ ˝ M . which clearly belongs to Op ` R n ˆ R n , End p S b S q ˘ as considered in subsec-tion 3.3. 20 roposition 3.9. We have, symb p { M s,t qp x, y, ξ, ζ q “ f s,t p x, y, ξ, ζ q ˝ ` { r ´ s ´ n ´ p ξ q b { r t ´ n ´ p ζ q ˘ where f s,t p x, y, ξ, ζ q “ | x ´ y | | ξ | b | ζ | ` i p s ` n ` q n ÿ j “ p x j ´ y j q ξ j b | ζ | ` i p t ` n ` q n ÿ j “ p y j ´ x j q| ξ | b ζ j ` i ` ρ p x ´ y q ρ p ξ q b | ζ | q ` i ` | ξ | b ρ p y ´ x q ρ p ζ q ˘ ´p s ` qp s ` n ` q id b| ζ | ´ p t ` qp t ` n ` q| ξ | b id ` p s ` n ` qp t ` n ` q n ÿ j “ ξ j b ζ j ` p s ` n ` q n ÿ j “ ξ j b ρ p e j q ρ p ζ q ` p t ` n ` q n ÿ j “ ρ p e j q ρ p ξ q b ζ j ` n ÿ j “ ρ p e j q ρ p ξ q b ρ p e j q ρ p ζ q . Proof.
First, symb p { R s b { R t qp ξ, ζ q “ { r ´ s ´ n p ξ q b { r t ´ n p ζ q Following (3.8) the composition formula for the symbols yields symb ` { R s b { R t ˝ | x ´ y | ˘ “ | x ´ y | { r ´ s ´ n p ξ q b { r t ´ n p ζ q` n ÿ j “ p x j ´ y j q ` i BB ξ j ˘ { r ´ s ´ n p ξ q b { r t ´ n p ζ q` n ÿ j “ p y j ´ x j q{ r ´ s ´ n p ξ q b ` i BB ζ j ˘ { r t ´ n p ζ q´ ∆ { r ´ s ´ n p ξ qb{ r t ´ n p ζ q` n ÿ j “ B j { r ´ s ´ n p ξ qbB j { r t ´ n p ζ q´{ r ´ s ´ n p ξ qb ∆ { r t ´ n p ζ q . Now use formulas (3.4) and (3.5) applied to { r and { r to get the result.Notice that f s,t is the symbol of a differential operator F s,t on R n ˆ R n .21 roposition 3.10. For s, t P C , let F s,t “ | x ´ y | ∆ x b ∆ y ´ p s ` n ` q n ÿ j “ p x j ´ y j q BB x j b ∆ y ´ p t ` n ` q n ÿ j “ p y j ´ x j q ∆ x b BB y j ´ ρ p x ´ y q { D x b ∆ y ´ x b ρ p y ´ x q { D y `p t ` qp t ` n ` q ∆ x b id ` p s ` qp s ` n ` q id b ∆ y ´ p s ` n ` qp t ` n ` q n ÿ j “ BB x j b BB y j ´ p s ` n ` q n ÿ j “ BB x j b ρ p e j q { D y ´ p t ` n ` q n ÿ j “ ρ p e j q { D x b BB y j ´ ` n ÿ j “ ρ p e j q { D x b ρ p e j q { D y ˘ . The symbol of the differential operator F s,t is equal to f s,t . The symbol calculus yields also the following theorem, which is the mainformula leading to the proof of Theorem 3.2.
Theorem 3.11.
The following identity holds for s, t P C ` { R s b { R t ˘ ˝ M “ c p s, t q F s,t ˝ ` { R s ` b { R t ` ˘ , (3.10) where c p s, t q “ p s ` qp s ` n ` qp t ` p t ` n ` q . Proof.
The identity for the symbols obtained in Proposition 3.8 is translatedas { c ´ s { c ´ t ` { R s b { R t ˘ ˝ M “ { c ´ s ` { c ´ t ` F s,t ˝ ` { R s ` b { R t ` ˘ . An elementary computation gives { c s ` { c s “ ´p s ` qp s ` n ` q and the theorem follows. 22 .5 The proof of the main theorem We now study the behaviour of the operators involved in the previous con-struction under the action of the conformal group G .The main observation is that up to a shift in the parameters, and upto a constant, the operator { R s (resp. { R t ) are essentially the Knapp-Steinoperators considered in Subsection 2.3.For λ, µ P C generic, compare (2.3) and (3.9) to get I λ “ { c λ ´ n { R λ ´ n , I µ “ { c µ ´ n { R µ ´ n . Change the normalization of the Knapp-Stein operator (but still keeping thenotation), that is redefine the Knapp-Stein operators by setting I λ “ { R λ ´ n , I µ “ { R µ ´ n . Moreover, set s “ ´ λ ´ , t “ ´ µ ´ E λ,µ “ F s,t “ F ´ λ ´ , ´ µ ´ . Notice that s ` n ` “ ´ λ ` n ´ , s ` “ ´ λ ´ , so that E λ,µ “ | x ´ y | ∆ x b ∆ y ` p λ ´ n ` q n ÿ j “ p x j ´ y j q BB x j b ∆ y ` p µ ´ n ` q n ÿ j “ p y j ´ x j q ∆ x b BB y j ´ ρ p x ´ y q { D x b ∆ y ´ x b ρ p x ´ y q { D y `p µ ´ n ` qp µ ` q ∆ x b id ` p λ ´ n ` qp λ ` q id b ∆ y ´ p λ ´ n ` qp µ ´ n ` q n ÿ j “ BB x j b BB y j ` p λ ´ n ` q n ÿ j “ BB x j b ρ p e j q { D y ` p µ ´ n ` q n ÿ j “ ρ p e j q { D x b BB y j ´ ` n ÿ j “ ρ p e j q { D x b ρ p e j q { D y ˘ . (3.11)23ow (3.10) can be rewritten as ` I n ´ λ ´ b I n ´ µ ´ ˘ ˝ M “ d p λ, µ q E λ,µ ˝ ` I n ´ λ b I n ´ µ ˘ , where d p λ, µ q “ p λ ´ n ` qp λ ` qp µ ´ n ` qp µ ` q . Theorem 3.12.
The differential operator E λ,µ satisfies, for any g P G E λ,µ ˝ ` π λ p g q b π µ p g q ˘ “ ` π λ ` p g q b π µ ` p g q ˘ ˝ E λ,µ . The equality holds when applied to fonctions f P C p R n ˆ R n , S b S q withcompact support and such that the action of g is defined on the support of f .Proof. As G is connected, Theorem 3.12 is equivalent to its infinitesimalversion, which we now formulate. Theorem 3.13.
For any X P g , E λ,µ ˝ ` dπ λ p X q b id ` id b dπ µ p X q ˘ “ ` dπ λ ` p X q b id ` id b dπ µ ` p X q ˘ ˝ E λ,µ . (3.12)A well-known and easy-to-prove result is that dπ λ p X q is a differentialoperator with End p S q -valued polynomials coefficients, hence preserves thespace S p R n , S q , so that both sides of (3.12) are well defined and are differen-tial operators on R n ˆ R n with End p S b S q -valued polynomial coefficients.In order to prove Theorem 3.13, let for X P g A λ,µ p X q “ E λ,µ ˝ ` dπ λ p X q b id ` id b dπ µ p X q ˘ ´ ` dπ λ ` p X q b id ` id b dπ µ ` p X q ˘ ˝ E λ,µ . We want to prove that A λ,µ p X q “ X P g , and in order to do it, wefirst prove the following weaker statement. Lemma 3.14.
For any X P g A λ,µ p X q ˝ ` I n ´ λ b I n ´ µ ˘ “ . (3.13) Proof.
It is sufficient to prove the results for p λ, µ q generic, so that we mayassume that λ, λ ` , n ´ λ, n ´ λ ´ I λ and same conditionson µ . Also assume that p λ, µ q is not a pole of the rational function d p λ, µ q .Thus for any X P g ` dπ λ ` p X q b id ` id b dπ µ ` p X q ˘ ˝ d p λ, µ q E λ,µ ˝ ` I n ´ λ b I n ´ µ ˘ ` dπ λ ` p X q b id ` id b dπ µ ` p X q ˘ ˝ ` I n ´ λ ´ b I n ´ µ ´ ˘ ˝ M “ ` I n ´ λ ´ b I n ´ µ ´ ˘ ˝ ` dπ n ´ λ ´ p X q b id ` id b dπ n ´ µ ´ p X q ˘ ˝ M “ ` I n ´ λ ´ b I n ´ µ ´ ˘ ˝ M ˝ ` dπ n ´ λ p X q b id ` id b dπ n ´ µ p X q ˘ “ d p λ, µ q E λ,µ ˝ ` I n ´ λ b I n ´ µ ˘ ˝ ` dπ n ´ λ p X q b id ` id b dπ n ´ µ p X q ˘˘ “ d p λ, µ q E λ,µ ˝ ` dπ λ p X q b id ` id b dπ µ p X q ˘ ˝ ` I n ´ λ b I n ´ µ ˘ , and (3.13) follows.The proof of Theorem 3.13 is achieved through the following lemma,valid in a more general context. Lemma 3.15.
Let V be a finite-dimensional vector space. Let D be a differ-ential operator acting on C p R p , V q with End p V q -valued polynomial coeffi-cients. Let K be a convolution operator on R p by a End p V q -valued tempereddistribution k . Assume that its Fourier transform p k coincides on a denseopen subset O Ă R p with an End p V q -valued smooth function and satisfiesfor any ξ P O p k p ξ q P GL p V q . Assume further that D ˝ K “ . Then D “ .Proof. Under the Fourier transform, the operator K corresponds to the mul-tiplication operator by p k , and the operator D corresponds to a differentialoperator p D “ ÿ I a I p ξ qB Iξ on R p with End p V q -valued polynomial coefficients. The assumption D ˝ K “ p D ˝ p K “
0, or in other words p D p p kψ q “ ψ P S p R p , V q .Let ξ P O , v P V and I a p -multi-index. There exists a smooth V -valued function ϕ with compact support included in O and such that in aneighbourhood of ξ ϕ p ξ q “ I ! p ξ ´ ξ q I v , so that B I ϕ p ξ q “ v . Now let ψ be defined on O by ψ p ξ q “ p k p ξ q ´ ϕ p ξ q and equal to 0 outside of O . The function ψ is a smooth function withcompact support on R p and0 “ p D p p kψ qp ξ q “ p D p ϕ qp ξ q “ a I p ξ q v . v P V , it follows that a I p ξ q “
0. As ξ wasarbitrary in O and a I is a polynomial, this implies a I ” I was arbitrary p D “
0. This finishes the proof of the lemma.For generic λ, µ , the operator K “ I n ´ λ b I n ´ µ satisfies the conditions ofthe lemma. Hence (3.12) holds true and Theorem 3.12 follows. r Ψ p k q Recall the study of the tensor product S b S under the action of M “ Spin p E q and in particular for k, ď k ď n , there is an M -intertwining mapΨ p k q : S b S ÝÑ Λ ˚ k p E q b C . Recall that τ ˚ k is the representation of M onΛ ˚ k p E q .For ν P C and k, ď k ď n , let π k ; ν “ Ind GP τ ˚ k b χ ν b C p R n , Λ ˚ k p E qq in the noncompact picture.Further let r Ψ p k q : C p R n ˆ R n , S b S q ÝÑ C p Λ ˚ k ` R n q b C ˘ , F ÞÝÑ ` Ψ p k q F p x, y q ˘ | x “ y . Let λ, µ P C . Form the spinorial representations π λ “ Ind GP ρ b χ λ b , π µ “ Ind GP ρ b χ µ b π λ b π µ . The following result is a consequence of thefonctoriality of the induction process. Proposition 4.1.
The map r Ψ p k q satisfies r Ψ p k q ˝ p π λ p g q b π µ p g qq “ π k ; λ ` µ p g q ˝ r Ψ p k q . For m P N , define the operator E p m q λ,µ : C p R n ˆ R n , S b S q ÝÑ C p R n ˆ R n , S b S q by E p m q λ,µ “ E λ ` m ´ ,µ ` m ´ ˝ ¨ ¨ ¨ ˝ E λ,µ . E p m q λ,µ satisfies, for any g P G E p m q λ,µ ˝ ` π λ p g q b π µ p g q ˘ “ ` π λ ` m p g q b π µ ` m p g q ˘ ˝ E p m q λ,µ . (4.1)Let B p m q k ; λ,µ “ r Ψ p k q ˝ E p m q λ,µ . Proposition 4.2. i q The operators B p m q k ; λ,µ : C ` R n ˆ R n , S b S ˘ Ñ C p R n , Λ ˚ k p R n q b C q are constant coefficients bi-differential operators and homogeneous of degree m . ii q For any g P G B p m q k ; λ,µ ˝ ` π λ p g q b π µ p g q ˘ “ π k ; λ ` µ ` m p g q ˝ B p m q k ; λ,µ . Proof. ii q is a direct consequence of the covariance property of the source oper-ators and of the map r Ψ p k q .Next apply ii q to the case where g is a translation by an element of R n .This implies that B p m q k ; λ,µ commutes to (diagonal) translations and hence hasconstant coefficients. Apply then to the case where g belongs to A acting bydilations of R n to get the homogeneity of degree 2 m for the operator B p m q k ; λ,µ .This completes the proof of i q .The definition of the SBDO B p m q k ; λ,µ yields a recurrence formula for theseoperators. Use the covariance relation (4.1) applied to diagonal transla-tions on R n ˆ R n to see that the coefficients of E p m q λ,µ are (operators valued)-functions of p x ´ y q . Let o E p m q λ,µ be the constant coefficients part of E p m q λ,µ . Proposition 4.3.
The SBDO B p m q k ; λ,µ satisfy the recurrence relation B p m q k ; λ,µ “ B p m ´ q k ; λ ` ,µ ` ˝ E λ,µ . Proof.
All coefficients of the difference E p m q λ,µ ´ o E p m q λ,µ vanish on the diagonalof R n ˆ R n . Hence r Ψ p k q ˝ E p m q λ,µ “ r Ψ p k q ˝ o E p m q λ,µ . Now E p m q λ,µ “ p E λ ` m ´ ,µ ` m ´ ˝ ¨ ¨ ¨ ˝ E λ ` ,µ ` q ˝ E λ,µ “ E p m ´ q λ ` ,µ ` ˝ E λ,µ . Hence B p m q k ; λ,µ “ r Ψ p k q ˝ E m ´ λ ` ,µ ` ˝ E λ,µ “ B p m ´ q k ; λ ` ,µ ` ˝ E λ,µ . .3 An example Let us write explicitly the SBDO for the case k “ m “ Theorem 4.4.
The operator B p q λ,µ : C p R n ˆ R n , S b S q ÝÑ C p R n q is given by B p q λ,µ ` v p¨q b w p¨¨q ˘ p x q “p µ ´ n ` qp µ ` q ` ∆ v p x q , w p x q ˘ `p λ ´ n ` qp λ ` q ` v p x q , ∆ w p x q ˘ ´ p λ ´ n ` qp µ ´ n ` q n ÿ j “ ˆ BB x j v p x q , BB y j w p x q ˙ ´ p λ ` µ ´ n ` q ` { Dv p x q , { D w p x q ˘ . (4.2) The operator B p q λ,µ satisfies, for any g P GB p q λ,µ ˝ ` π λ p g q b π µ p g q ˘ “ π λ ` µ ` p g q ˝ B p q λ,µ . Proof.
First notice that Ψ p q : S b S ÝÑ C is given byΨ p q p v b w q “ p v, w q , so that r Ψ p q : C p R n ˆ R n q ÝÑ C p R n q , is given by r Ψ p q ` v p¨q b w p¨¨q ˘ p x q “ ` v p x q , w p x q ˘ . Now use (3.11) and observe that by (1.2) ˆ BB x j v p x q , ρ p e j q { D w p x q ˙ “ ´ ˆ ρ p e j q BB x j v p x q , { D w p x q ˙ so that n ÿ j “ ˆ BB x j v p x q , ρ p e j q { D w p x q ˙ “ ´ ´ { Dv p x q , { D w p x q ¯ , and a similar result holds for ř nj “ ´ ρ p e j q { Dv p x q , BB y j w p x q ¯ . Also use (1.5)for k “ p q ˜ n ÿ ‘ j “ ´ ρ p e j q { D x v p . q b ρ p e j q { D y w p .. q ¯¸ p x q “ n ´ { D x v p x q , { D y w p x q ¯ . The final expression for B p q λ,µ is obtained by putting together the partialcomputations. 28 .4 The dimension n “ and the classical Rankin-Cohen brackets Let E “ R be the standard Euclidean space of dimension n “ e be the vector 1 (to distinguish it from the scalar 1). Let C (cid:96) p E q thecorresponding Clifford algebra which is isomorphic to the complex plane.Let C (cid:96) p E q be its complexification. The spin group Spin p E q is equal to t , ´ u .Let S “ C and define for v P S and x P R ρ p xe q v “ ixv and extend it as an action of C (cid:96) p E q on S , still denoted by ρ . Similarly, let S “ C and define for w P S and x P S ρ p xe q w “ ´ ixw Through the duality on p S , S q given by p v, w q ÞÝÑ vw , p ρ , S q is the dualClifford module of p ρ, S q , i.e. for x P R and v P S , w P S p ρ p x q v, w q “ ´p v, ρ p x q w q . The corresponding Dirac operators are { D “ i ddx , { D “ ´ i ddx . For a smooth S -valued (resp. S -valued) function v p x q (resp. w p x q ), p { Dv p x q , { D w p y qq “ dvdx p x q dw dy p y q Substituting these results in (4.2) yields B p q λ,µ “ µ p µ ` q B B x ` λ p λ ` q B B y ´ p λ ` qp µ ` q B B x B y (4.3)and this coincides (up to a scalar) to the degree two Rankin-Cohen operatorfor the group SL p , R q which is isomorphic to Spin p , q . See [2] Theorem10.7 and [12] for more general results in this direction. References [1] R. Beckmann, J.-L. Clerc,
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