Knapp-Stein Type Intertwining Operators for Symmetric Pairs II. -- The Translation Principle and Intertwining Operators for Spinors
aa r X i v : . [ m a t h . R T ] F e b KNAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRICPAIRS II. – THE TRANSLATION PRINCIPLE AND INTERTWININGOPERATORS FOR SPINORS
JAN M ¨OLLERS AND BENT ØRSTED
Abstract.
For a symmetric pair (
G, H ) of reductive groups we extend to a large class ofgeneralized principal series representations our previous construction of meromorphic familiesof symmetry breaking operators. These operators intertwine between a possibly vector-valued principal series of G and one for H and are given explicitly in terms of their integralkernels. As an application we give a complete classification of symmetry breaking operatorsfrom spinors on a Euclidean space to spinors on a hyperplane, intertwining for a double coverof the conformal group of the hyperplane. Introduction
In the study of representations of real reductive Lie groups, intertwining operators play adecisive role. The most prominent family of such operators is given by the standard Knapp–Stein operators which intertwine between two principal series representations of a group G .More recently, intertwining operators have been studied and used in the framework of branch-ing problems, i.e. the restriction of a representation to a subgroup H ⊆ G and its decomposi-tion. Here one is interested in operators from a representation of G to a representation of H ,intertwining for the subgroup. Such operators are also called symmetry breaking operators , aterm coined by T. Kobayashi in his program for branching problems (see e.g. [6]).Such symmetry breaking operators have been studied in great detail in the special case ofthe conformal groups corresponding to a Euclidean space and a hyperplane by Kobayashi–Speh [9]. In this case some of the operators turn out to be differential operators, namelyexactly the operators found by A. Juhl [4] in connection with his study of Q-curvature andholography in conformal geometry. Further connections to elliptic boundary value prob-lems [13] and automorphic forms [11] indicate the broad spectrum of applications that theseoperators provide.In our recent work with Y. Oshima [12] we generalized the construction of symmetrybreaking operators by Kobayashi–Speh to a large class of symmetric pairs ( G, H ) and spher-ical principal series representations, proving meromorphic dependence on the parameters ingeneral and generic uniqueness in some special cases. In this paper we shall further extendour construction to the case of vector-valued principal series representations; we establishmany of the properties, now for arbitrary vector bundles, in particular the meromorphic de-pendence on the parameters. The main argument is a version of a well-known translationprinciple, namely by tensoring with a finite-dimensional representation of the group. As an
Date : February 7, 2017.2010
Mathematics Subject Classification.
Primary 22E45; Secondary 47G10.
Key words and phrases.
Knapp–Stein intertwiners, intertwining operators, symmetry breaking operators,symmetric pairs, principal series, translation principle. application and illustration we give all details for the case of spinors on a Euclidean spacecarrying a representation of the spin cover of the conformal group; here we invoke a variationof the method of finding the eigenvalues on K -types of Knapp–Stein operators. The result isa complete classification of symmetry breaking operators from spinors on the Euclidean spaceto spinors on a hyperplane.Let us now explain our results in more detail. The translation principle.
Let G be a real reductive group and H ⊆ G a reductive sub-group. For parabolic subgroups P = M AN ⊆ G and P H = M H A H N H ⊆ H we form thegeneralized principal series representations (smooth normalized parabolic induction)Ind GP ( ξ ⊗ e λ ⊗ ) and Ind HP H ( η ⊗ e ν ⊗ ) , where ξ and η are finite-dimensional representations of M and M H , and λ ∈ a ∗ C , ν ∈ a ∗ H, C ,the complexified duals of the Lie algebras of A and A H . Consider the spaceHom H (Ind GP ( ξ ⊗ e λ ⊗ ) , Ind HP H ( η ⊗ e ν ⊗ ))of continuous H -intertwining operators. Realizing Ind GP ( ξ ⊗ e λ ⊗ ) and Ind HP H ( η ⊗ e ν ⊗ ) onthe spaces of smooth sections of the vector bundles V λ = G × P ( ξ ⊗ e λ + ρ ⊗ ) → G/P and W ν = H × P H ( η ⊗ e ν + ρ H ⊗ ) → H/P H , where ρ ∈ a ∗ and ρ H ∈ a ∗ H are the half sums of positive roots, one can identify continuous H -intertwining operators with their distribution kernels. More precisely, Kobayashi–Speh [9]showed that taking distribution kernels is a linear isomorphismHom H (Ind GP ( ξ ⊗ e λ ⊗ ) , Ind HP H ( η ⊗ e ν ⊗ )) ∼ → ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) , A K A , where V ∗ λ is the dual bundle of V λ and W ν the P H -representation defining W ν (see Section 1.2for the precise definition).Now suppose ( τ, E ) is an irreducible finite-dimensional representation of G . Then therestriction τ | P of τ to P contains a unique irreducible subrepresentation i : E ′ ֒ → E (seeLemma 2.1). With respect to the Langlands decomposition P = M AN this subrepresentationis of the form τ ′ ⊗ e µ ′ ⊗ for some irreducible finite-dimensional representation τ of M and µ ′ ∈ a ∗ . Let ( E ′ ) ∨ = Hom C ( E ′ , C ) denote the dual of E ′ . Theorem A (see Theorem 2.3 and Proposition 2.4) . For every P H -equivariant quotient p : E | P H ։ E ′′ with E ′′ ≃ τ ′′ ⊗ e µ ′′ ⊗ as P H -representations, there is a unique linear map Φ : Hom H (Ind GP ( ξ ⊗ e λ ⊗ ) , Ind HP H ( η ⊗ e ν ⊗ )) → Hom H (Ind GP (( ξ ⊗ τ ′ ) ⊗ e λ + µ ′ ⊗ ) , Ind HP H (( η ⊗ τ ′′ ) ⊗ e ν + µ ′′ ⊗ )) with the property that for every intertwining operator A with distribution kernel K A , thedistribution kernel K Φ( A ) of Φ( A ) is given by K Φ( A ) = ϕ ⊗ K A , the multiplication of K A withthe smooth section ϕ ∈ C ∞ ( G/P, G × P ( E ′ ) ∨ ) ⊗ E ′′ defined by ϕ ( g ) = p ◦ τ ( g ) ◦ i ∈ Hom C ( E ′ , E ′′ ) ≃ ( E ′ ) ∨ ⊗ E ′′ , g ∈ G. The translation principle allows to construct new intertwining operators from existing ones.But one can also reverse the roles and in some cases use the translation principle to classifyintertwining operators (see e.g. Theorem 3.8).
NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 3
Knapp–Stein type intertwining operators.
Now assume that (
G, H ) is a symmetricpair, i.e. H is an open subgroup of the fixed points G σ of an involution σ of G . For simplicitywe further assume that G is in the Harish-Chandra class. Let P = M AN ⊆ G be a σ -stable parabolic subgroup, then P H = P ∩ H is a parabolic subgroup of H and we write P H = M H A H N H for its Langlands decomposition.In our previous paper [12] with Y. Oshima we constructed meromorphic families of inter-twining operators between the spherical principal series representations Ind GP ( ⊗ e λ ⊗ ) andInd HP H ( ⊗ e ν ⊗ ). We now generalize this construction and obtain intertwining operatorsbetween vector-valued principal series representations using the translation principle.Assume that P and its opposite parabolic P are conjugate via the Weyl group, i.e. P =˜ w P ˜ w − , where ˜ w is a representative of the longest Weyl group element w . Then for anyfinite-dimensional representation ξ of M and α, β ∈ a ∗ C we consider the function K ξ,α,β ( g ) = ξ ( m ( ˜ w − g − )) − a ( ˜ w − g − ) α a ( ˜ w − g − σ ( g )) β , g ∈ G, where m ( g ) and a ( g ) are the densely defined projections of g ∈ N M AN onto the M - and A -component. For ξ = the trivial representation, these are the kernel functions constructedin [12]. Since a ( g ) is only defined on an open dense subset, it may happen that the factor a ( ˜ w − g − σ ( g )) is not defined for any g ∈ G , whence we additionally assume that the domainof definition for a ( ˜ w − g − σ ( g )) is not empty. In [12] we showed that in this case K ξ,α,β ( g ) isdefined on an open dense subset in G , and also gave a criterion to check this.Under the above assumptions, the translation principle combined with our previous resultsfrom [12] yields: Corollary B (see Corollary 2.9) . Assume that the finite-dimensional representation ξ of M is extendible to G (see Section 2.4 for the precise definition). Then the functions K ξ,α,β ( g ) extend to a meromorphic family of distributions K ξ,α,β ∈ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) , α, β ∈ a ∗ C , where (0.1) λ = − w α + σβ − w β + ρ, ν = − α | a H, C − ρ H . Therefore, they give rise to a meromorphic family of intertwining operators A ( ξ, α, β ) ∈ Hom H (Ind GP ( ˜ w ξ ⊗ e λ ⊗ ) , Ind HP H ( ξ | M H ⊗ e ν ⊗ )) . As in [12, Corollary B] this construction gives in particular lower bounds on multiplicities:dim Hom H (Ind GP ( ˜ w ξ ⊗ e λ ⊗ ) , Ind HP H ( ξ | M H ⊗ e ν ⊗ )) ≥ λ, ν ) of the form (0.1). Symmetry breaking operators for rank one orthogonal groups.
We illustrate thetranslation principle in two examples, the first one being scalar-valued principal series repre-sentations for the symmetric pair (
G, H ) = (O( n + 1 , , O( n, P and P H satisfy M H ≃ O( n − × O(1) ⊆ O( n ) × O(1) ≃ M, A H = A ≃ R + , N H ≃ R n − ⊆ R n ≃ N. We identify a ∗ C ≃ C such that ρ = n , and denote by sgn the non-trivial character of O (1) ⊆ M H ⊆ M . Then for δ, ε ∈ Z / Z and λ, ν ∈ C we consider intertwining operators between π λ,δ = Ind GP (sgn δ ⊗ e λ ⊗ ) and τ ν,ε = Ind HP H (sgn ε ⊗ e ν ⊗ ) . JAN M ¨OLLERS AND BENT ØRSTED
Tensoring with characters of G , it is easy to see thatHom H ( π λ,δ | H , τ ν,ε ) ≃ Hom H ( π λ, − δ | H , τ ν, − ε ) . For the pair (
G, H ), the restriction of a distribution kernel K ∈ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) to the open dense Bruhat cell in G/P , which is isomorphic to N ≃ R n , defines an isomorphismonto a subspace of D ′ ( R n ) (see Kobayashi–Speh [9, Theorem 3.16] or Theorem 1.2 for details).Let D ′ ( R n ) + λ,ν resp. D ′ ( R n ) − λ,ν denote the space of distribution kernels defining intertwiningoperators in Hom H ( π λ,δ | H , τ ν,ε ) with δ + ε ≡ δ + ε ≡ D ′ ( R n ) + λ,ν was classified by Kobayashi–Speh [9], and we briefly describe thisclassification, borrowing their notation. First, they construct a meromorphic family of distri-butions K A , + λ,ν ∈ D ′ ( R n ) + λ,ν , ( λ, ν ) ∈ C , given by K A , + λ,ν ( x ′ , x n ) = | x n | λ + ν − ( | x ′ | + x n ) − ν − n − , ( x ′ , x n ) ∈ R n − × R = R n . Then they show that every distribution in D ′ ( R n ) + λ,ν is given by K A , + λ,ν or a regularization ofit. By a detailed analysis of the meromorphic nature, the poles and all possible residues ofthe family K A , + λ,ν they obtain a complete description of D ′ ( R n ) + λ,ν for all ( λ, ν ) ∈ C . Moreprecisely, let L even = { ( − ρ − i, − ρ H − j ) : i, j ∈ N , i − j ∈ N } , then the corresponding statement for symmetry breaking operators is: For δ + ε ≡ H ( π λ,δ | H , τ ν,ε ) = dim D ′ ( R n ) + λ,ν = ( λ, ν ) ∈ L even ,1 for ( λ, ν ) ∈ C − L even ,and every intertwining operator is given by the distribution kernel K A , + λ,ν or a regularizationof it.We apply the translation principle to the kernels K A , + λ,ν to obtain a meromorphic family K A , − λ,ν ∈ D ′ ( R n ) − λ,ν given by K A , − λ,ν ( x ) = x n · K A , + λ − ,ν ( x ) = sgn( x n ) | x n | λ + ν − ( | x ′ | + x n ) − ν − n − , x ∈ R n . In Theorem 3.6 we derive from the poles and residues of K A , + λ,ν all poles and all possible residuesof the family K A , − λ,ν and use them to classify intertwining operators. For the statement let L odd = { ( − ρ − i, − ρ H − j ) : i, j ∈ N , i − j ∈ N + 1 } . Theorem C (see Theorems 3.6 and 3.8) . For δ + ε ≡ we have dim Hom H ( π λ,δ | H , τ ν,ε ) = ( for ( λ, ν ) ∈ L odd , for ( λ, ν ) ∈ C − L odd ,and every intertwining operator is given by the distribution kernel K A , − λ,ν or a regularizationof it. We remark that for λ − ν = − − ℓ , ℓ ∈ N , there exists a residue of K A , − λ,ν which issupported at the origin in R n , inducing a differential intertwining operator. In this setting G/P ≃ S n and H/P H ≃ S n − , and the differential intertwining operators form the family ofodd order conformally invariant differential operators C ∞ ( S n , V λ ) → C ∞ ( S n − , W ν ) studied NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 5 previously by Juhl [4] (see also [8]). The even order Juhl operators were already obtained asresidues of the family K A , + λ,ν by Kobayashi–Speh [9]. Symmetry breaking operators for rank one pin groups.
The second illustration ofthe translation principle is for the symmetric pair ( e G, e H ) = (Pin( n + 1 , , Pin( n, p, q ) denotes a certain double cover of the group O( p, q ) (see Appendix A for details) sothat we have compatible double covers e G → G and e H → H with G and H as in the previoussection. The preimages of the parabolic subgroups P and P H under the covering maps are e P ≃ f M AN and e P H ≃ f M H A H N H with A , N , A H and N H as in the previous section and f M H ≃ Pin( n − × O(1) ⊆ Pin( n ) × O(1) ≃ f M .
The fundamental representations of the Lie algebra so ( n ) of M are the exterior power repre-sentations ∧ k R n and the spin representations. Symmetry breaking operators for the exteriorpower representations are recently being investigated in detail by Kobayashi–Speh (see alsoFischmann–Juhl–Somberg [3] and Kobayashi–Kubo–Pevzner [7] for the case of differentialsymmetry breaking operators). Here we focus on the fundamental spin representations.The group Pin( n ) can be realized inside the Clifford algebra Cl( n ) with n generators (seeAppendix A for details). The fundamental spin representations of Pin( n ) are the restric-tions of irreducible representations of the complex Clifford algebra Cl( n ; C ) = Cl( n ) ⊗ R C toPin( n ), and therefore we do not distinguish between the representations of Cl( n ; C ) and theirrestrictions to Pin( n ). For even n the the Clifford algebra Cl( n ; C ) has a unique irreduciblerepresentation, and for odd n it has two inequivalent irreducible representations. Let ( ζ n , S n )be an irreducible representation of Cl( n ; C ) and ( ζ n − , S n − ) an irreducible representation ofCl( n − C ). If sgn denotes the non-trivial representation of O(1), we have for all λ, ν ∈ C and δ, ε ∈ Z / Z principal series representations /π λ,δ = Ind e G e P (( ζ n ⊗ sgn δ ) ⊗ e λ ⊗ ) and /τ ν,ε = Ind e H e P H (( ζ n − ⊗ sgn ε ) ⊗ e ν ⊗ ) , and we study intertwining operators in the spaceHom e H ( /π λ,δ | e H , /τ ν,ε ) . As above, taking distribution kernels and restricting them to the open dense Bruhat cell in
G/P identifies the space of intertwining operators with a subspace D ′ ( R n ; Hom C ( S n , S n − )) ± λ,ν ⊆ D ′ ( R n ; Hom C ( S n , S n − )) ≃ D ′ ( R n ) ⊗ Hom C ( S n , S n − ) , where the sign + represents δ + ε ≡ − represents δ + ε ≡ K A , ± λ,ν ∈ D ′ ( R n ) ± λ,ν yields mero-morphic families of distribution kernels P /K A , ± λ,ν ∈ D ′ ( R n ; Hom C ( S n , S n − )) ± λ,ν given by P /K A , ± λ,ν ( x ) = ( P ζ n ( x )) · K A , ∓ λ − ,ν + ( x ) , where ζ n ( x ) ∈ End C ( S n ) is the value of the representation ζ n of the Clifford algebra Cl( n ; C )at the vector x ∈ R n ⊆ Cl( n ) ⊆ Cl( n, C ) and 0 = P ∈ Hom
Pin( n − ([ ζ n ⊗ det] | Pin( n − , ζ n − ).(Note that the spin representation ζ n − of Pin( n −
1) occurs in the restriction [ ζ n ⊗ det] | Pin( n − with multiplicity one, where det : Pin( n ) → O( n ) → {± } denotes the determinant charac-ter.) JAN M ¨OLLERS AND BENT ØRSTED
To state our result on the classification of intertwining operators, let /L even = { ( − ρ − − i, − ρ H − − j ) : i, j ∈ N , i − j ∈ N } ,/L odd = { ( − ρ − − i, − ρ H − − j ) : i, j ∈ N , i − j ∈ N + 1 } . Theorem D (see Theorems 4.3, 4.4 and 5.5) . (1) For δ + ε ≡ we have dim Hom e H ( /π λ,δ | e H , /τ ν,ε ) = ( for ( λ, ν ) ∈ /L even , for ( λ, ν ) ∈ C − /L even ,and every intertwining operator is given by the distribution kernel P /K A , + λ,ν or a regu-larization of it.(2) For δ + ε ≡ we have dim Hom e H ( /π λ,δ | e H , /τ ν,ε ) = ( for ( λ, ν ) ∈ /L odd , for ( λ, ν ) ∈ C − /L odd ,and every intertwining operator is given by the distribution kernel P /K A , − λ,ν or a regu-larization of it. In Theorems 4.3 and 4.4 we determine all poles and residues of the meromorphic families
P /K A , ± λ,ν , and hence give an explicit description of the distribution kernels of all intertwiningoperators between spinor-valued principal series representations.We remark that for λ − ν = − − ℓ resp. λ − ν = − − ℓ , ℓ ∈ N , there exists aresidue of P /K A , + λ,ν resp. P /K A , − λ,ν which is supported at the origin in R n , inducing a differentialintertwining operator. These families of spinor-valued differential intertwining operators werepreviously obtained by Kobayashi–Ørsted–Somberg–Souˇcek [8], and it was conjectured thatthese are all differential intertwining operators in this setting. Our classification confirms thisconjecture (see Remark 4.6).In contrast to the proof of Theorem C which only uses the translation principle, we employthe method developed in [10] for the proof of Theorem D. This method describes intertwiningoperators between the underlying Harish-Chandra modules of principal series representationsin terms of their action on the different K -types. The explicit knowledge of the action on K -types also allows us to determine the dimensions of intertwining operators between theirreducible constituents of /π λ,δ and /τ ν,ε at reducibility points.The representation /π λ,δ is reducible if and only if λ = ± ( ρ + + i ), i ∈ N . More precisely, for λ = − ρ − − i the representation /π λ,δ has a finite-dimensional irreducible subrepresentation F δ ( i ) and the quotient T δ ( i ) = /π λ,δ / F δ ( i ) is irreducible. The composition factors at λ = ρ + + i can be described in terms of F δ ( i ) and T δ ( i ) by tensoring with the determinantcharacter (see Lemma 5.6 for the precise statement). We use the analogous notation for thecomposition factors F ′ ε ( j ) and T ′ ε ( j ) of /τ ν,ε at ν = − ρ H − − j , j ∈ N . Theorem E (see Theorem 5.7) . For π ∈ {T δ ( i ) , F δ ( i ) } and τ ∈ {T ′ ε ( j ) , F ′ ε ( j ) } the multiplic-ities dim Hom e H ( π | e H , τ ) are given by NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 7 π τ F ′ ε ( j ) T ′ ε ( j ) F δ ( i ) 1 0 T δ ( i ) 0 1 for ≤ j ≤ i , i + j ≡ δ + ε (2) , π τ F ′ ε ( j ) T ′ ε ( j ) F δ ( i ) 0 0 T δ ( i ) 1 0 otherwise. We remark that the multiplicities are the same as in the case of spherical principal series(see [9, Theorem 1.2]).
Structure of the paper.
In Section 1 we fix the notation for (generalized) principal seriesrepresentations of real reductive groups, explain how to describe intertwining operators be-tween them in terms of invariant distributions, and recall the construction of Knapp–Steintype intertwining operators for symmetric pairs from our joint work with Y. Oshima [12].Section 2 explains the idea of the translation principle in detail and contains the proofs ofTheorem A and Corollary B. We then apply the translation principle in two different situ-ations. Firstly, in Section 3 we construct and classify intertwining operators between prin-cipal series representations of (
G, H ) = (O( n + 1 , , O( n, e G, e H ) = (Pin( n + 1 , , Pin( n, Notation. N = { , , , . . . } , ( λ ) n = λ ( λ + 1) · · · ( λ + n − A − B = { a ∈ A : a / ∈ B } .1. Preliminaries
We recall the basic facts about (generalized) principal series representations of real reduc-tive groups, symmetry breaking operators, and their construction for symmetric pairs. Moredetails can be found in [9, 12].1.1.
Generalized principal series representations.
Let G be a real reductive group and P ⊆ G a parabolic subgroup with Langlands decomposition P = M AN . We write g , m , a and n for the Lie algebras of G , M , A and N . Then A = exp( a ) and N = exp( n ). Let n denotethe nilradical opposite to n and N = exp( n ) the corresponding closed connected subgroup of G . Then P = M AN ⊆ G is the parabolic subgroup opposite to P .The multiplication map N × M × A × N → G is a diffeomorphism onto an open densesubset of G , and for g ∈ N M AN we define n ( g ) ∈ N , m ( g ) ∈ M , a ( g ) ∈ A and n ( g ) ∈ N by g = n ( g ) m ( g ) a ( g ) n ( g ).We consider representations of G which are parabolically induced from finite-dimensionalrepresentations of P . Let ( ξ, V ) be a finite-dimensional representation of M , λ ∈ a ∗ C anddenote by the trivial representation of N , then ξ ⊗ e λ ⊗ is a finite-dimensional representationof P = M AN . Write ρ = tr ad n ∈ a ∗ for half the sum of all positive roots and let V λ = ξ ⊗ e λ + ρ ⊗ . We define the generalized principal series representation Ind GP ( ξ ⊗ e λ ⊗ ) as JAN M ¨OLLERS AND BENT ØRSTED the left-regular representation on the space C ∞ ( G, V λ ) P = { f ∈ C ∞ ( G, V ) : f ( gman ) = a − λ − ρ ξ ( m ) − f ( g ) ∀ g ∈ G, man ∈ M AN } . If we write V λ = G × P V λ → G/P for the homogeneous vector bundle associated to the representation V λ of P , then C ∞ ( G, V λ ) P can be identified with the space C ∞ ( G/P, V λ ) of smooth sections of V λ .1.2. Distribution sections of vector bundles.
Let V ∗ λ = ξ ∨ ⊗ e − λ + ρ ⊗ , where ξ ∨ is thecontragredient representation of ξ , and write V ∗ λ = G × P V ∗ λ for the dual bundle of V λ . Wedefine the space D ′ ( G/P, V λ ) of distribution sections of the bundle V λ as the (topological)dual of C ∞ ( G/P, V ∗ λ ): D ′ ( G/P, V λ ) = C ∞ ( G/P, V ∗ λ ) ′ . Note that since
G/P is compact, smooth sections on
G/P are automatically compactly sup-ported. Then C ∞ ( G/P, V λ ) ≃ C ∞ ( G, V λ ) P embeds G -equivariantly into D ′ ( G/P, V λ ) by f T f , where h T f , ϕ i = Z K h f ( k ) , ϕ ( k ) i dk ∀ ϕ ∈ C ∞ ( G, V ∗ λ ) P , and dk denotes the normalized Haar measure on K .Let ( τ, E ) be another finite-dimensional representation of P and E = G × P E the corre-sponding vector bundle over G/P . Then every smooth section f ∈ C ∞ ( G/P, E ) defines acontinuous linear multiplication operator D ′ ( G/P, V λ ) → D ′ ( G/P,
E ⊗ V λ ) , u f ⊗ u, which is dual to the composition C ∞ ( G, E ∨ ⊗ V ∗ λ ) P f ⊗ → C ∞ ( G, E ⊗ E ∨ ⊗ V ∗ λ ) P c ∗ → C ∞ ( G, V ∗ λ ) P of the pointwise tensor product f ⊗ and the push-forward by the contraction map c : E ⊗ E ∨ ⊗ V ∗ λ → V ∗ λ .1.3. Symmetry breaking operators.
Now let H ⊆ G be a reductive subgroup of G and P H = M H A H N H ⊆ H a parabolic subgroup of H . Similarly, we define Ind HP H ( η ⊗ e ν ⊗ ) for afinite-dimensional representation ( η, W ) of M H and ν ∈ a ∗ H, C as the left-regular representationof H on C ∞ ( H, W ν ) P H , where W ν = η ⊗ e ν + ρ H ⊗ , ρ H = tr ad n H . Then C ∞ ( H, W ν ) P H identifies with the space C ∞ ( H/P H , W ν ) of smooth sections of the vector bundle W ν = H × P H W ν → H/P H .In this paper we study continuous H -intertwining operatorsInd GP ( ξ ⊗ e λ ⊗ ) → Ind HP H ( η ⊗ e ν ⊗ ) . By the Schwartz Kernel Theorem such maps are identified with H -invariant distributionsections of some vector bundle over G/P × H/P H . Using the isomorphism ∆( H ) \ ( G × H ) ≃ G which is induced by G × H → G, ( g, h ) g − h , invariant distributions on G/P × H/P H reduceto invariant distributions on G/P (see [9, Section 3.2]):
Proposition 1.1 ([9, Proposition 3.2]) . We have natural isomorphisms of vector spaces
Hom H (Ind GP ( ξ ⊗ e λ ⊗ ) , Ind HP H ( η ⊗ e ν ⊗ )) ≃ D ′ ( G/P × H/P H , V ∗ λ ⊗ W ν ) ∆( H ) ≃ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 9
Under the isomorphism an H -intertwining operator A : Ind GP ( ξ ⊗ e λ ⊗ ) → Ind HP H ( η ⊗ e ν ⊗ )maps to the distribution kernel K A ∈ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) such that Af ( h ) = Z G/P c ( f ( x ) ⊗ K A ( h − x )) dx, f ∈ C ∞ ( G/P, V λ ) , where c : V λ ⊗ V ∗ λ ⊗ W ν → W ν is the contraction map, and the integral has to be understoodin the distribution sense.Sometimes it is convenient to work on the open dense Bruhat cell in G/P , because it isisomorphic to the vector space n . Under certain conditions on the parabolic subgroups P and P H the restriction of the invariant distribution sections in Proposition 1.1 to the open denseBruhat cell is injective: Theorem 1.2 ([9, Theorem 3.16]) . Assume that P H = P ∩ H, M H = M ∩ H, A H = A ∩ H, N H = N ∩ H. If additionally G = P H N P then the restriction to n ≃ N ֒ → G/P defines a linear isomorphism ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) → D ′ ( n , V ∗ λ ⊗ W ν ) M H A H , n H . Here the action of M H A H on D ′ ( n , V ∗ λ ⊗ W ν ) is the obvious action induced by the actionsof M H A H on n , V ∗ λ and W ν , and the action of n H is induced by the infinitesimal action on n viewed as subset of the generalized flag variety G/P .1.4.
Symmetric pairs and Knapp–Stein type intertwining operators.
For symmet-ric pairs (
G, H ) we provided in [12] explicit expressions of invariant distributions definingsymmetry breaking operators, which we briefly recall.Let σ be an involution of G and let H be an open subgroup of G σ , the fixed points of σ .Then ( G, H ) forms a symmetric pair. We make the following two additional assumptions: P and P are conjugate via the Weyl group(G) P is σ -stable . (H)Then (G) implies P = ˜ w P ˜ w − , where ˜ w is a representative of the longest Weyl groupelement w . Further, (H) implies that P H = P ∩ H is a parabolic subgroup of H .Recall the a -projection g a ( g ) ∈ A from Section 1.1 which is defined on the open densesubset N M AN ⊆ G . For α, β ∈ a ∗ C sufficiently positive we define K α,β ( g ) = a ( ˜ w − g − ) α a ( ˜ w − g − σ ( g )) β , g ∈ G. Since the second factor might not be defined for any g ∈ G , we make the additional assumption(D) The domain of definition for K α,β is non-empty.In [12, Proposition 2.5] we showed that in this case the domain of definition for K α,β is alreadyopen and dense in G , and gave a criterion to check this.In [12] we studied the meromorphic continuation of K α,β in the parameters α, β ∈ a ∗ C , andproved that they give rise to intertwining operators between spherical principal series: Theorem 1.3 ([12, Theorems 3.1 & 3.3]) . Under the assumptions (G) , (H) and (D) , thefunctions K α,β extend to a meromorphic family of distributions K α,β ∈ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) , where (1.1) λ = − w α + σβ − w β + ρ, ν = − α | a H, C − ρ H . Therefore, they give rise to a meromorphic family of intertwining operators A ( α, β ) : Ind GP ( ⊗ e λ ⊗ ) → Ind HP H ( ⊗ e ν ⊗ ) . The Translation Principle
We describe a technique, called the translation principle , which allows to obtain new sym-metry breaking operators from existing ones by tensoring with finite-dimensional representa-tions of G .2.1. General technique.
Fix a principal series representation Ind GP ( ξ ⊗ e λ ⊗ ) of G andlet ( τ, E ) be a finite-dimensional representation of G , then there is a natural G -equivariantisomorphism(2.1) ι τ : Ind GP (( ξ ⊗ e λ ⊗ ) ⊗ E | P ) ∼ → Ind GP ( ξ ⊗ e λ ⊗ ) ⊗ E. When we view both sides as ( V ⊗ E )-valued functions on G , this isomorphism is given by( ι τ f )( g ) = (id ⊗ τ ( g )) f ( g ) , g ∈ G. Now, for any P -stable subspace E ′ ⊂ E we have a natural injective mapInd GP (( ξ ⊗ e λ ⊗ ) ⊗ E ′ ) ֒ → Ind GP (( ξ ⊗ e λ ⊗ ) ⊗ E | P ) . Suppose that N acts trivially on E ′ and A acts by a fixed character e µ ′ , µ ∈ a ∗ , then the P -action on E ′ can be written as τ ′ ⊗ e µ ′ ⊗ . The above map becomes(2.2) Ind GP (( ξ ⊗ τ ′ ) ⊗ e λ + µ ′ ⊗ ) ֒ → Ind GP (( ξ ⊗ e λ ⊗ ) ⊗ E | P ) . Assuming irreducibility of τ and τ ′ , there is essentially only one choice of such a P -stablesubspace E ′ ⊆ E : Lemma 2.1.
For every irreducible finite-dimensional representation ( τ, E ) of G the restric-tion τ | P contains a unique irreducible subrepresentation E ′ . Moreover, E ′ is generated by thehighest weight space of E , and P = M AN acts on E ′ by τ ′ ⊗ e µ ′ ⊗ , where τ ′ is an irreduciblerepresentation of M , µ ′ ∈ a ∗ and the trivial representation of N .Proof. It suffices to treat the case of G connected since M meets all connected componentsof G . We first fix some notation. Let t ⊆ m be a Cartan subalgebra, then c = t ⊕ a is aCartan subalgebra of g and c C is a Cartan subalgebra of g C . Choose any system of positiveroots Σ + ( g C , c C ) ⊆ Σ( g C , c C ) such that the non-zero restrictions of positive roots to a are theroots of n . Then the non-zero restrictions of positive roots in m to t C form a positive systemof roots Σ + ( m C , t C ) ⊆ Σ( m C , t C ). We consider highest weights with respect to these positivesystems.Now let E ′ ⊆ E be any irreducible subrepresentation for τ | P . Since M is reductive, E ′ decomposes into the direct sum E ′ = E ′ ⊕ · · · ⊕ E ′ m of irreducible M -representations E ′ i ofhighest weight λ i ∈ t ∗ C . Since M and A commute, A acts by a character µ i ∈ a ∗ on E ′ i . Now,let 1 ≤ i ≤ m such that λ i + µ i is maximal among the λ j + µ j , 1 ≤ j ≤ m . Then it is easy tosee that τ | N is trivial on E ′ i . Hence, E ′ i ⊆ E is stable under P . Since E ′ was assumed to beirreducible for P , we have E ′ = E ′ i and hence N acts trivially on E ′ .To show that E ′ is unique, we simply observe that a highest weight vector for the action of M on E ′ is automatically a highest weight vector for the action of G on E which is unique NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 11 (up to scalar multiples). Hence E ′ is the P -subrepresentation of E generated by the highestweight space. (cid:3) Further, in the case of minimal parabolic subgroups, essentially every irreducible finite-dimensional representation of M extends to G : Lemma 2.2 ([14, Theorem 2.1]) . Assume that G is a linear connected reductive Lie group and P ⊆ G is minimal parabolic. Then every irreducible finite-dimensional representation ( τ ′ , E ′ ) of M is conjugate via the Weyl group to a representation that occurs as a direct summand inan irreducible finite-dimensional representation ( τ, E ) of G and on which N acts trivially. Similarly, for a fixed principal series representation Ind HP H ( η ⊗ e ν ⊗ ) we have an isomor-phism(2.3) Ind HP H ( η ⊗ e ν ⊗ ) ⊗ E ∼ → Ind HP H (( η ⊗ e ν ⊗ ) ⊗ E | P H ) . We also take a P H -quotient space E ։ E ′′ on which N H acts trivially and A H acts by acharacter e µ ′′ . Note that such a quotient always exists since the contragredient representation E ∨ of E possesses a P H -stable subspace on which N H acts trivially by Lemma 2.1. However,in contrast to Lemma 2.1, there might be several possibilities for E ′′ . Denoting the P H -actionon E ′′ by τ ′′ ⊗ e µ ′′ ⊗ , we get a map(2.4) Ind HP H (( η ⊗ e ν ⊗ ) ⊗ E | P H ) ։ Ind HP H (( η ⊗ τ ′′ ) ⊗ e ν + µ ′′ ⊗ ) . Now suppose that an H -intertwining operator A : Ind GP ( ξ ⊗ e λ ⊗ ) → Ind HP H ( η ⊗ e ν ⊗ )is given, and form the tensor product(2.5) A ⊗ id E : Ind GP ( ξ ⊗ e λ ⊗ ) ⊗ E → Ind HP H ( η ⊗ e ν ⊗ ) ⊗ E. Then we obtain an H -intertwining operatorΦ( A ) : Ind GP (( ξ ⊗ τ ′ ) ⊗ e λ + µ ′ ⊗ ) → Ind HP H (( η ⊗ τ ′′ ) ⊗ e ν + µ ′′ ⊗ )by composing the maps (2.2), (2.1), (2.5), (2.3) and (2.4), namelyInd GP (( ξ ⊗ τ ′ ) ⊗ e λ + µ ′ ⊗ ) ֒ → Ind GP (( ξ ⊗ e λ ⊗ ) ⊗ E | P ) ∼ → Ind GP ( ξ ⊗ e λ ⊗ ) ⊗ E → Ind HP H ( η ⊗ e ν ⊗ ) ⊗ E (2.6) ∼ → Ind HP H (( η ⊗ e ν ⊗ ) ⊗ E | P H ) ։ Ind HP H (( η ⊗ τ ′′ ) ⊗ e ν + µ ′′ ⊗ ) . This proves:
Theorem 2.3.
Let ( τ, E ) be a finite-dimensional G -representation, E ′ ⊆ E a P -stable sub-space with E ′ | P = τ ′ ⊗ e µ ′ ⊗ and E ։ E ′′ a P H -equivariant quotient with E ′′ | P H = τ ′′ ⊗ e µ ′′ ⊗ .Then (2.6) defines a linear map Φ : Hom H (Ind GP ( ξ ⊗ e λ ⊗ ) , Ind HP H ( η ⊗ e ν ⊗ )) → Hom H (Ind GP (( ξ ⊗ τ ′ ) ⊗ e λ + µ ′ ⊗ ) , Ind HP H (( η ⊗ τ ′′ ) ⊗ e ν + µ ′′ ⊗ )) for all finite-dimensional representations ξ of M and η of M H and all λ ∈ a ∗ C , ν ∈ a ∗ H, C . Integral kernels.
Recall that H -intertwining operators are given by distribution ker-nels (see Proposition 1.1). Let us see how the integral kernel behaves under the translationprinciple. Suppose that A : Ind GP ( ξ ⊗ e λ ⊗ ) → Ind HP H ( η ⊗ e ν ⊗ )is given by a the distribution kernel K A ∈ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) in the sense that Af ( h ) = Z G/P c ( f ( x ) ⊗ K A ( h − x )) dx, f ∈ C ∞ ( G/P, V λ ) , where c denotes the contraction map V λ ⊗ V ∗ λ ⊗ W ν → W ν and the integral is meant in thedistribution sense (see Section 1.3 for details). Write i : E ′ → E for the inclusion map and p : E → E ′′ for the quotient map, and let E ′ = G × P E ′ and E ′′ = H × P H E ′′ denote thecorresponding homogeneous vector bundles over G/P and
H/P H . Let u ∈ C ∞ ( G/P, E ′ ⊗ V λ ),then Φ( A ) u ∈ C ∞ ( H/P H , E ′′ ⊗ W ν ) is given byΦ( A ) f ( h )= (cid:16)(cid:0) p ◦ τ ( h − ) (cid:1) ⊗ id W ν (cid:17) Z G/P (id E ⊗ c ) (cid:18)(cid:16) ( τ ( x ) ◦ i ) ⊗ id V λ (cid:17)(cid:0) f ( x ) (cid:1) ⊗ K A ( h − x ) (cid:19) dx ! = Z G/P c (cid:16)(cid:0) p ◦ τ ( h − x ) ◦ i (cid:1) f ( x ) ⊗ K ( h − x ) (cid:17) dx. This implies:
Proposition 2.4.
The integral kernel Φ( K A ) = K Φ( A ) of Φ( A ) is given by Φ( K A ) = ϕ ⊗ K A , where ϕ ∈ C ∞ ( G/P, ( E ′ ) ∨ ) ⊗ E ′′ ≃ C ∞ ( G, ( E ′ ) ∨ ⊗ E ′′ ) P is given by ϕ ( g ) = p ◦ τ ( g ) ◦ i ∈ Hom C ( E ′ , E ′′ ) ≃ ( E ′ ) ∨ ⊗ E ′′ . Remark 2.5.
Since the operator Φ is given by multiplication with the fixed smooth function ϕ , and the multiplication map D ′ ( G/P, V ∗ λ ) ⊗ W ν → D ′ ( G/P, V ∗ λ ⊗ ( E ′ ) ∨ ) ⊗ ( W ν ⊗ E ′′ ) , K ϕ ⊗ K is continuous, the operator Φ maps holomorphic families of distributions to holomorphicfamilies. More precisely, if K z ∈ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) depends holomorphically on z ∈ Ω ⊆ C n , then Φ( K z ) depends holomorphically on z ∈ Ω. More generally, if K z dependsmeromorphically on z ∈ Ω with poles in the set Σ ⊆ Ω, then Φ( K z ) depends meromorphicallyon z ∈ Ω and its poles are contained in Σ. However, it may of course happen that K z has apole at z = z whereas Φ( K z ) is regular at z = z since the multiplication by ϕ can have anon-trivial kernel. Remark 2.6.
Since ϕ ( g ) is a matrix coefficient of a finite-dimensional repesentation, it is ob-viously smooth in g ∈ G . In view of Theorem 1.2 one can in some cases study the distributionkernels by their restriction to n ≃ N ֒ → G/P . On n the function ϕ ( g ) is actually a polynomial.In fact, the nilpotency of n implies that there exists N ∈ N such that τ ( X ) · · · τ ( X n ) = 0 forall X , . . . , X n ∈ n and n ≥ N . This shows that ϕ | n is a polynomial of degree at most N . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 13
Reformulation using P . We can also use the opposite parabolic subgroup P insteadof P in the above procedure. Assume there exists an element ˜ w ∈ K such that ˜ w P ˜ w − = P and ˜ w A ˜ w − = A . Then we have a G -equivariant isomorphismInd GP ( ξ ⊗ e λ ⊗ ) ≃ Ind GP ( ˜ w − ξ ⊗ e w − λ ⊗ ) , f f ( · ˜ w − ) . Here, ˜ w − ξ denotes the representation of M on V given by ( ˜ w − ξ )( m ) = ξ ( ˜ w m ˜ w − ). Supposethat an H -intertwining operator A : Ind GP ( ξ ⊗ e λ ⊗ ) → Ind HP H ( η ⊗ e ν ⊗ )is given. Composing with the above ismorphism we haveInd GP ( ˜ w − ξ ⊗ e w − λ ⊗ ) → Ind HP H ( η ⊗ e ν ⊗ )Then in a similar way, using P instead of P , we obtain an H -intertwining operatorInd GP (( ˜ w − ξ ⊗ τ ′ ) ⊗ e w − λ + µ ′ ⊗ ) → Ind HP H (( η ⊗ τ ′′ ) ⊗ e ν + µ ′′ ⊗ )for every P -stable subspace i : E ′ ֒ → E with E ′ ≃ τ ′ ⊗ e µ ′ ⊗ and every P H -stable quotient p : E → E ′′ . Composing with the map f f ( · ˜ w ), we getΨ( A ) : Ind GP (( ξ ⊗ ˜ w τ ′ ) ⊗ e λ + w µ ′ ⊗ ) → Ind HP H (( η ⊗ τ ′′ ) ⊗ e ν + µ ′′ ⊗ ) . Moreover, if A is given by the distribution kernel K A ∈ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) , then wesimilarly see that Ψ( A ) has distribution kernel Ψ( K A ) = K Ψ( A ) given byΨ( K A ) = ( p ◦ τ ( · ˜ w ) ◦ i ) ⊗ K A . Now assume additionally that P H ⊆ P with M H A H ⊆ M A and N H ⊆ N , then E ′ and E ′′ can be chosen compatibly. More precisely, let E ′ ⊆ E be the unique irreducible P -subrepresentation (see Lemma 2.1), then E ′ is the lowest a -weight space and E ′ ≃ τ ′ ⊗ e µ ′ ⊗ .Write E for the direct sum of all other a -weight spaces, then E = E ′ ⊕ E and the projection E ։ E ′ is P H -equivariant. Hence we can take E ′′ = E ′ : Corollary 2.7.
Assume that ˜ w P ˜ w − = P and that M H A H ⊆ M A and N H ⊆ N . Let ( τ, E ) be an irreducible finite-dimensional G -representation and E ′ ⊆ E the unique irreducible P -subrepresentation with E ′ = τ ′ ⊗ e µ ′ ⊗ . Then for all finite-dimensional representations ξ of M and η of M H and all λ ∈ a ∗ C , ν ∈ a ∗ H, C we obtain a linear map Ψ : Hom H (Ind GP ( ξ ⊗ e λ ⊗ ) , Ind HP H ( η ⊗ e ν ⊗ )) → Hom H (Ind GP (( ξ ⊗ ˜ w τ ′ ) ⊗ e λ + w µ ′ ⊗ ) , Ind HP H (( η ⊗ τ ′ | M H ) ⊗ e ν + µ ′ | a H ⊗ )) which maps an intertwining operator A with distribution kernel K A to the intertwining oper-ator Ψ( A ) with distribution kernel Ψ( K A ) = K Ψ( A ) given by Ψ( K A ) = ψ ⊗ K A , where ψ ∈ C ∞ ( G/P, ( E ′ ) ∨ ) ⊗ E ′ ≃ C ∞ ( G, ( E ′ ) ∨ ⊗ E ′ ) P is given by ψ ( g ) = τ ′ ( m ( ˜ w − g − )) − · a ( ˜ w − g − ) − µ ′ ∈ End C ( E ′ ) ≃ ( E ′ ) ∨ ⊗ E ′ . Proof.
It remains to show the formula for ψ ( g ). Write˜ w − g − = n ( ˜ w − g − ) m ( ˜ w − g − ) a ( ˜ w − g − ) n ( ˜ w − g − ) ∈ N M AN, then we have g ˜ w = n ( ˜ w − g − ) − m ( ˜ w − g − ) − a ( ˜ w − g − ) − n ( ˜ w − g − ) − ∈ N M AN .
Since N acts trivially on E ′ we have τ ( n ) ◦ i = i for n ∈ N , and similarly p ◦ τ ( n ) = p for n ∈ N H ⊆ N . Hence p ◦ τ ( g ˜ w ) ◦ i = τ ′ ( m ( ˜ w − g − ) − ) · a ( ˜ w − g − ) − µ ′ . (cid:3) Remark 2.8.
We note that the function ψ ( g ) resembles the integral kernel of the standardKnapp–Stein intertwining operator A ( λ ) : Ind GP ( τ ′ ⊗ e λ ⊗ ) → Ind GP ( ˜ w τ ′ ⊗ e w λ ⊗ ) , Af ( g ) = Z N f ( g ˜ w n ) dn. More precisely, it is shown in [5, Chapter VII, §
7] that Af ( g ) = Z N τ ′ ( m ( ˜ w − n )) a ( ˜ w − n ) λ − ρ f ( gn ) dn. Application to Knapp–Stein type intertwining operators.
We now specialize tothe setting of Section 1.4. In this case, all assumptions of Corollary 2.7 are satisfied and wecan apply it to the family A ( α, β ) of intertwining operators A ( α, β ) : Ind GP ( ⊗ e λ ⊗ ) → Ind HP H ( ⊗ e ν ⊗ ) ,λ and ν given by (1.1), with distribution kernels K α,β ( g ) = a ( ˜ w − g − ) α a ( ˜ w − g − σ ( g )) β , g ∈ G. We call an irreducible finite-dimensional representation ξ of M extendible if there exists afinite-dimensional irreducible representation ( τ, E ) of G such that E N = { v ∈ E : τ ( n ) v = v ∀ n ∈ N } ≃ ξ as M -representations. Corollary 2.9.
Assume conditions (G) , (H) and (D) . Then for every extendible irreduciblefinite-dimensional M -representation ξ the functions K ξ,α,β ( g ) = ξ ( m ( ˜ w − g − )) − a ( ˜ w − g − ) α a ( ˜ w − g − σ ( g )) β , g ∈ G, extend to a meromorphic family of distributions K ξ,α,β ∈ ( D ′ ( G/P, V ∗ λ ) ⊗ W ν ) ∆( P H ) , for λ and ν given by (1.1) and V λ = G × P ( ˜ w ξ ⊗ e λ + ρ ⊗ ) , W ν = ξ | M H ⊗ e ν + ρ H ⊗ . Therefore,they give rise to a meromorphic family of intertwining operators A ( ξ, α, β ) : Ind GP ( ˜ w ξ ⊗ e λ ⊗ ) → Ind HP H ( ξ ⊗ e ν ⊗ ) . Proof.
Let ( τ, E ) be an irreducible finite-dimensional representation of G with E N ≃ ξ as M -representations, then E ′ = E N ⊆ E is P -stable of the form E ′ ≃ ξ ⊗ e µ ⊗ . Now thestatement follows from Theorem 1.3 and Corollary 2.7. (cid:3) Remark 2.10.
For (
G, H ) = (O( n + 1 , , O( n, w of w that centralizes M . Hence, ˜ w ξ = ξ for all irreducible finite-dimensional representations ξ of M . By Lemma 2.2 either ξ or ˜ w ξ = ξ is extendible, so that in this case all irreduciblefinite-dimensional M -representations are extendible. NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 15 Example: ( G, H ) = (O( n + 1 , , O( n, Parabolic subgroups and the symmetric pair.
Let G = O( n + 1 , n ≥
2, realizedas the subgroup of GL( n + 2 , R ) preserving the indefinite bilinear form R n +2 → R , x x + · · · + x n +1 − x n +2 . We choose the (minimal) parabolic subgroup P = P min = M AN ⊆ G such that a = R H with H = n and n = g α for α ∈ a ∗ with α ( H ) = 1. Then M = ε m ε : m ∈ O( n ) , ε = ± ≃ O( n ) × Z / Z . Let m = diag( − , , . . . , , −
1) and identify m ∈ O( n ) with diag(1 , m, ∈ M , then M =O( n ) ∪ m O( n ). We further identify n = g − α ∼ = R n by R n → n , x x ⊤ x n x − x ⊤ and use this identification to parametrize N = exp( n ) by R n → n exp −−→ N , x n x = | x | / x ⊤ | x | / x n x −| x | / − x ⊤ − | x | / . The group M = O( n ) ∪ m O( n ) acts on x ∈ n ∼ = R n by the adjoint action as follows:Ad( m ) x = mx ( m ∈ O( n )) , Ad( m ) x = − x. Further, P is conjugate to its opposite parabolic P = M AN by˜ w = diag(1 , . . . , , − . Let us identify a ∗ C ∼ = C by λ λ ( H ), so that ρ = n . Lemma 3.1.
For x ∈ R n : ˜ w − n x ∈ N M AN ⇔ x ∈ R n − { } , and in this case for λ ∈ a ∗ C ≃ C we have a ( ˜ w − n x ) λ = | x | λ , m ( ˜ w − n x ) = n − xx ⊤ | x | . Now let σ be the involution of G given by conjugation with the matrix diag(1 , . . . , , − , H = G σ ≃ O( n,
1) and (
G, H ) forms a symmetric pair. It is easy to see that the pair(
G, H ) satisfies the assumptions in Theorem 1.2 and we write P H = P ∩ H = M H A H N H .Then M H = M ∩ H = O( n − ∪ m O( n −
1) and a H = a = R H . Further, under the identification n ≃ R n the involution σ acts by σ ( x ) = ( x , . . . , x n − , − x n ) , x ∈ R n , and therefore the subalgebra n H = n σ is given by the standard embedding of R n − into R n as the first n − Principal series representations and symmetry breaking operators.
Denote bysgn : M → {± } the character of M = O( n ) ∪ m O( n ) given by sgn( m ) = 1 for m ∈ O( n )and sgn( m ) = −
1. Abusing notation we also write sgn for the corresponding character of M H . For δ, ε ∈ Z / Z and λ, ν ∈ a ∗ C ≃ C we define the scalar principal series representations(smooth normalized parabolic induction) π λ,δ = Ind GP (sgn δ ⊗ e λ ⊗ ) and τ ν,ε = Ind HP H (sgn ε ⊗ e ν ⊗ ) . We consider the space Hom H ( π λ,δ | H , τ ν,ε )of symmetry breaking operators between π λ,δ and τ ν,ε . By Theorem 1.2 every such operatoris given by a distribution kernel K ∈ D ′ ( n ) M H A H , n H satisfying certain invariance conditionsfor the action of M H A H and n H . Since m ∈ M H acts by K ( x ) ( − δ + ε K ( − x ), it is clearthat D ′ ( n ) M H A H , n H only depends on the parity of δ + ε . Identfying n ≃ R n as above we write D ′ ( R n ) + λ,ν resp. D ′ ( R n ) + λ,ν for the space D ′ ( R n ) M H A H , n H with δ + ε ≡ δ + ε ≡ H ( π λ,δ | H , τ ν,ε ) ∼ → ( D ′ ( R n ) + λ,ν for δ + ε ≡ D ′ ( R n ) − λ,ν for δ + ε ≡ Construction of symmetry breaking operators.
In this section we describe allintertwining operators in Hom H ( π λ,δ , τ ν,ε ) for all λ, ν ∈ C , δ, ε ∈ Z / Z . We note that thenotation is essentially due to Kobayashi–Speh [9].3.3.1. Spherical principal series.
For δ = ε = 0 the representations π λ,δ and τ ν,ε are spherical,i.e. possess a vector invariant under a maximal compact subgroup. In this setting, Section 1.4provides a meromorphic family of intertwining operators A λ,ν ∈ Hom H ( π λ, | H , τ ν, ) given bythe distribution kernels K A , + λ,ν ( x ) = | x n | λ + ν − ( | x ′ | + x n ) − ν − n − , x ∈ R n . By analyzing the poles and residues of K A , + λ,ν explicitly, Kobayashi–Speh [9] completely deter-mine the space D ′ ( R n ) + λ,ν for all λ, ν ∈ C . (Note that our parameters ( λ, ν ) are normalizedsuch that π λ,δ and τ ν,ε are unitary for λ, ν ∈ i R . Therefore, in our notation one has to replace NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 17 ( λ, ν ) by ( λ − ρ, ν − ρ H ) to obtain Kobayashi–Speh’s notation, see [9].) To summarize theirresults let \\ + = { ( λ, ν ) : λ + ν = − − k, k ∈ N } ,// + = { ( λ, ν ) : λ − ν = − − ℓ, ℓ ∈ N } ,L even = { ( λ, ν ) = ( − ρ − i, − ρ H − j ) : i, j ∈ N , i − j ∈ N } ⊆ // + . For ( λ, ν ) ∈ \\ + with λ + ν = − − k we further define e K B , + λ,ν ( x ) = 1Γ( ( λ − ν + )) ( | x ′ | + x n ) − ν − n − δ (2 k ) ( x n )= k X i =0 ( − i (2 k )!( ν + n − ) i i !(2 k − i )!Γ( ( λ − ν + )) | x ′ | − n − ν − i δ (2 k − i ) ( x n ) , and for ( λ, ν ) ∈ // + with λ − ν = − − ℓ we put e K C , + λ,ν ( x ) = ℓ X j =0 ℓ − j ( ν − ℓ ) ℓ − j j !(2 ℓ − j )! (∆ j R n − ∂ ℓ − jn δ )( x ) . Note that both e K B , + λ,ν and e K C , + λ,ν depend holomorphically on ν ∈ C (or λ ∈ C ). Theorem 3.2.
The renormalized distribution e K A , + λ,ν ( x ) = K A , + λ,ν ( x )Γ( ( λ + ν + ))Γ( ( λ − ν + )) depends holomorphically on ( λ, ν ) ∈ C and vanishes only for ( λ, ν ) ∈ L even . More precisely,(1) For ( λ, ν ) ∈ C − ( \\ + ∪ // + ) we have supp e K A , + λ,ν = R n .(2) For ( λ, ν ) ∈ \\ + − // + with λ + ν = − − k , k ∈ N , we have e K A , + λ,ν ( x ) = ( − k k !(2 k )! e K B , + λ,ν ( x ) . In particular, supp e K A , + λ,ν = R n − .(3) For ( λ, ν ) ∈ // + − L even with λ − ν = − − ℓ , ℓ ∈ N , we have e K A , + λ,ν ( x ) = ( − ℓ ℓ ! π n − ℓ Γ( ν + n − ) e K C , + λ,ν ( x ) . In particular, supp e K A , + λ,ν = { } .(4) For ( − ρ − i, − ρ H − j ) ∈ L even , ℓ = i − j , restricting the function ( λ, ν ) e K A , + λ,ν totwo different complex hyperplanes in C through ( − ρ − i, − ρ H − j ) and renormalizinggives two holomorphic families of distributions: e K C , + λ,ν ( x ) = ( − ℓ ℓ ℓ ! π n − Γ( ν + ρ H ) e K A , + λ,ν ( x ) ( λ − ν = − − ℓ ) , ee K A , + λ,ν ( x ) = Γ( ( λ − ν + )) e K A , + λ,ν ( x ) ( ν = − ρ H − j ) . Their special values at ( − ρ − i, − ρ H − j ) satisfy supp e K C , + − ρ − i, − ρ H − j = { } , supp ee K A , + − ρ − i, − ρ H − j = ( R n for n even, R n − for n odd,more precisely, for n odd and k = i + j + n − ee K A , + − ρ − i, − ρ H − j = ( − k k !(2 k )! ( | x ′ | + x n ) j δ (2 k ) ( x n ) . Proof.
This is a summary of the results of [9]. (cid:3)
These results can be used to describe the space D ′ ( R n ) + λ,ν : Theorem 3.3.
We have D ′ ( R n ) + λ,ν = ( C e K A , + λ,ν for ( λ, ν ) ∈ C − L even , C ee K A , + λ,ν ⊕ C e K C , + λ,ν for ( λ, ν ) ∈ L even .Proof. See [9, Theorem 1.9] and also [10, Theorem 4.9]. (cid:3)
Scalar principal series.
We use the translation principle to describe D ′ ( R n ) − λ,ν in termsof D ′ ( R n ) + λ,ν .Let ( τ, E ) be the defining representation of G = O( n + 1 ,
1) on E = C n +2 . Then i : E ′ = C ֒ → E, z ( z, , . . . , , z ) ⊤ is the maximal subspace on which N acts trivially, and E ′ | P = sgn ⊗ e α ⊗ . For the P H -quotient space we choose p : E ։ C , ( z , . . . , z n +2 ) z n +1 , then E ′′ | P H = ⊗ ⊗ . Now let us compute the ( E ′ ) ∗ ⊗ E ′′ -valued function p ◦ τ ( g ) ◦ i on N . For g = n x , x ∈ R n , we have p ◦ τ ( g ) ◦ i (1) ⊤ = p (1 + | x | , x, − | x | ) ⊤ = 2 x n . Hence, by Theorem 2.3, we get a linear map x n : D ′ ( R n ) ± λ,ν → D ′ ( R n ) ∓ λ +1 ,ν , K ( x ) x n K ( x ) . Applying this map to the meromorphic family of distributions K A , + λ,ν ( x ) we obtain anothermeromorphic family K A , − λ,ν ( x ) = x n · K A , + λ − ,ν = sgn( x n ) | x n | λ + ν − ( | x ′ | + x n ) − ν − n − , x ∈ R n . To normalize K A , − λ,ν ( x ) so that it becomes holomorphic in ( λ, ν ) ∈ C we first consider thekernel of the map x n : D ′ ( R n ) + λ,ν → D ′ ( R n ) − λ +1 ,ν . Lemma 3.4. { K ∈ D ′ ( R n ) + λ,ν : x n K = 0 } = C e K B , + λ,ν for λ + ν = − , C e K C , + λ,ν for λ − ν = − , { } else. NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 19
Note that for ( λ, ν ) = ( − ,
0) we have e K B , + λ,ν ( x ) = e K C , + λ,ν ( x ) = δ ( x ). Proof. If K ∈ D ′ ( R n ) + λ,ν such that x n K = 0 then clearly supp( K ) ⊆ R n − . Hence, K = M X m =0 u m ( x ′ ) δ ( m ) ( x n )for some distributions u m ∈ D ′ ( R n − ). Since x n δ ( m ) ( x n ) = − mδ ( m − ( x n ) we must have u m = 0 for m >
0, so that K ( x ) = u ( x ′ ) δ ( x n ). By the classification in Theorems 3.2 and 3.3the only possibilities for such K in the space D ′ ( R n ) + λ,ν are K ( x ) = e K B , + λ,ν ( x ) = | x ′ | − n − ν δ ( x n )Γ( − ν )with λ + ν = − (i.e. k = 0) and K ( x ) = e K C , + λ,ν ( x ) = δ ( x )with λ − ν = − (i.e. ℓ = 0). (cid:3) For the full classification of D ′ ( R n ) − λ,ν we also need to understand the kernel of the map x n : D ′ ( R n ) − λ,ν → D ′ ( R n ) + λ +1 ,ν . Lemma 3.5. { K ∈ D ′ ( R n ) − λ,ν : x n K = 0 } = { } . Proof.
As in the previous proof any K with x n K = 0 must be of the form K ( x ) = u ( x ′ ) δ ( x n ).Now, if K ∈ D ′ ( R n ) − λ,ν then by the invariance under m ∈ M H we have u ( − x ) = − u ( x ) andsince δ ( − x n ) = δ ( x n ) this implies u ( − x ′ ) = − u ( x ′ ) . On the other hand, invariance of K under O( n − ⊆ M H implies u ( mx ′ ) = u ( x ′ ) for all m ∈ O( n − m = − , hence u ( − x ′ ) = u ( x ′ ) . This shows that u = 0 and the proof is complete. (cid:3) To state the analogue of Theorem 3.2 for K A , − λ,ν ( x ) let \\ − = { ( λ, ν ) : λ + ν = − − k, k ∈ N } ,// − = { ( λ, ν ) : λ − ν = − − ℓ, ℓ ∈ N } ,L odd = { ( λ, ν ) = ( − ρ − i − , − ρ H − j ) : i, j ∈ N , i − j ∈ N } ⊆ // − . For ( λ, ν ) ∈ \\ − with λ + ν = − − k we further define e K B , − λ,ν ( x ) = 1Γ( ( λ − ν + )) ( | x ′ | + x n ) − ν − n − δ (2 k +1) ( x n )= k X i =0 ( − i (2 k + 1)!( ν + n − ) i i !(2 k − i + 1)!Γ( ( λ − ν + )) | x ′ | − n − ν − i δ (2 k − i +1) ( x n ) , and for ( λ, ν ) ∈ // − with λ − ν = − − ℓ we put e K C , − λ,ν ( x ) = ℓ X j =0 ℓ − j ( ν − ℓ ) ℓ − j j !(2 ℓ − j + 1)! (∆ j R n − ∂ ℓ − j +1 n δ )( x ) . Theorem 3.6.
The renormalized distribution e K A , − λ,ν ( x ) = K A , − λ,ν ( x )Γ( ( λ + ν + ))Γ( ( λ − ν + )) depends holomorphically on ( λ, ν ) ∈ C and vanishes only for ( λ, ν ) ∈ L odd . More precisely,(1) For ( λ, ν ) ∈ C − ( \\ − ∪ // − ) we have supp e K A , − λ,ν = R n .(2) For ( λ, ν ) ∈ \\ − − // − with λ + ν = − − k , k ∈ N , we have e K A , − λ,ν ( x ) = ( − k +1 k !(2 k + 1)! e K B , − λ,ν ( x ) . In particular, supp e K A , − λ,ν = R n − .(3) For ( λ, ν ) ∈ // − − L odd with λ − ν = − − ℓ , ℓ ∈ N , we have e K A , − λ,ν ( x ) = ( − ℓ +1 ℓ ! π n − ℓ Γ( ν + n − ) e K C , − λ,ν ( x ) . In particular, supp e K A , − λ,ν = { } .(4) For ( − ρ − i, − ρ H − j ) ∈ L odd , ℓ = i − j − , restricting the function ( λ, ν ) e K A , − λ,ν totwo different complex hyperplanes in C through ( − ρ − i, − ρ H − j ) and renormalizinggives two holomorphic families of distributions: e K C , − λ,ν ( x ) = ( − ℓ ℓ ℓ ! π n − Γ( ν + ρ H ) e K A , − λ,ν ( x ) , ( λ − ν = − − ℓ ) , ee K A , − λ,ν ( x ) = Γ( ( λ − ν + )) e K A , − λ,ν ( x ) , ( ν = − ρ H − j ) . Their special values at ( − ρ − i, − ρ H − j ) satisfy supp e K C , −− ρ − i, − ρ H − j = { } , supp ee K A , −− ρ − i, − ρ H − j = ( R n for n even, R n − for n odd,more precisely, for n odd and k = n + i + j − ee K A , − λ,ν ( x ) = ( − k +1 k !(2 k + 1)! ( | x ′ | + x n ) j δ (2 k +1) ( x n ) , Proof.
We first apply the map x n : D ′ ( R n ) + λ − ,ν → D ′ ( R n ) − λ,ν to the holomorphic family e K A , + λ − ,ν and obtain a holomorphic family of distributions x n e K A , + λ − ,ν ( x ) = K A , − λ,ν ( x )Γ( ( λ + ν − ))Γ( ( λ − ν − )) . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 21
By Theorem 3.2 and Lemma 3.4 we have x n e K A , + λ − ,ν = 0 if and only if λ + ν = or λ − ν = or ( λ − , ν ) ∈ L even . We can therefore renormalize the kernels e K A , − λ,ν ( x ) = x n e K A , + λ − ,ν ( x ) ( λ + ν − ) · ( λ − ν − ) = K A , − λ,ν ( x )Γ( ( λ + ν + ))Γ( ( λ − ν + )) . This shows that e K A , − λ,ν ( x ) depends holomorphically on ( λ, ν ) ∈ C . Next, we apply the map(3.1) x n : D ′ ( R n ) − λ,ν → D ′ ( R n ) + λ +1 ,ν to e K A , − λ,ν ( x ) and find x n e K A , − λ,ν = e K A , + λ +1 ,ν . By Lemma 3.5 the map (3.1) is injective, hence e K A , − λ,ν = 0 if and only if e K A , + λ +1 ,ν = 0, which isonly the case for ( λ + 1 , ν ) ∈ L even , i.e. ( λ, ν ) ∈ L odd .(1) Let ( λ, ν ) ∈ C − ( \\ − ∪ // − ). Then ( λ + 1 , ν ) ∈ C − ( \\ + ∪ // + ) and hencesupp( x n e K A , − λ,ν ) = supp( e K A , + λ +1 ,ν ) = R n . This implies supp( e K A , − λ,ν ) = R n .(2) Let ( λ, ν ) ∈ \\ − − // − with λ + ν = − − k , then it is easy to see, using x n δ ( m ) ( x n ) = − mδ ( m − ( x n ), that x n e K B , + λ − ,ν = 2( k + 1)( ν + k + 1) e K B , − λ,ν . Hence, e K A , − λ,ν = 1 ( λ + ν − ) · ( λ − ν − ) x n e K A , + λ − ,ν = 1( k + 1)( ν + k + 1) ( − k +1 ( k + 1)!(2 k + 2)! x n e K B , + λ − ,ν = ( − k +1 k !(2 k + 1)! e K B , − λ,ν . (3) For ( λ, ν ) ∈ // − − L odd the same method as in (2) applies to show (3). Here we usethat x n e K C , + λ − ,ν = − ν − ℓ − e K C , − λ,ν for λ − ν = − − ℓ .(4) Now let ( − ρ − i, − ρ H − j ) ∈ L odd . The first statement about e K C , − λ,ν ( x ) follows imme-diately from (3). For the second statement note that ( − ρ − ( i + 1) , − ρ H − j ) ∈ L even and we can use Theorem 3.2 (4). Restricting to ν = − ρ H − j we have ee K A , − λ,ν ( x ) = Γ( ( λ − ν + )) e K A , − λ,ν ( x )= Γ( ( λ − ν + )) x n e K A , + λ − ,ν ( x ) ( λ + ν − ) · ( λ − ν − ) = x n ee K A , + λ − ,ν ( x ) ( λ + ν − ) . The denominator is equal to ( λ + ν − ) = − ( n + i + j ) = − ( k +1) = 0, whence ee K A , − λ,ν depends holomorphically on λ ∈ C . Moreover, for even n we have supp ee K A , + λ − ,ν = R n so that supp( x n ee K A , + λ − ,ν ) = R n and hence supp ee K A , − λ,ν = R n . For odd n it follows from x n δ ( m ) ( x n ) = − mδ ( m − ( x n ) that ee K A , − λ,ν ( x ) = − k + 1 · ( − k +1 ( k + 1)!(2 k + 2)! x n ( | x ′ | + x n ) j δ (2 k +2) ( x n )= ( − k +1 k !(2 k + 1)! ( | x ′ | + x n ) j δ (2 k +1) ( x n ) . (cid:3) Remark 3.7.
The differential intertwining operators belonging to the distribution kernels e K C , + λ,ν are the even order conformally covariant differential operators found by Juhl [4], some-times also referred to as Juhl operators . Kobayashi–Speh showed that they are obtainedas residue families of the non-local intertwining operators with kernels e K A , + λ,ν . Theorem 3.6proves that also the odd order Juhl operators with integral kernels e K C , − λ,ν can be obtained inthis way. Further, the translation principle explains the following relation between even andodd order Juhl operators: x n e K C , + λ,ν = − λ + ν + ) e K C , − λ +1 ,ν and x n e K C , − λ,ν = − e K C , + λ +1 ,ν . Using the translation principle we can finally determine the space D ′ ( R n ) − λ,ν completely: Theorem 3.8.
We have (3.2) D ′ ( R n ) − λ,ν = ( C e K A , − λ,ν for ( λ, ν ) ∈ C − L odd , C ee K A , − λ,ν ⊕ C e K C , − λ,ν for ( λ, ν ) ∈ L odd .Proof. In the proof of Theorem 3.6 we showed that the map x n : D ′ ( R n ) − λ,ν → D ′ ( R n ) + λ +1 ,ν is injective and hence(3.3) dim D ′ ( R n ) − λ,ν ≤ dim D ′ ( R n ) + λ +1 ,ν . For ( λ, ν ) ∈ C − L odd we have 0 = e K A , − λ,ν ∈ D ′ ( R n ) − λ,ν , and hence the left hand side of (3.3) is ≥
1. For ( λ, ν ) ∈ L odd we have ee K A , − λ,ν , e K C , − λ,ν ∈ D ′ ( R n ) − λ,ν and these distributions are linearlyindependent since supp e K C , − λ,ν = { } ( supp ee K A , − λ,ν . This shows that the left hand side of (3.3) is ≥
2. Now note that ( λ, ν ) ∈ L odd if and onlyif ( λ + 1 , ν ) ∈ L even , and therefore (3.3) is actually an equality for all ( λ, ν ) ∈ C , thanks toTheorem 3.3. This finishes the proof. (cid:3) Example: ( e G, e H ) = (Pin( n + 1 , , Pin( n, e G = Pin( n + 1 ,
1) and e H = Pin( n,
1) induced from fundamental spin representations of f M ≃ Pin( n ) × Z / Z and f M H ≃ Pin( n − × Z / Z using the translation principle. NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 23
Parabolic subgroups and the symmetric pair.
We consider the two-fold covering q : e G = Pin( n + 1 , → O( n + 1 ,
1) = G, realized inside the Clifford algebra Cl( n + 1 ,
1) of R n +1 , . For details on the definition andproperties of the pin groups and Clifford algebras we refer the reader to Appendix A.1. Inwhat follows we will write e S = q − ( S ) for any subgroup S ⊆ G .We choose the same parabolic subgroup P = M AN ⊆ G as in Section 3.1, then f M =Pin( n ) · { , m = e e n +2 } , where the embedding of Pin( n ) into Pin( n + 1 ,
1) is the restrictionof the embedding of Clifford algebras Cl( n ) ֒ → Cl( n + 1 ,
1) induced by R n ֒ → R n +1 , , e i e i +1 (1 ≤ i ≤ n ) . We note that m = e e n +2 commutes with Pin( n ) and hence f M ≃ Pin( n ) × Z / Z . Both A and N split in the cover, so that e A ≃ A × Z / Z and e N ≃ N × Z / Z , and we identify A and N with e A , e N ⊆ e G . Then e P = f M AN is the Langlands decomposition of the parabolicsubgroup e P of e G .For the Weyl group element w we choose the representative(4.1) ˜ w = ± e n +2 with the sign yet to be determined. Since q ( ˜ w ) = diag(1 , . . . , , − w corre-sponds to the representative chosen in Section 3.1 for the group G . Hence, by Lemma 3.1 wefind q ( m ( ˜ w − n − x ) − ) = n − xx ⊤ | x | , x ∈ R n − { } . The matrix n − xx ⊤ | x | ∈ O( n ) is the orthogonal reflection in R n at the hyperplane orthogonalto x , and the corresponding elements in the cover Pin( n ) ⊆ Cl( n ) are ± x | x | . Therefore, wemay choose the sign in (4.1) so that under the identification f M ≃ Pin( n ) × Z / Z we have m ( ˜ w − n − x ) − = − x | x | ∈ Pin( n ) ⊆ f M .
Further we note that˜ w m ˜ w − = α ( m ) = det( m ) m ( m ∈ Pin( n )) , ˜ w m ˜ w − = − m , where α denotes the canonical automorphism of the Clifford algebra Cl( n ) and det : Pin( n ) → O( n ) → {± } the determinant character of Pin( n ).The symmetric pair ( G, H ) = (O( n + 1 , , O( n, e G, e H ) = (Pin( n + 1 , , Pin( n, e H = e G ∩ Cl( n, , where the embedding of Cl( n,
1) into Cl( n + 1 ,
1) is induced by the embedding R n, ֒ → R n +1 , , x ( x , . . . , x n , , x n +1 ) . Further, e P H = e P ∩ e H = f M H AN H is a parabolic subgroup of e H with f M H = Pin( n − ·{ , m } . Principal series representations and symmetry breaking operators.
For thefundamental spin representations of Pin( n ) we use the notation introduced in Appendix A.2.The group Pin( n ) has for even n one fundamental spin representation, and for odd n twofundamental spin representations. For n even we have ζ n ⊗ det ≃ ζ n for the fundamental spinrepresentation ζ n , and for n odd and ζ n any fundamental spin representation of Pin( n ) therepresentations ζ n and ζ n ⊗ det are non-equivalent.Denote by sgn : { , m } → {± } the non-trivial character of the two-element group { , m } . Then for δ ∈ Z / Z and a fundamental spin representation ζ n of Pin( n ) we con-sider the representation ζ n ⊗ sgn δ of f M = Pin( n ) × { , m } . Note that(4.2) ˜ w ( ζ n ⊗ sgn δ ) = [ ζ n ⊗ det] ⊗ sgn − δ . For ( ζ n , S n ) a fundamental spin representation of Pin( n ), δ ∈ Z / Z and λ ∈ a ∗ C ≃ C weform the principal series representations /π λ,δ = Ind e G e P (( ζ n ⊗ sgn δ ) ⊗ e λ ⊗ ) . Similarly, for ( ζ n − , S n − ) a fundamental spin representation of Pin( n − δ ∈ Z / Z and ν ∈ a ∗ H, C ≃ C we form principal series representations of e H : /τ ν,ε = Ind e H e P H (( ζ n − ⊗ sgn ε ) ⊗ e ν ⊗ ) . The main result of this section is the construction of symmetry breaking operators(4.3) A ∈ Hom e H ( /π λ,δ | e H , /τ ν,ε ) . In Section 5 we then show that this construction actually gives a full classification of symmetrybreaking operators between spinor-valued principal series.By Theorem 1.2 every intertwining operator in (4.3) is uniquely determined by its dis-tribution kernel K ∈ D ′ ( R n ; Hom C ( S n , S n − )). As explained in Section 3.2, the space ofdistribution kernels describing intertwining operators in (4.3) only depends on ( λ, ν ) ∈ C and the parity of δ + ε . We therefore denote by D ′ ( R n ; Hom C ( S n , S n − )) ± λ,ν ⊆ D ′ ( R n ; Hom C ( S n , S n − ))the corresponding space of distribution kernels of intertwining operators, where the sign +describes kernels for δ + ε ≡ − describes kernels for δ + ε ≡ Construction of symmetry breaking operators.
We extend the representation( ζ n , S n ) of Pin( n ) to a representation of the Clifford algebra Cl( n ; C ) and write ( ζ, S ) =( ζ n , S n ) for short (see Appendix A.2. Consider the representation τ ′ = [ ζ ⊗ det] ⊗ of f M ≃ Pin( n ) × Z / Z . By Lemma 2.2 there exists a representation ( τ, E ) of e G = Pin( n + 1 , E ′ ⊆ E invariant under f M AN such that E ′ ≃ τ ′ ⊗ e µ ′ ⊗ or E ′ ≃ ˜ w τ ′ ⊗ e µ ′ ⊗ for some µ ′ ∈ a ∗ . By possibly replacing ( τ, E ) by its twist ( τ ◦ θ, E ) by theCartan involution θ we may assume that E ′ ≃ τ ′ ⊗ e µ ′ ⊗ . It is easy to see that µ ′ = − α .By (4.2) we have w ( τ ′ ⊗ e µ ′ ⊗ ) ≃ ( ζ ⊗ sgn) ⊗ e − µ ′ ⊗ . Further, using Lemma 3.1 we have the following expression for the translation kernel: τ ′ ( m ( ˜ w − n − x ) − ) · a ( ˜ w − n − x ) − µ ′ = [ ζ ⊗ det]( − x | x | ) · | x | − µ ′ = ζ ( x ) . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 25
Then Corollary 2.7 gives linear mapsHom H ( π λ,δ | H , τ ν,ε ) → Hom e H ( /π λ + , − δ , Ind e H e P H (([ ζ ⊗ det] ⊗ sgn ε ) ⊗ e ν − ⊗ ))which are on the level of integral kernels given by(4.4) D ′ ( R n ) → D ′ ( R n ; End C ( S )) , K ( x ) ζ ( x ) · K ( x ) . To obtain intertwining operators into /τ ν,ε note that ζ n − occurs in the restriction [ ζ n ⊗ det] | Pin( n − with multiplicity one, and fix a projection P : [ ζ n ⊗ det] | Pin( n − → ζ n − . Thenwe have a linear map Hom H ( π λ,δ | H , τ ν,ε ) → Hom e H ( /π λ + , − δ , /τ ν − ,ε )which is on the level of integral kernels given by D ′ ( R n ) ± λ,ν → D ′ ( R n ; Hom C ( S n , S n − )) ∓ λ + ,ν − , K ( x ) ( P ζ ( x )) · K ( x ) . By Theorems 3.3 and 3.8 all symmetry breaking operators in Hom H ( π λ,δ | H , τ ν,ε ) are ob-tained from the meromorphic family of kernels K A , ± λ,ν ∈ D ′ ( R n ) ± λ,ν . We therefore consider theHom C ( S n , S n − )-valued kernels P /K A , ± λ,ν ( x ) = ( P ζ ( x )) · K A , ∓ λ − ,ν + ( x ) . For n odd the restriction of the Pin( n )-representation ζ n ⊗ det to Pin( n −
1) is isomorphicto ζ n − , so the map P is an isomorphism. Therefore, we can as well study the End C ( S )-valueddistributions /K A , ± λ,ν ( x ) = ζ ( x ) · K A , ∓ λ − ,ν + ( x ) . For n even the restriction of the Pin( n )-representation ζ n ⊗ det to Pin( n −
1) is isomorphicto the direct sum of ζ n − and ζ n − ⊗ det. The next result shows that the poles of P /K A , ± λ,ν ∈D ′ ( R n ; Hom C ( S n , S n − )) and /K A , ± λ,ν ∈ D ′ ( R n ; End C ( S )) agree and that the residues of P /K A , ± λ,ν agree with the composition of the residues of /K A , ± λ,ν with P . Note that /K A , ± λ,ν ( x ) = n X i =1 (cid:16) x i · K A , ∓ λ − ,ν + ( x ) (cid:17) e i , where e i = ζ n ( e i ) ∈ End C ( S n ). Proposition 4.1.
Assume n is even and let u ∈ D ′ ( R n ; End C ( S n )) be a distribution of theform u ( x ) = n X i =1 u i ( x ) e i with u i ∈ D ′ ( R n ) . Then supp u = supp P u , in particular
P u = 0 if and only if u = 0 .Proof. We have (
P u )( x ) = n X i =1 u i ( x ) · ( P ◦ e i ) . We claim that the operators P ◦ e , . . . , P ◦ e n are linearly independent, then the statementfollows. Decompose ζ n ⊗ det into irreducible Pin( n − S n = S n ⊕ S n , where P : S n → S n − is an isomorphism and S n = ker P . Now note that for 1 ≤ i ≤ n − P ◦ e i = P ◦ ζ n ( e i ) = − P ◦ [ ζ n ⊗ det]( e i ) = − ζ n − ( e i ) ◦ P and therefore P ◦ e i vanishes on S n . On the other hand, e n = ζ n ( e n ) maps S n to S n , so that P ◦ e n vanishes on S n . Now assume that n X i =1 λ i P ◦ e i = 0 . Restricting to S n this implies n − X i =1 λ i ζ n − ( e i ) = 0and hence λ = . . . = λ n − = 0 since ζ n − ( e ) , . . . , ζ n − ( e n − ) are linearly independent. Thenalso λ n = 0 and the proof is complete. (cid:3) To find the right normalization making /K A , ± λ,ν ( x ) holomorphically dependent on ( λ, ν ) ∈ C we first study the kernel in D ′ ( R n ) ± λ,ν of the map (4.4). Note that ζ ( x ) = P ni =1 x i e i , andthe operators e , . . . , e n ∈ End C ( S ) are linearly independent. Hence, for any distribution u ∈ D ′ ( R n ) we have ζ · u = 0 ⇔ x i u = 0 (1 ≤ i ≤ n )and(4.5) supp( ζ · u ) = n [ i =1 supp( x i u ) . Lemma 4.2.
We have { K ∈ D ′ ( R n ) + λ,ν : ζ · K = 0 } = ( C e K C , + λ,ν = C δ for λ − ν = − , { } else, { K ∈ D ′ ( R n ) − λ,ν : ζ · K = 0 } = { } . Proof.
It is easy to see that multiples of K = δ are the only distributions with x i K = 0 forall 1 ≤ i ≤ n . Then the claim follows from Theorems 3.2, 3.3, 3.6 and 3.8. (cid:3) To state the analogues of Theorems 3.2 and 3.6 for /K A , ± λ,ν ( x ) we write / D R n − and / D n forthe operators / D R n − = n − X i =1 e i ∂∂x i , / D n = e n ∂∂x n . The distributions /K A , + λ,ν ( x ) . Let \\ + = { ( λ, ν ) : λ + ν = − − k, k ∈ N } ,// + = { ( λ, ν ) : λ − ν = − − ℓ, k ∈ N } ,/L even = { ( − ρ − − i, − ρ H − − j ) : i, j ∈ N , i − j ∈ N } ⊆ // + . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 27
For ( λ, ν ) ∈ \\ + with λ + ν = − − k we further define e /K B , + λ,ν ( x ) = 1Γ( ( λ − ν + )) (cid:16) ζ ( x ′ )( | x ′ | + x n ) − ν − n δ (2 k +1) ( x n ) − (2 k + 1)( | x ′ | + x n ) − ν − n / D n δ (2 k − ( x n ) (cid:17) , and for ( λ, ν ) ∈ // + with λ − ν = − − ℓ we put e /K C , + λ,ν ( x ) = ℓ − X j =0 ℓ − j − ( ν + − ℓ ) ℓ − j − j !(2 ℓ − j − j R n − ∂ ℓ − j − n / D R n − δ )( x )+ ℓ X j =0 ℓ − j ( ν + − ℓ ) ℓ − j j !(2 ℓ − j )! (∆ j R n − ∂ ℓ − j − n / D n δ )( x ) . Note that both e /K B , + λ,ν and e /K C , + λ,ν depend holomorphically on λ ∈ C (or ν ∈ C ). Theorem 4.3.
The renormalized distribution e /K A , + λ,ν ( x ) = /K A , + λ,ν ( x )Γ( ( λ + ν + ))Γ( ( λ − ν + )) depends holomorphically on ( λ, ν ) ∈ C and vanishes only for ( λ, ν ) ∈ /L even . More precisely,(1) For ( λ, ν ) ∈ C − ( \\ + ∪ // + ) we have supp e /K A , + λ,ν = R n .(2) For ( λ, ν ) ∈ \\ + − // + with λ + ν = − − k , k ∈ N , we have e /K A , + λ,ν ( x ) = ( − k +1 k !(2 k + 1)! e /K B , + λ,ν ( x ) . In particular, supp e /K A , + λ,ν = R n − .(3) For ( λ, ν ) ∈ // + − /L even with λ − ν = − − ℓ , ℓ ∈ N , we have e /K A , + λ,ν ( x ) = ( − ℓ ℓ ! π n − ℓ Γ( ν + n ) e /K C , + λ,ν ( x ) . In particular, supp e /K A , + λ,ν = { } .(4) For ( − ρ − − i, − ρ H − − j ) ∈ /L even , ℓ = i − j , restricting the function ( λ, ν ) e /K A , + λ,ν to two different complex hyperplanes in C through ( − ρ − − i, − ρ H − − j ) andrenormalizing gives two holomorphic families of distributions: e /K C , + λ,ν ( x ) = ( − ℓ ℓ ℓ ! π n − Γ( ν + ρ H + ) e /K A , + λ,ν ( x ) , ( λ − ν = − − ( i − j )) , ee /K A , + λ,ν ( x ) = Γ( ( λ − ν + )) e /K A , + λ,ν ( x ) , ( ν = − ρ H − − j ) . Their special values at ( λ, ν ) = ( − ρ − − i, − ρ H − − j ) satisfy supp e /K C , + − ρ − − i, − ρ H − − j = { } , supp ee /K A , + − ρ − − i, − ρ H − − j = ( R n for n even, R n − for n odd,more precisely, for n odd and k = n + i + j − ee /K A , + − ρ − − i, − ρ H − − j = ( − k +1 k !(2 k + 1)! ( | x ′ | + x n ) j (cid:16) ζ ( x ′ ) δ (2 k +1) ( x n ) − (2 k + 1) / D n δ (2 k − ( x n ) (cid:17) . Proof.
By Theorem 3.6 the distribution e /K A , + λ,ν ( x ) = ζ ( x ) · e K A , − λ − ,ν + ( x ) = ζ ( x ) · K A , − λ − ,ν + ( x )Γ( ( λ + ν + ))Γ( ( λ − ν + ))depends holomorphically on ( λ, ν ) ∈ C , and by Lemma 4.2 it vanishes if and only if ( λ − , ν + ) ∈ L odd , i.e. ( λ, ν ) ∈ /L even . The rest is similar to the proof of Theorem 4.4 whichwe carry out in detail, some arguments are even easier since in this case we do not need torenormalize the kernel and we have( λ, ν ) ∈ \\ + ⇔ ( λ − , ν + ) ∈ \\ − , ( λ, ν ) ∈ // + ⇔ ( λ − , ν + ) ∈ // − . (cid:3) The distributions /K A , − λ,ν ( x ) . Let \\ − = { ( λ, ν ) : λ + ν = − − k, k ∈ N } ,// − = { ( λ, ν ) : λ − ν = − − ℓ, k ∈ N } ,/L odd = { ( − ρ − − i, − ρ H − − j ) : i, j ∈ N , i − j ∈ N + 1 } ⊆ // − . For ( λ, ν ) ∈ \\ − with λ + ν = − − k we further define e /K B , − λ,ν ( x ) = 1Γ( ( λ − ν + )) (cid:16) ζ ( x ′ )( | x ′ | + x n ) − ν − n δ (2 k ) ( x n ) − k ( | x ′ | + x n ) − ν − n / D n δ (2 k − ( x n ) (cid:17) , and for ( λ, ν ) ∈ // − with λ − ν = − − ℓ we put e /K C , − λ,ν ( x ) = ℓ X j =0 ℓ − j ( ν − ℓ − ) ℓ − j j !(2 ℓ − j )! (∆ j R n − ∂ ℓ − jn / D R n − δ )( x )+ ℓ X j =0 ℓ − j +1 ( ν − ℓ − ) ℓ − j +1 j !(2 ℓ − j + 1)! (∆ j R n − ∂ ℓ − jn / D n δ )( x ) . Note that both e /K B , − λ,ν and e /K C , − λ,ν depend holomorphically on λ ∈ C (or ν ∈ C ). Theorem 4.4.
The renormalized distribution e /K A , − λ,ν ( x ) = /K A , − λ,ν ( x )Γ( ( λ + ν + ))Γ( ( λ − ν + )) NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 29 depends holomorphically on ( λ, ν ) ∈ C and vanishes only for ( λ, ν ) ∈ /L odd . More precisely,(1) For ( λ, ν ) ∈ C − ( \\ − ∪ // − ) we have supp e /K A , − λ,ν = R n .(2) For ( λ, ν ) ∈ \\ − − // − with λ + ν = − − k , k ∈ N , we have e /K A , − λ,ν ( x ) = ( − k k !(2 k )! e /K B , − λ,ν ( x ) . In particular, supp e /K A , − λ,ν = R n − .(3) For ( λ, ν ) ∈ // − − /L odd with λ − ν = − − ℓ , ℓ ∈ N , we have e /K A , − λ,ν ( x ) = ( − ℓ +1 ℓ ! π n − ℓ +1 Γ( ν + n ) e /K C , − λ,ν ( x ) . In particular, supp e /K A , − λ,ν = { } .(4) For ( − ρ − − i, − ρ H − − j ) ∈ /L odd , ℓ = i − j − , restricting the function ( λ, ν ) e /K A , − λ,ν to two different complex hyperplanes in C through ( − ρ − − i, − ρ H − − j ) and renormalizing gives two holomorphic families of distributions: e /K C , − λ,ν ( x ) = ( − ℓ +1 ℓ +1 ℓ ! π n − Γ( ν + ρ H + ) e /K A , − λ,ν ( x ) , ( λ − ν = − − ( i − j )) , ee /K A , − λ,ν ( x ) = Γ( ( λ − ν + )) e /K A , − λ,ν ( x ) , ( ν = − ρ H − − j ) . Their special values at ( λ, ν ) = ( − ρ − − i, − ρ H − − j ) satisfy supp e /K C , −− ρ − − i, − ρ H − − j = { } , supp ee /K A , −− ρ − − i, − ρ H − − j = ( R n for n even, R n − for n odd,more precisely, for n odd and k = n + i + j : ee /K A , −− ρ − − i, − ρ H − − j = ( − k k !(2 k )! ( | x ′ | + x n ) j (cid:16) ζ ( x ′ ) δ (2 k ) ( x n ) − k / D n δ (2 k − ( x n ) (cid:17) . Proof.
By Theorem 3.2 the distribution ζ ( x ) · e K A , + λ − ,ν + ( x ) = ζ ( x ) · K A , + λ − ,ν + ( x )Γ( ( λ + ν + ))Γ( ( λ − ν − ))depends holomorphically on ( λ, ν ) ∈ C , and by Lemma 4.2 it vanishes if and only if λ − ν = or ( λ − , ν + ) ∈ L even . Renormalizing shows that e /K A , − λ,ν ( x ) = ζ ( x ) · e K A , + λ − ,ν + ( x ) ( λ − ν − ) = ζ ( x ) · K A , + λ − ,ν + ( x )Γ( ( λ + ν + ))Γ( ( λ − ν + )) depends holomorphically on ( λ, ν ) ∈ C . To prove the remaining statements, note that ζ ( x ) · e /K A , − λ,ν ( x ) = ζ ( x ) · e K A , + λ − ,ν + ( x ) ( λ − ν − ) = − | x | · e K A , + λ − ,ν + ( x ) ( λ − ν − ) · id S = − e K A , + λ + ,ν − ( x ) · id S and hence(4.6) supp e K A , + λ + ,ν − ⊆ supp e /K A , − λ,ν . (1) Let ( λ, ν ) ∈ C − ( \\ − ∪ // − ), then ( λ + , ν − ) ∈ C − ( \\ + ∪ // + ) and hencesupp e K A , + λ + ,ν + = R n . Now (4.6) implies supp e /K A , − λ,ν = R n .(2) Let ( λ, ν ) ∈ \\ − − // − with λ + ν = − − k , then ( λ − , ν + ) ∈ \\ + and byTheorem 3.2 (2) and using x n δ (2 k ) ( x n ) = − kδ (2 k − ( x n ) we have e /K A , − λ,ν ( x ) = ζ ( x ) · e K A , + λ − ,ν + ( x ) ( λ − ν − ) = ( − k k !(2 k )! · ζ ( x ) · e K B , + λ − ,ν + ( λ − ν − )= ( − k k !(2 k )! e /K B , − λ,ν ( x ) . (3) Let ( λ, ν ) ∈ // − − /L odd with λ − ν = − − ℓ , then ( λ − , ν + ) ∈ // + with( λ − ) − ( ν + ) = − − ℓ + 1) and by Theorem 3.2 (3) and using [∆ j R n − , x i ] =2 j ∆ j − ∂∂x i we have e /K A , − λ,ν ( x ) = ζ ( x ) · e K A , + λ − ,ν + ( x ) ( λ − ν − ) = ( − ℓ +1 ( ℓ + 1)! π n − ℓ +2 Γ( ν + n ) · ζ ( x ) · e K C , + λ − ,ν + ( − ℓ − − ℓ +1 ℓ ! π n − ℓ +1 Γ( ν + n ) e /K C , − λ,ν ( x ) . (4) Now let ( − ρ − − i, − ρ H − − j ) ∈ /L odd . The first statement about e /K C , − λ,ν ( x ) followsimmediately from (3). For the second statement note that ( − ρ − ( i + 1) , − ρ H − j ) ∈ L even and we can use Theorem 3.2 (4). Restricting to ν = − ρ H − − j we have ee /K A , − λ,ν ( x ) = Γ( ( λ − ν + )) e /K A , − λ,ν = Γ( ( λ − ν + )) ζ ( x ) · e K A , + λ − ,ν + ( x ) ( λ − ν − ) = ζ ( x ) · ee K A , + λ − ,ν + ( x )which clearly depends holomorphically on λ ∈ C . Moreover, supp ee K A , + − ρ − ( i +1) , − ρ H − j = R n resp. R n − so that supp x i ee K A , + − ρ − ( i +1) , − ρ H − j = R n resp. R n − for 1 ≤ i ≤ n −
1, andhence supp ee /K A , −− ρ − − i, − ρ H − − j = R n resp. R n − by (4.5). For odd n and 2 k = n + i + j NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 31 we further have by Theorem 3.2 (4) ee /K A , −− ρ − − i, − ρ H − − j ( x ) = ζ ( x ) · ee K A , + − ρ − ( i +1) , − ρ H − j ( x )= ( − k k !(2 k )! ζ ( x ) · ( | x ′ | + x n ) j δ (2 k ) ( x n )= ( − k k !(2 k )! ( | x ′ | + x n ) j (cid:16) ζ ( x ′ ) δ (2 k ) ( x n ) − k / D n δ (2 k − (cid:17) . (cid:3) Remark 4.5.
The identity ζ ( x ) · e /K A , − λ,ν ( x ) = − e K A , + λ + ,ν − ( x ) · id S that was used in the previous proof can be explained by again applying the translation prin-ciple, now with the dual representation ( τ ′ ) ∗ , and then projecting onto the trivial constituentin τ ′ ⊗ ( τ ′ ) ∗ = End C ( τ ′ ) which is spanned by id S . Remark 4.6.
The intertwining differential operators C ∞ ( R n ; S ) → C ∞ ( R n − , S ) with in-tegral kernel e /K C , ± λ,ν ( x ) have been obtained before [8, Theorem 5.7] and it was conjecturedthat these operators exhaust the space of all intertwining differential operators. Theorem 5.5confirms this conjecture.5. The compact picture of symmetry breaking operators between spinors
We use the method developed in [10] to show that the symmetry breaking operators foundin Section 4 between spinor-valued principal series span the space of all symmetry breakingoperators. The notation will be as in the previous section.5.1.
Reduction to the pair ( e G , e H ) . To simplify computations we first reduce the studyof symmetry breaking operators for the pair ( e G, e H ) = (Pin( n + 1 , , Pin( n, e G , e H ) = (Pin( n + 1 , , Pin( n, ), wherePin( p, q ) = Pin( p, q ) ++ ∪ Pin( p, q ) − + . We refer the reader to Appendix A.1 for the definition of Pin( p, q ) ±± . Note that e P = e P ∩ e G = f M AN is a parabolic subgroup of e G with f M = Pin( n ). Similarly, e P H, = e P H ∩ e H = f M H, AN H is a parabolic subgroup of e H with f M H, = Pin( n − e G / e P ≃ e G/ e P and e H / e P H, ≃ e H/ e P H , and hence /π λ,δ | e G ≃ /π λ = Ind e G e P ( ζ n ⊗ e λ ⊗ ) ,/τ ν,ε | e H ≃ /τ ν = Ind e H e P H, ( ζ n − ⊗ e ν ⊗ ) . Note that the restrictions are independent of δ and ε . Lemma 5.1.
For λ, ν ∈ C and δ, ε ∈ Z / Z we have dim Hom e H ( /π λ,δ , /τ ν,ε ) + dim Hom e H ( /π λ, − δ , /τ ν,ε ) = dim Hom e H ( /π λ , /τ ν ) . Proof.
The proof of the two identities is analogous to [7, Theorem 2.10] and uses e G/ e G ≃ f M / f M ≃ Z / Z and e H/ e H ≃ f M H / f M H, ≃ Z / Z . (cid:3) In the remaining part of this section we determine dim Hom e H ( /π λ , /τ ν ) using the techniquedeveloped in [10].5.2. Harish-Chandra modules.
Let θ denote the Cartan involution of G = O( n + 1 , , . . . , , − θ lifts to a Cartan involutionof e G which restricts to Cartan involutions of e G , e H and e H . This gives the following maximalcompact subgroups: e K = e G θ = Pin( n + 1) , e K H, = e H θ = Pin( n ) . The embedding e K H, ⊆ e K is given by the embedding Cl( n ) ⊆ Cl( n + 1) induced from thestandard embedding R n ⊆ R n +1 , x ′ ( x ′ , /π λ and /τ ν which we denote by ( /π λ ) HC and ( /τ ν ) HC .As vector space ( /π λ ) HC is the space of all e K -finite vectors in /π λ , or equivalently the sum ofall irreducible e K -subrepresentations of /π λ . Hence, ( /π λ ) HC clearly carries a e K -action. It isfurther invariant under the action of the Lie algebra g of e G and hence carries the structureof a ( g , e K )-module.Since ( /π λ ) HC is dense in /π λ , we obtain an injective mapHom e H ( /π λ | e H , /τ ν ) ֒ → Hom ( h , e K H, ) (( /π λ ) HC | ( h , e K H, ) , ( /τ ν ) HC ) . Using the method in [10] we determine in this section the dimension of the space of inter-twining operators between the Harish-Chandra modules. This gives an upper bound for thedimension of the space of intertwining operators between the representations /π λ and /τ ν . Fromthe explicit construction of intertwining operators in Section 4 we also have a lower bound,and it turns out that these bounds agree, so that the injective map in fact is a bijection.5.3. K -types. We study the representations /π λ and /τ ν in the compact picture, i.e. onsections of spin bundles over e K / f M ≃ S n and e K H, / f M H, ≃ S n − . More precisely, /π λ | e K = C ∞ ( e K × f M ζ n ) . By Frobenius Reciprocity and using the classical branching rules for spin groups we easilyobtain ( /π λ ) HC | e K ≃ ∞ M i =0 ( ζ + n +1 ,i ⊕ ζ − n +1 ,i ) for n even, ∞ M i =0 ζ n +1 ,i for n odd,where ζ ( ± ) n +1 ,i denote the higher spin representations of Pin( n + 1) (see Appendix A.3). ByAppendix A.4 ζ ± n +1 ,i | e K H, ≃ i M j =0 ζ n,j ( n even), ζ n +1 ,i | e K H, ≃ i M j =0 ( ζ + n,j ⊕ ζ − n,j ) ( n odd). NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 33
We use the notation α = ( i, ± ) and α ′ = j for even n , and α = i , α ′ = ( j, ± ) for odd n , andwrite α ′ ⊂ α if j ≤ i . Denote by E ( α ) the subspace of C ∞ ( e K × f M ζ n ) which is isomorphic tothe e K -representation ζ ± n +1 ,i if α = ( i, ± ) and ζ n +1 ,i if α = i , and similar for the correspondingsubspace E ′ ( α ′ ) of C ∞ ( e K H, × f M H, ζ n − ). In this notation( /π λ ) HC | e K = M α E ( α ) , ( /τ ν ) HC | e K H, = M α ′ E ′ ( α ′ ) . Further, E ( α ) | e K H, = M α ′ ⊂ α E ( α, α ′ ) , where E ( α, α ′ ) ≃ E ′ ( α ′ ). We fix a non-zero e K H, -intertwining operator R α,α ′ : E ( α, α ′ ) ∼ → E ′ ( α ′ )for each pair ( α, α ′ ) with α ′ ⊂ α .5.4. Scalar identities for symmetry breaking operators.
Write g = k ⊕ s for theeigenspace decomposition of g under θ and h = k H ⊕ s H for the one of h . We identify s ≃ R n +1 via R n +1 ∼ −→ s , Y (cid:18) n +1 YY ⊤ (cid:19) . Consider the map ω : s C → C ∞ ( e K / f M ) = C ∞ ( S n ) given by ω ( Y )( x ) = Y ⊤ x, x ∈ S n , Y ∈ C n +1 . Multiplication by ω ( Y ) defines an operator M ( ω ( Y )) : C ∞ ( e K × f M ζ n ) → C ∞ ( e K × f M ζ n ).From the weights of s C it is easy to see that M ( ω ( Y )) maps E ( α ) to E ( β ) if and only if α = ( i, ± ) and β ∈ { ( i + 1 , ± ) , ( i, ∓ ) , ( i − , ± ) } ( n even), α = i and β ∈ { i + 1 , i, i − } ( n odd).Write α ↔ β if this is the case.We use the same notation for α ′ and β ′ and write M ( ω ′ ( Y )) for multiplication by ω ′ ( Y )( x ) = Y ⊤ x , x ∈ S n − , Y ∈ C n . Further write ( α, α ′ ) ↔ ( β, β ′ ) if α ′ ⊂ α , β ′ ⊂ β and α ↔ β , α ′ ↔ β ′ .For ( α, α ′ ) ↔ ( β, β ′ ) define ω β,β ′ α,α ′ ( Y ) = proj E ( β,β ′ ) ◦ M ( ω ( Y )) | E ( α,α ′ ) ,ω β ′ α ′ ( Y ) = proj E ′ ( β ′ ) ◦ M ( ω ′ ( Y )) | E ′ ( α ′ ) . Then both R β,β ′ ◦ ω β,β ′ α,α ′ and ω β ′ α ′ ◦ R α,α ′ are e K H, -intertwining operators s H, C ⊗E ( α, α ′ ) → E ′ ( β ′ ).Since E ′ ( β ′ ) occurs in s H, C ⊗ E ( α, α ′ ) with multiplicity at most one, we have(5.1) R β,β ′ ◦ ω β,β ′ α,α ′ ( Y ) = λ β,β ′ α,α ′ · ω β ′ α ′ ( Y ) ◦ R α,α ′ for some constant λ β,β ′ α,α ′ ∈ C .Now, every e K H, -intertwining operator T : ( /π λ ) HC → ( /τ ν ) HC is uniquely determined by itsrestriction to the subspaces E ( α, α ′ ) on which it is given by(5.2) T | E ( α,α ′ ) = t α,α ′ · R α,α ′ for some scalars t α,α ′ ∈ C . In [10] it is proven that T is ( h , e K H, )-intertwining if and only iffor all ( α, α ′ ) and α ′ ↔ β ′ we have(5.3) (2 ν + σ ′ β ′ − σ ′ α ′ ) t α,α ′ = X β ( α,α ′ ) ↔ ( β,β ′ ) λ β,β ′ α,α ′ (2 λ + σ β − σ α ) t β,β ′ . Here the constants σ α and σ ′ α ′ are essentially the eigenvalues of the Casimir element on the K -types E ( α ) and E ′ ( α ′ ). From [1, Section 3.a] it follows that σ β − σ α = i + n + 1 for ( α, β ) = (( i, ± ) , ( i + 1 , ± )) resp. ( i, i + 1),0 for ( α, β ) = (( i, ± ) , ( i, ∓ )) resp. ( i, i ), − i − n + 1 for ( α, β ) = (( i, ± ) , ( i − , ± )) resp. ( i, i − σ ′ β ′ − σ ′ α ′ . Hence, the only missing constants in (5.3) are the numbers λ β,β ′ α,α ′ which we now compute after choosing explicit operators R α,α ′ .5.5. Explicit embeddings of K -types. We realize the K -types ζ ( ± ) n +1 ,i explicitly inside /π λ .Assume first that n is even, then the f M -representation ζ n is equivalent to the restrictionof the e K -representation ζ n +1 = ζ + n +1 to f M . Hence /π λ | e K = Ind e K f M ( ζ n +1 | f M ) ≃ Ind e K f M ( ) ⊗ ζ n +1 = C ∞ ( e K / f M , S n +1 ) . The underlying Harish-Chandra module is in this picture given by restrictions of S n +1 -valuedpolynomials on R n +1 to the sphere e K / f M ≃ S n ⊆ R n +1 . Denote by M i ( R n +1 , S n +1 ) thespace of monogenic polynomials of degree i (see Appendix A.3 for details). By the Fischerdecomposition (A.2) we have C ∞ ( S n , S n +1 ) HC = ∞ M i =0 (cid:16) M i ( R n +1 , S n +1 ) ⊕ x M i ( R n +1 , S n +1 ) (cid:17) , where we identify a homogeneous polynomial φ : R n +1 → S n +1 with its restriction to theunit sphere S n ⊆ R n +1 . Then M i ( R n +1 , S n +1 ) carries the representation ζ + n +1 ,i = ζ n +1 ,i and x M i ( R n +1 , S n +1 ) carries the representation ζ − n +1 ,i = ζ n +1 ,i ⊗ det. In the notation ofSection 5.3 this means E ( α ) = ( M i ( R n +1 , S n +1 ) for α = ( i, +), x M i ( R n +1 , S n +1 ) for α = ( i, − ).Now let n be odd, then the restriction of the e K -representation ζ n +1 to f M decomposesinto ζ + n ⊕ ζ − n , where ζ + n = ζ n and ζ − n = ζ n ⊗ det. Hence /π λ | e K = Ind e K f M ( ζ + n ) ⊆ Ind e K f M ( ζ n +1 | f M ) ≃ C ∞ ( e K / f M , S n +1 ) . Again, the underlying Harish-Chandra module is given by C ∞ ( S n , S n +1 ) HC = ∞ M i =0 (cid:16) M i ( R n +1 , S n +1 ) ⊕ x M i ( R n +1 , S n +1 ) (cid:17) NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 35 but in this case M i ( R n +1 , S n +1 ) and x M i ( R n +1 , S n +1 ) are isomorphic e K -representations,more precisely they both are equivalent to ζ n +1 ,i . A short calculation using Lemma A.2shows that the K -types belonging to /π λ are given by M i ( R n +1 , S n +1 ) → M i ( R n +1 , S n +1 ) ⊕ x M i ( R n +1 , S n +1 ) , φ φ + xφ ∨ , where φ ∨ ( x ) = γ ( φ ( x )), γ : S n +1 → S n +1 being the map defined in (A.1) that intertwines ζ n +1 with ζ n +1 ⊗ det. In the notation of Section 5.3 this reads E ( α ) = { φ + xφ ∨ : φ ∈ M i ( R n +1 , S n +1 ) } for α = i .Using the same identifications for the e K H, -types E ′ ( α ′ ) we now fix for each pair α ′ ⊂ α a e K H, -intertwining operator S α,α ′ : E ′ ( α ′ ) ֒ → E ( α, α ′ )as follows: For n even set S α,α ′ ( φ + x ′ φ ∨ ) = I j → i ( φ ) , α = ( i, +) , α ′ = j,S α,α ′ ( φ + x ′ φ ∨ ) = xI j → i ( φ ∨ ) , α = ( i, − ) , α ′ = j, and for n odd we put S α,α ′ ( φ ) = I j → i ( φ ) + xI j → i ( φ ) ∨ , α = i, α ′ = ( j, +) ,S α,α ′ ( x ′ φ ) = I j → i ( φ ) ∨ + xI j → i ( φ ) , α = i, α ′ = ( j, − ) . Then S α,α ′ is a e K H, -equivariant isomorphism and we can define the e K H, -intertwining oper-ator R α,α ′ : E ( α, α ′ ) → E ′ ( α ′ ) by R α,α ′ = S − α,α ′ . For this particular choice of operators R α,α ′ we can now compute the proportionality constants λ β,β ′ α,α ′ defined in (5.1). Lemma 5.2.
Let ≤ j ≤ i .(1) For n even and ( α, α ′ ) = (( i, ± ) , j ) we have λ β,β ′ α,α ′ = n +2 j − n +2 i +1 ( β, β ′ ) = (( i + 1 , ± ) , j + 1) , ± n +2 j − n +2 i − n +2 i +1) √− β, β ′ ) = (( i, ∓ ) , j + 1) , − n +2 j − n +2 i − ( β, β ′ ) = (( i − , ± ) , j + 1) , ∓ i − j +1 n +2 i +1 √− β, β ′ ) = (( i + 1 , ± ) , j ) , ( n +2 i )( n +2 j − n +2 i − n +2 i +1) ( β, β ′ ) = (( i, ∓ ) , j ) , ± n + i + j − n +2 i − √− β, β ′ ) = (( i − , ± ) , j ) , − ( i − j +1)( i − j +2)( n +2 i +1)( n +2 j − ( β, β ′ ) = (( i + 1 , ± ) , j − , ± i − j +1)( n + i + j − n +2 i − n +2 i +1)( n +2 j − √− β, β ′ ) = (( i, ∓ ) , j − , ( n + i + j − n + i + j − n +2 i − n +2 j − ( β, β ′ ) = (( i − , ± ) , j − . (2) For n odd and ( α, α ′ ) = ( i, ( j, ± )) we have λ β,β ′ α,α ′ = n +2 j − n +2 i +1 ( β, β ′ ) = ( i + 1 , ( j + 1 , ± )) , ∓ n +2 j − n +2 i − n +2 i +1) ( β, β ′ ) = ( i, ( j + 1 , ± )) , − n +2 j − n +2 i − ( β, β ′ ) = ( i − , ( j + 1 , ± )) , ± i − j +1 n +2 i +1 ( β, β ′ ) = ( i + 1 , ( j, ∓ )) , ( n +2 i )( n +2 j − n +2 i − n +2 i +1) ( β, β ′ ) = ( i, ( j, ∓ )) , ∓ n + i + j − n +2 i − ( β, β ′ ) = ( i − , ( j, ∓ )) , − ( i − j +1)( i − j +2)( n +2 i +1)( n +2 j − ( β, β ′ ) = ( i + 1 , ( j − , ± )) , ∓ i − j +1)( n + i + j − n +2 i − n +2 i +1)( n +2 j − ( β, β ′ ) = ( i, ( j − , ± )) , ( n + i + j − n + i + j − n +2 i − n +2 j − ( β, β ′ ) = ( i − , ( j − , ± )) . Proof.
Since S α,α ′ = R − α,α ′ the defining equation (5.1) for λ β,β ′ α,α ′ is equivalent to ω β,β ′ α,α ′ ( Y ) ◦ S α,α ′ = λ β,β ′ α,α ′ · S β,β ′ ◦ ω β ′ α ′ ( Y ) , Y ∈ s H, C . Let first n be even and α = ( i, +), α ′ = j . Then for Y = e k , 1 ≤ k ≤ n , and φ + x ′ φ ∨ ∈ E ′ ( α ′ ), φ ∈ M j ( R n ; S n ) } , we have ω ( Y )( φ + x ′ φ ∨ ) = x k φ + x ′ ( x k φ ) ∨ = ( φ + k − x ′ φ k + φ − k ) + x ′ ( φ + k − x ′ φ k + φ − k ) ∨ = ( φ + k + x ′ ( φ + k ) ∨ ) − (( φ k ) ∨ + x ′ ( φ k ) ∨∨ ) + ( φ − k + x ′ ( φ − k ) ∨ ) . Hence, ω β ′ α ′ ( Y ) φ = φ + k β ′ = j + 1 , − ( φ k ) ∨ β ′ = j,φ − k β ′ = j − S β,β ′ ◦ ω β ′ α ′ ( Y ) φ = I j +1 → i +1 ( φ + k ) ( β, β ′ ) = (( i + 1 , +) , j + 1) ,xI j +1 → i (( φ + k ) ∨ ) ( β, β ′ ) = (( i, − ) , j + 1) ,I j +1 → i − ( φ + k ) ( β, β ′ ) = (( i − , +) , j + 1) , − I j → i +1 (( φ k ) ∨ ) ( β, β ′ ) = (( i + 1 , +) , j ) , − xI j → i ( φ k ) ( β, β ′ ) = (( i, − ) , j ) , − I j → i − (( φ k ) ∨ ) ( β, β ′ ) = (( i − , +) , j ) ,I j − → i +1 ( φ − k ) ( β, β ′ ) = (( i + 1 , +) , j − ,xI j − → i (( φ − k ) ∨ ) ( β, β ′ ) = (( i, − ) , j − ,I j − → i − ( φ − k ) ( β, β ′ ) = (( i − , +) , j − . Note that φ ∨ = −√− e n +1 φ by Lemma A.1 (2). On the other hand, S α,α ′ φ = I j → i ( φ )and ω ( Y ) S α,α ′ φ = x k I j → i ( φ ) = ( I j → i ( φ )) + k − x ( I j → i ( φ )) k + ( I j → i ( φ )) − k . Then Lemma A.5 gives the constants λ β,β ′ α,α ′ for ( α, α ′ ) = (( i, +) , j ). The case α = ( i, − ) and α ′ = j is treated similarly. The verification for odd n is left to the reader. (cid:3) NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 37
Multiplicities for symmetry breaking operators between spinor-valued prin-cipal series.
Inserting the explicit constants λ β,β ′ α,α ′ determined in Lemma 5.2 into the scalaridentities (5.3) we obtain an explicit characterization of symmetry breaking operators in termsof scalar identities. We assume that n is even for the statement of these identities, the caseof odd n is similar. Corollary 5.3.
Assume n is even. Then a e K H, -intertwining operator T : ( /π λ ) HC → ( /τ ν ) HC is ( h , e K H, ) -intertwining if and only if the scalars t ( i, ± ) ,j defined by (5.2) satisfy the followingthree relations: ( n + 2 i − n + 2 i + 1)( ν + ρ H + + j ) t ( i, ± ) ,j (5.4) = ( n + 2 i − n + 2 j − λ + ρ + + i ) t ( i +1 , ± ) ,j +1 ± n + 2 j − √− λt ( i, ∓ ) ,j +1 − ( n + 2 i + 1)( n + 2 j − λ − ρ + − i ) t ( i − , ± ) ,j +1 , ( n + 2 i − n + 2 i + 1) νt ( i, ± ) ,j (5.5) = ∓ ( i − j + 1)( n + 2 i − √− λ + ρ + + i ) t ( i +1 , ± ) ,j + ( n + 2 i )( n + 2 j − λt ( i, ∓ ) ,j ± ( n + 2 i + 1)( n + i + j − √− λ − ρ + − i ) t ( i − , ± ) ,j , ( n + 2 i − n + 2 i + 1)( n + 2 j − ν − ρ H + − j ) t ( i, ± ) ,j (5.6) = − ( i − j + 1)( i − j + 2)( n + 2 i − λ + ρ + + i ) t ( i +1 , ± ) ,j − ± i − j + 1)( n + i + j − √− λt ( i, ∓ ) ,j − + ( n + 2 i + 1)( n + i + j − n + i + j − λ − ρ + − i ) t ( i − , ± ) ,j − . As previously carried out in [10, Section 4.3] for similar relations, we solve this system toobtain the following result about multiplicities of symmetry breaking operators:
Theorem 5.4. dim Hom ( h , e K H, ) (( /π λ ) HC | ( h , e K H, ) , ( /τ ν ) HC ) = ( for ( λ, ν ) ∈ /L = /L even ∪ /L odd , else.Proof. We assume n is even, the case of n odd is treated similarly. It is more convenient towork with the scalars s ± i,j = t ( i, +) ,j ± ( − i − j t ( i, − ) ,j , then the identities in Corollary 5.3 become( n + 2 i − n + 2 i + 1)( ν + ρ H + + j ) s ± i,j (5.7) = ( n + 2 i − n + 2 j − λ + ρ + + i ) s ± i +1 ,j +1 ± ( − i − j +1 n + 2 j − √− λs ± i,j +1 − ( n + 2 i + 1)( n + 2 j − λ − ρ + − i ) s ± i − ,j +1 , (cid:16) ( n + 2 i − n + 2 i + 1) ν ∓ ( − i − j ( n + 2 i )( n + 2 j − λ (cid:17) s ± i,j (5.8) = − ( i − j + 1)( n + 2 i − √− λ + ρ + + i ) s ± i +1 ,j + ( n + 2 i + 1)( n + i + j − √− λ − ρ + − i ) s ± i − ,j , ( n + 2 i − n + 2 i + 1)( n + 2 j − ν − ρ H + − j ) s ± i,j (5.9) = − ( i − j + 1)( i − j + 2)( n + 2 i − λ + ρ + + i ) s ± i +1 ,j − ± ( − i − j +1 i − j + 1)( n + i + j − √− λs ± i,j − + ( n + 2 i + 1)( n + i + j − n + i + j − λ − ρ + − i ) s ± i − ,j − . Note that each identity only involves scalars s ± i,j with one choice of sign ± , so that the systemof equations degenerates into two systems of equations, one for s + i,j and one for s − i,j . Fixing asign ± we have to find the dimension of the space of tuples ( s ± i,j ) ≤ j ≤ i satisfying the identities(5.7)–(5.9). Let us first visualize the e K - e K H, -type picture in a diagram: ✲✻ s s ss ss ss ss ss ss s ij Here the dot at position ( i, j ) represents the scalar s ± i,j . Now, each identity is a linear relationbetween certain neighboring dots in this diagram, visualized as s ss s ❅❅❅ (cid:0)(cid:0)(cid:0) ( i, j )(5.7) s s s ( i, j )(5.8) s ss s (cid:0)(cid:0)(cid:0) ❅❅❅ ( i, j )(5.9)Note that the only coefficients in the identities that can possibly vanish are those involving λ and ν . Identity (5.7) at ( i, i ) gives(5.10) ( n + 2 j − λ + ρ + + i ) s ± i +1 ,i +1 = ( n + 2 i + 1)( ν + ρ H + + i ) s ± i,i . It is easy to see that the dimension of the space of diagonal sequences ( s ± i,i ) satisfying (5.10)is equal to 1 if ( λ, ν ) ∈ C − /L and equal to 2 if ( λ, ν ) ∈ /L (see [10, Lemma 4.4] for the sameargument in a similar setting). NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 39
Step 1.
Now let us first assume that λ / ∈ − ρ − − N , in particular ( λ, ν ) ∈ C − /L .Then identity (5.8) can be used to define s ± i +1 ,j in terms of s ± i,j and s ± i − ,j since the coefficient( λ + ρ + + i ) of s ± i +1 ,j never vanishes. This shows that each diagonal sequence ( s ± i,i ) uniquelydetermines the numbers s ± i,j for all 0 ≤ j ≤ i and hence, for both signs ± the dimension ofthe space of sequences ( s ± i,j ) satisfying (5.7)–(5.9) is 1. Step 2.
Now assume λ = − ρ − − k , k ≥
0, but ( λ, ν ) ∈ C − /L . Then the extensionargument from Step 1 using (5.8) still works for i > k and hence any diagonal sequenceuniquely determines the numbers s ± i,j for j > k . Next, we can repeatedly use (5.7) with j ≤ k to define s ± i,j in terms of s ± i +1 ,j +1 , s ± i,j +1 and s ± i − ,j +1 . Note that the coefficient ( ν + ρ H + + j )of s ± i,j never vanishes since j ≤ k and ( λ, ν ) / ∈ /L . Therefore, also in this case, for both signs ± the space of sequences ( s ± i,j ) satisfying (5.7)–(5.9) is one-dimensional. Step 3.
Finally assume ( λ, ν ) = ( − ρ − − k, − ρ H − − ℓ ) ∈ /L , 0 ≤ ℓ ≤ k . We onlydiscuss the case where k − ℓ ∈ N , the case k − ℓ ∈ N + 1 is treated similarly. We divide the e K - e K H, -type picture into four regions accoding to whether i ≤ k or i > k and j ≤ ℓ or j > ℓ : ✲✻ s s s s s ss s s s ss s s ss s ss ss ∗ ∗ ∗ ∗ ij k k + 1 ℓℓ + 1Then from the diagonal identity (5.10) it follows that the diagonal entries in the upper leftregion are 0 whereas the diagonal entries in the upper right and the lower left region canbe chosen independently (as indicated by the stars and zeros). This shows that the space ofdiagonal sequences satisfying (5.10) is two-dimensional. Using (5.8) as before shows that infact all scalars in the upper left region have to be 0. Now we study what happens near theintersection of the two dashed lines. For instance, we have two identities that relate s ± k,ℓ and s ± k − ,ℓ , namely (5.8) for ( i, j ) = ( k, ℓ ):(5.11) (cid:16) ( n + 2 k − n + 2 ℓ ) ∓ ( n + 2 k )( n + 2 ℓ − (cid:17) s ± k,ℓ = 2( n + 2 k )( n + k + ℓ − √− s ± k − ,ℓ , and (5.9) for ( i, j ) = ( k, ℓ + 1):(5.12) ± ( k − ℓ ) s ± k,ℓ = ( n + 2 k )( n + k + ℓ − √− s ± k − ,ℓ . Let us first consider the positive sign scalars s + i,j . In this case the coefficient of s + k,ℓ in (5.11)is ( n + 2 k − n + 2 ℓ ) − ( n + 2 k )( n + 2 ℓ −
1) = 2( k − ℓ )so that (5.11) is a multiple of (5.12). Hence, the two identities (5.11) and (5.12) are dependentand do not force s ± k,ℓ and s ± k − ,ℓ to be 0. As above one can uniquely extend any diagonalsequence to the whole lower left region s + i,j with i ≤ k , j ≤ ℓ . For the negative sign scalars s − i,j the opposite is true: the relations (5.11) and (5.12) are independent and hence s − k − ,ℓ = s − k,ℓ =0. Extending the zeros in the other direction forces all scalars s − i,j in the lower left region tobe 0. The same happens for the upper right region, where (5.7) for ( i, j ) = ( k + 1 , ℓ ) and(5.8) for ( i, j ) = ( k + 1 , ℓ + 1) are two dependent resp. independent relations for s + k +1 ,ℓ +1 and s + k +2 ,ℓ +1 resp. s − k +1 ,ℓ +1 and s − k +2 ,ℓ +1 . This shows that the space of scalars in the upper left,lower left and upper right region satisfying (5.7)–(5.9) is two-dimensional for s + i,j and trivialfor s − i,j . It remains to extend such tuples to the lower right region. Assuming that we alreadyhave defined s ± i,j in the first three regions, there are two linear relations for s ± k +1 ,ℓ and s ± k +2 ,ℓ ,namely (5.8) for ( i, j ) = ( k + 1 , ℓ ):(5.13) (cid:16) ( n + 2 k + 3)( n + 2 ℓ ) ± ( n + 2 k + 2)( n + 2 ℓ − (cid:17) s ± k +1 ,ℓ − k − ℓ + 2) √− s ± k +2 ,ℓ = ∗ , and (5.9) for ( i, j ) = ( k + 1 , ℓ + 1):(5.14) ∓ ( n + k + ℓ + 1) s ± k +1 ,ℓ − ( k − ℓ + 2) √− s ± k +2 ,ℓ = ∗ . Here we write ∗ for the right hand side which is a linear combination of scalars from the otherthree regions which we already specified. Now, for the negative sign scalars the coefficient of s − k +1 ,ℓ in (5.13) is equal to( n + 2 k + 3)( n + 2 ℓ ) − ( n + 2 k + 2)( n + 2 ℓ −
1) = 2( n + k + ℓ + 1)and the two relations are dependent. This means that every choice of s − k +1 ,ℓ ∈ C can beuniquely extended to s − i,j , 0 ≤ j ≤ i , using the extension methods from (1) and (2) (with s − i,j = 0 if i ≤ k or j > ℓ ). Hence, the dimension of the space of tuples ( s − i,j ) satisfying(5.7)–(5.9) is 1. For the positive sign scalars, it is easy to see that the relations (5.13) and(5.14) are independent, and therefore s + k +1 ,ℓ and s + k +2 ,ℓ are uniquely determined by the chosenscalars in the other three regions. Together with the extension techniques outlined in Step 1and 2 this implies that the dimension of the space of tuples ( s + i,j ) satisfying (5.7)–(5.9) is 2.This proves the claim. (cid:3) Classification of symmetry breaking operators between spinor-valued princi-pal series.
Combining Theorems 4.3, 4.4 and Proposition 4.1 with Lemma 5.1 and Theo-rem 5.4 we obtain a full classification of symmetry breaking operators between spinor-valuedprincipal series representations:
Theorem 5.5.
We have D ′ ( R n ; Hom C ( S n , S n − )) + λ,ν = C (cid:16) P e K A , + λ,ν (cid:17) for ( λ, ν ) ∈ C − /L even , C (cid:16) P ee K A , + λ,ν (cid:17) ⊕ C (cid:16) P e K C , + λ,ν (cid:17) for ( λ, ν ) ∈ /L even , D ′ ( R n ; Hom C ( S n , S n − )) − λ,ν = C (cid:16) P e K A , − λ,ν (cid:17) for ( λ, ν ) ∈ C − /L odd , C (cid:16) P ee K A , − λ,ν (cid:17) ⊕ C (cid:16) P e K C , − λ,ν (cid:17) for ( λ, ν ) ∈ /L odd . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 41
Symmetry breaking operators at reducibility points.
We study symmetry break-ing operators between irreducible constituents of /π λ,δ and /τ ν,ε at reducibility points. For thiswe first describe the irreducible constituents. Lemma 5.6. (1) The representation /π λ,δ is reducible if and only if λ ∈ ± ( ρ + + N ) .(2) For λ = − ρ − − i , i ∈ N , the representation /π λ,δ has a unique irreducible subrep-resentation F δ ( i ) which is finite-dimensional, and the quotient T δ ( i ) = /π λ,δ / F δ ( i ) isirreducible.(3) For λ = ρ + + i , i ∈ N , the representation /π λ,δ has a unique irreducible subrepresen-tation isomorphic to T − δ ( i ) ⊗ det an irreducible quotient isomorphic to F − δ ( i ) ⊗ det ,where det : Pin( n + 1 , → O( n + 1 , → {± } is the determinant character.Proof. We first describe the K -type decomposition of /π λ,δ . The maximal compact subgroup e K of e G is a semidirect product of Pin( n + 1) with the two-element group which acts on Pin( n +1) via the canonical automorphism α of the Clifford algebra Cl( n + 1) (see Appendix A.1for details). As remarked in Section 5.3, the restriction of /π λ,δ to Pin( n + 1) ≃ e K ⊆ e K decomposes into a multiplicity-free direct sum of higher spin representations. For n even, itis the direct sum of ζ ± n +1 ,i , i ≥
0, and it is easy to see that ζ + n +1 ,i ⊕ ζ − n +1 ,i extends uniquely toan irreducible representation e ζ n +1 ,i of e K . For n odd, the restriction of /π λ,δ to e K is the directsum of ζ n +1 ,i , i ≥
0, and there are two inequivalent ways of extending ζ n +1 ,i to an irreducible e K -representation. For fixed δ ∈ Z / Z let e ζ n +1 ,i denote the extension which occurs in /π λ,δ .Then, for both even and odd n we have( /π λ,δ ) HC = ∞ M i =0 e ζ n +1 ,i . The Lie algebra action of g maps e ζ n +1 ,i into the direct sum of e ζ n +1 ,i +1 , e ζ n +1 ,i and e ζ n +1 ,i − .From [1] and the computations in Section 5.4 it follows that one can reach e ζ n +1 ,i +1 if andonly if 2 λ + 2 i + n + 1 = 0, and one can reach e ζ n +1 ,i − if and only if 2 λ − i − n + 1 = 0.Then statements (1) and (2) follow with T δ ( i ) HC ≃ ∞ M k = i +1 e ζ n +1 ,k and F δ ( i ) HC ≃ i M k =0 e ζ n +1 ,k . To prove (3) observe that the standard intertwining operators in this situation (see e.g. Re-mark 2.8) Ind e G e P (( ζ n ⊗ sgn δ ) ⊗ e λ ⊗ ) → Ind e G e P (([ ζ n ⊗ det] ⊗ sgn − δ ) ⊗ e − λ ⊗ )map irreducible quotients to irreducible subrepresentations. Since the character det of f M extends to e G we haveInd e G e P (([ ζ n ⊗ det] ⊗ sgn − δ ) ⊗ e − λ ⊗ ) ≃ Ind e G e P (( ζ n ⊗ sgn − δ ) ⊗ e − λ ⊗ ) ⊗ det , and the claim follows by specializing to λ = ± ( ρ + + i ). (cid:3) Note that the representations F δ ( i ) and T δ ( i ) depend on the chosen spin-representation ζ of f M that /π λ,δ is induced from. However, as in the case of the full principal series, themultiplicities of intertwining operators turn out to be independent of ζ . Denote by T ′ ε ( j ) and F ′ ε ( j ) the corresponding composition factors of /τ ν,ε at ν = ± ( ρ H + + j ), j ∈ N . Theorem 5.7.
For π ∈ {T δ ( i ) , F δ ( i ) } and τ ∈ {T ′ ε ( j ) , F ′ ε ( j ) } the multiplicities dim Hom e H ( π | e H , τ ) are given by π τ F ′ ε ( j ) T ′ ε ( j ) F δ ( i ) 1 0 T δ ( i ) 0 1 for ≤ j ≤ i , i + j ≡ δ + ε (2) , π τ F ′ ε ( j ) T ′ ε ( j ) F δ ( i ) 0 0 T δ ( i ) 1 0 otherwise.Proof. We only treat the case of even n , for odd n similar arguments can be used. Write( /π λ,δ ) HC ≃ ∞ M i =0 e ζ n +1 ,i and ( /τ ν,ε ) HC ≃ ∞ M j =0 e ζ n,j for the decomposition into irreducible representations of e K and e K H . Then, as in the proof ofTheorem 5.4, an intertwining operator ( /π λ,δ ) HC → ( /τ ν,ε ) HC is given by a sequence ( s i,j ) ≤ j ≤ i of scalars, describing the action of the operator between e ζ n +1 ,i and e ζ n,j . Analyzing the actionof e H/ e H it is easy to see that intertwining operators are described by the scalars s i,j = s + i,j if δ + ε ≡ s i,j = s − i,j if δ + ε ≡ T δ ( i ) → T ′ ε ( j ) are intertwining operators /π λ,δ → /τ ν, − ε ⊗ det for λ = − ρ − − i and ν = ρ H + + j which vanish on the finite-dimensional subrepresentation of /π λ,δ (hence fac-tor to the quotient T δ ( i )) and whose image is contained in the irreducible subrepresentation T ′ ε ( j ) ⊆ /τ ν, − ε ⊗ det. By replacing the spin representation ζ n − of f M H , that /τ ν, − ε is in-duced from, by ζ n − ⊗ det we may as well consider intertwining operators /π λ,δ → /τ ν, − ε .Then the multiplicity dim Hom ( h , e K H ) ( T δ ( i ) HC , T ′ ε ( j )) is the dimension of the space of se-quences ( s k,ℓ ) ≤ ℓ ≤ k satisfying (5.7), (5.8) and (5.9) (with + for δ + (1 − ε ) ≡ − for δ + (1 − ε ) ≡ s k,ℓ = 0 whenever k ≤ i or ℓ ≤ j . By Theorem 5.5 there isup to scalar multiples a unique intertwiner /π λ,δ → /τ ν, − ε , and the arguments in the proof ofTheorem 5.4 show that for this intertwiner we have s k,ℓ = 0 whenever k ≤ i or ℓ ≤ j if andonly if j ≤ i and i + j ≡ δ + ε (2). The other multiplicities are computed similarly. (cid:3) Appendix A. Clifford algebras, pin groups and their representations
We recall the basic definitions for Clifford algebras, pin groups and their representations.Most of the results are well-known or follow easily from the standard literature.A.1.
Clifford algebras and pin groups.
For p, q ≥ R p,q = ( R p + q , Q p,q ), where Q = Q p,q is the following quadratic form on R p + q : Q ( x ) = − x − · · · − x p + x p +1 + · · · + x p + q . Abusing notation, we also write Q ( x, y ) for the associated symmetric bilinear form on R p + q . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 43
We define the Clifford algebra Cl( p, q ) to be the unital R -algebra generated by R p,q subjectto the relation xy + yx = 2 Q ( x, y ) . For the standard basis vectors e , . . . , e p + q ∈ R p,q this implies e i = Q ( e i ) , e i e j + e j e i = 0 ( i = j ) . For ( p, q ) = ( n,
0) we also write Cl( n ) = Cl( n,
0) for short. The Clifford algebra Cl( p, q ) hasa natural grading into even and odd elementsCl( p, q ) = Cl( p, q ) even ⊕ Cl( p, q ) odd . The map α of Cl( p, q ) which acts by +1 on Cl( p, q ) even and by − p, q ) odd is an algebrainvolution called the canonical automorphism .Denote by Cl( p, q ; C ) = Cl( p, q ) ⊗ R C and Cl( n ; C ) = Cl( n ) ⊗ R C the complexifications ofCl( p, q ) and Cl( n ). Abusing notation, we will also use Q for the extension of the symmetric R -bilinear form Q p,q on R p + q to a symmetric C -bilinear form on C p + q . Note that Cl( p, q ; C ) ≃ Cl( p + q ; C ) as C -algebras.We define the groups Pin( p, q ) and Spin( p, q ) byPin( p, q ) = { v · · · v k : k ≥ , v i ∈ R p,q , Q ( v i ) = ± } ⊆ Cl( p, q ) , Spin( p, q ) = Pin( p, q ) ∩ Cl( p, q ) even . Further, put Pin( n ) = Pin( n,
0) and Spin( n ) = Spin( n, n ) is connected andPin( n ) = Spin( n ) ∪ Pin( n ) − , where Pin( n ) − = Spin( n ) · e . For p, q > p, q )has four connected componentsPin( p, q ) ++ , Pin( p, q ) + − , Pin( p, q ) − + and Pin( p, q ) −− , where the first + resp. − means that the number of v i ’s with Q ( v i ) = 1 in the product of anelement x = v · · · v k ∈ Pin( p, q ) is even resp. odd, and the second + resp. − the same forthe number of v i ’s with Q ( v i ) = −
1. Then clearly Spin( p, q ) = Pin( p, q ) ++ ∪ Pin( p, q ) −− , soSpin( p, q ) has two connected components.For all p, q ≥ p, q ) is a double cover of the indefinite orthogonal groupO( p, q ), the covering map being q : Pin( p, q ) → O( p, q ) , q ( x ) y = α ( x ) yx − . The restriction of q to Spin( p, q ) induces a double covering q : Spin( p, q ) → SO( p, q ) . A.2.
Clifford modules and fundamental spin representations.
We now describe theirreducible representations of Cl( n ; C ). For n = 2 m even there is only one irreducible repre-sentation, and for n = 2 m + 1 odd there are two. To construct these we choose the maximalisotropic subspaces W = C w ⊕ · · · ⊕ C w m and W ′ = C w ′ ⊕ · · · ⊕ C w ′ m of C n , where w i = ( e i − + √− e i ), w ′ i = ( e i − − √− e i ). Then C n = ( W ⊕ W ′ for n even, W ⊕ W ′ ⊕ C e m +1 for n odd. We define an irreducible representation ζ n of Cl( n ; C ) on S n = V • W by ζ n ( w ) ξ ∧ . . . ∧ ξ k = w ∧ ξ ∧ . . . ∧ ξ k ,ζ n ( w ′ ) ξ ∧ . . . ∧ ξ k = 2 k X i =1 ( − i − Q ( w ′ , ξ i ) ξ ∧ . . . ∧ b ξ i ∧ . . . ∧ ξ k , for w ∈ W , w ′ ∈ W ′ , and in case n is odd additionally ζ n ( e m +1 ) ξ ∧ . . . ∧ ξ k = ( − k √− ξ ∧ . . . ∧ ξ k . Then for n even ( ζ n , S n ) is the unique irreducible representation of Cl( n ; C ) and ζ n ◦ α ≃ ζ n ,where α is the canonical automorphism of Cl( n ; C ). More precisely, the map(A.1) γ : S n → S n , γ ( ξ ∧ . . . ∧ ξ k ) = ( − k ξ ∧ . . . ∧ ξ k intertwines ζ n ◦ α and ζ n . For n odd, ζ + n = ζ n and ζ − n = ζ n ◦ α are inequivalent and we obtaintwo irreducible inequivalent representations ( ζ ± n , S n ) of Cl( n ; C ). For x ∈ C n we write x = ζ n ( x )whenever the representation ζ n is clear from the context.The restriction of any irreducible complex representation of the Clifford algebra Cl( n ; C )to Pin( n ) ⊆ Cl( n ) ⊆ Cl( n ; C ) defines an irreducible representation of Pin( n ). These represen-tations are called fundamental spin representations and will also be denoted by ( ζ n , S n ) for n even and ( ζ ± n , S n ) for n odd. Note that for n even ζ n ⊗ det ≃ ζ n and for n odd ζ ± n ⊗ det ≃ ζ ∓ n ,where det is the one-dimensional representation of Pin( n ) given by the determinant characterdet : Pin( n ) → O( n ) → {± } .For n even the restriction of ζ n to Spin( n ) ⊆ Pin( n ) decomposes into the direct sum of twoirreducible representations of Spin( n ) according to the decomposition V • W = V even W ⊕ V odd W. They have highest weights ( , . . . , , ± ) in the standard notation. For n odd the restrictionsof ζ + n and ζ − n to Spin( n ) define equivalent irreducible representations of Spin( n ) whose highestweight is ( , . . . , ).A.3. Higher spin representations on monogenic polynomials.
Let i ≥
0. For n even the direct sum of the two irreducible Spin( n )-representations with highest weight ( i + , , . . . , , ± ) extends uniquely to an irreducible representation of Pin( n ) which we denoteby ζ n,i . Note that ζ n,i ⊗ det ≃ ζ n,i . For n odd the irreducible Spin( n )-representation withhighest weight ( i + , , . . . , ) has two inequivalent extensions to Pin( n ) which we denote by ζ ± n,i .We realize the representations ζ ( ± ) n,i as monogenic polynomials on R n . For each i ≥ n ) acts on the space Pol i ( R n ; S n ) of S n -valued homogeneous polynomials on R n ofdegree i by ( g · φ )( x ) = ζ n ( g ) φ ( q ( g ) − x ) , g ∈ Pin( n ) , x ∈ R n . We consider the Dirac operator / D on Pol( R n ; S n ) = L ∞ i =0 Pol i ( R n ; S n ) given by/ D φ ( x ) = n X k =1 e k ∂φ∂x k ( x ) . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 45
Then the space M i ( R n ; S n ) = { φ ∈ Pol i ( R n ; S n ) : / D φ = 0 } of homogeneous monogenic polynomials of degree i is invariant under the action of Pin( n )and defines an irreducible representation ζ n,i of Pin( n ). For n even, ζ n,i ⊗ det ≃ ζ n,i , theisomorphism being M i ( R n ; S n ) → M i ( R n ; S n ) φ γ ◦ φ. For n odd, ζ + n,i = ζ n,i and ζ − n,i = ζ n,i ⊗ det define inequivalent irreducible representations( ζ ± n,i , M i ( R n ; S n )) of Pin( n ).A.4. Branching laws.
We give the explicit branching laws for the restriction of the Pin( n +1)-representations ζ ( ± ) n +1 ,i to Pin( n ).A.4.1. Fundamental spin representations.
We use the explicit realizations of the representa-tions ζ ( ± ) n given in Section A.2. Then S n +1 = S n for even n , and S n +1 = S n ⊕ ( S n ∧ w m +1 )for odd n = 2 m + 1. Recall the map γ : S n → S n from (A.1).For n even, the restriction of ( ζ ± n +1 , S n +1 ) to Cl( n ; C ) stays irreducible and is isomorphicto ( ζ n , S n ): Lemma A.1.
Assume n = 2 m is even.(1) The explicit isomorphisms ζ n ≃ ζ ± n +1 | Cl( n ; C ) are given by ( ζ n , S n ) → ( ζ + n +1 , S n +1 ) , ω ω, ( ζ n , S n ) → ( ζ − n +1 , S n +1 ) , ω γ ( ω ) . (2) ζ n +1 ( e n +1 ) = √− γ .Proof. This follows immediately from the definition of ζ n and ζ ± n +1 and the fact that γ inter-twines ζ n with ζ n ◦ α . (cid:3) For n odd, the restriction of ( ζ n +1 , S n +1 ) to Cl( n ; C ) decomposes into the direct sum of( ζ + n , S n ) and ( ζ − n , S n ): Lemma A.2.
Assume n = 2 m + 1 is odd.(1) The explicit embeddings ζ ± n ֒ → ζ n +1 | Cl( n ; C ) are given by ( ζ + n , S n ) ֒ → ( ζ n +1 , S n +1 ) , ω ω − √− ω ∧ w m +1 , ( ζ − n , S n ) ֒ → ( ζ n +1 , S n +1 ) , ω γ ( ω ) + √− γ ( ω ) ∧ w m +1 . In particular, the image of the embedding ζ ± n ֒ → ζ n +1 is equal to S ± n +1 = { ω ∓ √− ω ∧ w m +1 : ω ∈ S n } . (2) ζ n +1 ( e n +1 ) | S ± n +1 = ∓ γ . In particular, the map (id ± ζ n +1 ( e n +1 ) ◦ γ ) is the canonicalprojection S n +1 → S ± n +1 .Proof. It is easy to see that for ω ∈ S n ⊆ S n +1 we have ζ n +1 ( e n ) ω = γ ( ω ) ∧ w m +1 , ζ n +1 ( e n )( ω ∧ w m +1 ) = − γ ( ω ) ,ζ n +1 ( e n +1 ) ω = −√− γ ( ω ) ∧ w m +1 , ζ n +1 ( e n +1 )( ω ∧ w m +1 ) = −√− γ ( ω ) , then the claims follow. (cid:3) For the representations ζ ( ± ) n +1 of Pin( n + 1) this implies ζ ± n +1 | Pin( n ) ≃ ζ n ( n even), ζ n +1 | Pin( n ) ≃ ζ + n ⊕ ζ − n ( n odd).A.4.2. Higher spin representations.
Using classical branching laws for the pair ( so ( n +1) , so ( n ))of Lie algebras, it is easy to see that the branching laws for the higher spin representationsare ζ ± n +1 ,i | Pin( n ) ≃ i M j =0 ζ n,j ( n even), ζ n +1 ,i | Pin( n ) ≃ i M j =0 (cid:16) ζ + n,j ⊕ ζ − n,j (cid:17) ( n odd).To make this branching explicit in the realizations on monogenic polynomials, we use theclassical Gegenbauer polynomials C λn ( z ) given by (see e.g. [2, Chapter 10.9, (18)]) C λn ( z ) = ⌊ n/ ⌋ X m =0 ( − m ( λ ) n − m m !( n − m )! (2 z ) n − m . The polynomial u = C λn satisfies the differential equation (see [2, Chapter 10.9, (14)])(1 − z ) u ′′ − (2 λ + 1) zu ′ + n ( n + 2 λ ) u = 0 . Lemma A.3.
Let ( ζ n +1 , S n +1 ) be a fundamental spin representation of Pin( n +1) and assume ( ζ n , S n ) occurs in the restriction of ζ n +1 to Pin( n ) . If we identify S n with a subspace of S n +1 ,then for every ≤ j ≤ i the map I j → i : M j ( R n ; S n ) → M i ( R n +1 , S n +1 ) ,I j → i φ ( x ′ , x n +1 ) = ( n + i + j − | x | i − j φ ( x ′ ) C n − + ji − j ( x n +1 | x | )+ ( n + 2 j − | x | i − j − x ′ e n +1 φ ( x ′ ) C n +12 + ji − j − ( x n +1 | x | ) is Pin( n ) -intertwining, where x = ( x ′ , x n +1 ) ∈ R n +1 .Proof. We first show the intertwining property. Let g ∈ Pin( n ), then ( q ( g ) x ) ′ = q ( g ) x ′ and( q ( g ) x ) n +1 = x n +1 . Further, ζ n +1 ( g ) ◦ x ′ e n +1 = ζ n +1 ( g ) ◦ ζ n +1 ( x ′ ) ◦ ζ n +1 ( e n +1 )= ζ n +1 ( α ( g ) x ′ g − ) ◦ ζ n +1 ( α ( g ) e n +1 g − ) ◦ ζ n +1 ( g )= ζ n +1 ( q ( g ) x ′ ) ◦ ζ n +1 ( q ( g ) e n +1 ) ◦ ζ n +1 ( g )= ( q ( g ) x ′ ) e n +1 ◦ ζ n +1 ( g ) . Then the intertwining property follows. It remains to show that I j → i φ is monogenic, and forthis we abbreviate p = ( n + i + j − C n − + ji − j and q = ( n + 2 j − C n +12 + ji − j − . Then I j → i φ ( x ′ , x n +1 ) = | x | i − j φ ( x ′ ) p ( x n +1 | x | ) + | x | i − j − x ′ e n +1 φ ( x ′ ) q ( x n +1 | x | ) . NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 47
Applying the Dirac operator to I j → i φ yields, after a short computation:/ D ( I j → i φ )( x ′ , x n +1 )= n X k =1 e k ∂ ( I j → i φ ) ∂x i ( x ′ , x n +1 ) + e n +1 ∂ ( I j → i φ ) ∂x n +1 ( x ′ , x n +1 )= | x | i − j − x ′ φ ( x ′ ) (cid:16) ( i − j ) p ( z ) − zp ′ ( z ) + ( i − j − zq ( z ) + (1 − z ) q ′ ( z ) (cid:17) + | x | i − j − e n +1 φ ( x ′ ) (cid:16) ( i − j ) zp ( z ) + (1 − z ) p ′ ( z ) − ( n + i + j − q ( z )+ ( i − j − z q ( z ) + (1 − z ) zq ′ ( z ) (cid:17) , where z = x n +1 | x | . This vanishes if and only if( i − j ) p − zp ′ = − ( i − j − zq − (1 − z ) q ′ , ( i − j ) zp + (1 − z ) p ′ = ( n + i + j − q − ( k − j − z q − (1 − z ) zq ′ . Multiplying the first equation with z and subtracting it from the second one yields p ′ = ( n + i + j − q. If we now insert this into the first equation we obtain(1 − z ) p ′′ − ( n + 2 j ) zp ′ + ( i − j )( n + i + j − p = 0 , which is the Gegenbauer differential equation with solution p ( z ) = C n − + ji − j ( z ), the classicalGegenbauer polynomial. Finally, using (see [2, Chapter 10.9, (23)]) ddz C λn ( z ) = 2 λC λ +1 n − ( z )and renormalizing p and q shows that indeed / D ( I j → i φ ) = 0 and the proof is complete. (cid:3) A.5.
Multiplication with coordinates.
By the Fischer decomposition we have(A.2) Pol( R n ; S n ) = ∞ M i,j =0 x j M i ( R n ; S n ) . Note that x = −| x | . We now study how the product of a monogenic polynomial with acoordinate function x k decomposes according to this decomposition. Lemma A.4.
Let φ ∈ M i ( R n ; S n ) and ≤ k ≤ n . Then x k φ ∈ M i +1 ( R n ; S n ) ⊕ x M i ( R n ; S n ) ⊕ | x | M i − ( R n ; S n ) . More precisely, x k φ = φ + k − xφ k + | x | φ − k with φ + k = x k φ + 1 n + 2 i (cid:18) xe k φ − | x | ∂φ∂x k (cid:19) ∈ M i +1 ( R n ; S n ) ,φ k = 1 n + 2 i (cid:18) e k φ − n + 2 i − x ∂φ∂x k (cid:19) ∈ M i ( R n ; S n ) ,φ − k = 1 n + 2 i − ∂φ∂x k ∈ M i − ( R n ; S n ) . Proof.
This follows from the following identities which are easily verified:/ D ( x k φ ) = e k φ, / D ( xe k φ ) = − ( n + 2 i ) e k φ + 2 x ∂φ∂x k , / D ( e k φ ) = − ∂φ∂x k , / D ( x ∂φ∂x k ) = − ( n + 2 i − ∂φ∂x k , / D ∂φ∂x k = 0 , / D ( | x | ∂φ∂x k ) = 2 x ∂φ∂x k . (cid:3) Now, recall the intertwining map I j → i : M j ( R n ; S n ) → M i ( R n +1 , S n +1 ) from Lemma A.3.The next result relates the decompositions of the polynomials x k ( I j → i φ )( x ′ , x n +1 ) and x k φ ( x ′ )for 1 ≤ k ≤ n . Lemma A.5.
For ≤ j ≤ i and ≤ k ≤ n we have ( I j → i φ ) + k = n + 2 j − n + 2 i + 1 I j +1 → i +1 ( φ + k ) + i − j + 1 n + 2 i + 1 I j → i +1 ( e n +1 φ k ) − ( i − j + 1)( i − j + 2)( n + 2 i + 1)( n + 2 j − I j − → i +1 ( φ − k ) , ( I j → i φ ) k = − n + 2 j − n + 2 i − n + 2 i + 1) I j +1 → i ( e n +1 φ + k )+ ( n + 2 i )( n + 2 j − n + 2 i − n + 2 i + 1) I j → i ( φ k ) − i − j + 1)( n + i + j − n + 2 i − n + 2 i + 1)( n + 2 j − I j − → i ( e n +1 φ − k ) , ( I j → i φ ) − k = − n + 2 j − n + 2 i − I j +1 → i − ( φ + k ) − n + i + j − n + 2 i − I j → i − ( e n +1 φ k )+ ( n + i + j − n + i + j − n + 2 i − n + 2 j − I j − → i − ( φ − k ) . Note that we identify S n with a subspace of S n +1 , so that e n +1 S n is also a subspace of S n +1 which is invariant under Pin( n ). More precisely, if Pin( n ) acts on S n by ζ n , then it acts on e n +1 S n by ζ n ⊗ det. Proof.
This a lenghty but elementary computation involving several identities for Gegenbauerpolynomials. We provide these identities and leave the computation to the reader. First, wehave ddz C λn = 2 λC λ +1 n − (A.3) 2 λzC λ +1 n − ( n + 1) C λn +1 − λC λ +1 n − = 0(A.4) 2 λC λ +1 n − ( n + 2 λ ) C λn = 2 λzC λ +1 n − (A.5) 2 λ (1 − z ) C λ +1 n − = ( n + 2 λ ) zC λn − ( n + 1) C λn +1 (A.6) ( λ − C λn +1 − ( n + λ ) C λ − n +1 = ( λ − C λn − . (A.7)(A.3) is [2, Chapter 10.9, (23)], (A.4) follows from [2, Chapter 10.9, (24)] and (A.3), (A.5) isa consequence of [2, Chapter 10.9, (25)] and (A.3), (A.6) is [2, Chapter 10.9, (35)] and (A.7) NAPP–STEIN TYPE INTERTWINING OPERATORS FOR SYMMETRIC PAIRS II 49 is [2, Chapter 10.9, (36)]. Moreover, (A.5) combined with (A.6) implies(A.8) 4 λ ( λ + 1)(1 − z ) C λ +2 n − − λ (2 λ + 1) zC λ +1 n − + n (2 λ + n ) C λn = 0 , (A.8) and (A.5) give(A.9) 4 λ ( λ + 1)(1 − z ) C λ +2 n − + (2 λ + n )(2 λ + n + 1) C λn − λ (2 λ + 1) C λ +1 n = 0 , and (A.4) together with (A.8) implies (cid:3) (A.10) 4 λ ( λ + 1)(1 − z ) C λ +2 n − − λ (2 λ + 1) C λ +1 n − + n ( n − C λn = 0 . References [1] Thomas Branson, Gestur ´Olafsson, and Bent Ørsted,
Spectrum generating operators and intertwiningoperators for representations induced from a maximal parabolic subgroup , J. Funct. Anal. (1996),no. 1, 163–205.[2] Arthur Erd´elyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,
Higher transcendentalfunctions. Vol. II , Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, Based on notes left byHarry Bateman, Reprint of the 1953 original.[3] Matthias Fischmann, Andreas Juhl, and Petr Somberg,
Conformal symmetry breaking differential opera-tors on differential forms , (2016), preprint, available at arXiv:1605.04517.[4] Andreas Juhl,
Families of conformally covariant differential operators, Q -curvature and holography ,Progress in Mathematics, vol. 275, Birkh¨auser Verlag, Basel, 2009.[5] Anthony W. Knapp, Representation theory of semisimple groups , Princeton Mathematical Series, vol. 36,Princeton University Press, Princeton, NJ, 1986, An overview based on examples.[6] Toshiyuki Kobayashi,
A program for branching problems in the representation theory of real reductivegroups , Representations of reductive groups, Progr. Math., vol. 312, Birkh¨auser/Springer, Cham, 2015,pp. 277–322.[7] Toshiyuki Kobayashi, Toshihisa Kubo, and Michael Pevzner,
Conformal symmetry breaking operators fordifferential forms on spheres , Lecture Notes in Mathematics, vol. 2170, Springer, Singapore, 2016.[8] Toshiyuki Kobayashi, Bent Ørsted, Petr Somberg, and Vladim´ır Souˇcek,
Branching laws for Verma mod-ules and applications in parabolic geometry. I , Adv. Math. (2015), 1796–1852.[9] Toshiyuki Kobayashi and Birgit Speh,
Symmetry breaking for representations of rank one orthogonalgroups , Mem. Amer. Math. Soc. (2015), no. 1126.[10] Jan M¨ollers and Bent Ørsted,
The compact picture of symmetry breaking operators for rank one orthogonaland unitary groups , (2014), preprint, available at arXiv:1404.1171.[11] Jan M¨ollers and Bent Ørsted,
Estimates for the restriction of automorphic forms on hyperbolic manifoldsto compact geodesic cycles , (2014), to appear in Int. Math. Res. Not. IMRN, published online first atdoi:10.1093/imrn/rnw119.[12] Jan M¨ollers, Bent Ørsted, and Yoshiki Oshima,
Knapp–Stein type intertwining operators for symmetricpairs , Adv. Math. (2016), 256–306.[13] Jan M¨ollers, Bent Ørsted, and Genkai Zhang,
On boundary value problems for some conformally invariantdifferential operators , Comm. Partial Differential Equations (2016), no. 4, 609–643.[14] Nolan R. Wallach, Cyclic vectors and irreducibility for principal series representations. , Trans. Amer.Math. Soc. (1971), 107–113.
Department Mathematik, FAU Erlangen–N¨urnberg, Cauerstr. 11, 91058 Erlangen, Germany
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