Knapp-Stein Type Intertwining Operators for Symmetric Pairs II. -- The Translation Principle and Intertwining Operators for Spinors
Jan Frahm, Bent {\O}rsted
TL;DR
This paper extends the construction of symmetry breaking operators for reductive group pairs, explicitly describing their integral kernels and classifying operators between spinor representations in a conformal setting.
Contribution
It introduces a broad class of meromorphic symmetry breaking operators for generalized principal series and classifies spinor intertwining operators for conformal groups.
Findings
Explicit integral kernel formulas for symmetry breaking operators.
Complete classification of spinor intertwining operators.
Extension of previous constructions to a larger class of representations.
Abstract
For a symmetric pair of reductive groups we extend to a large class of generalized principal series representations our previous construction of meromorphic families of symmetry breaking operators. These operators intertwine between a possibly vector-valued principal series of and one for and are given explicitly in terms of their integral kernels. As an application we give a complete classification of symmetry breaking operators from spinors on a Euclidean space to spinors on a hyperplane, intertwining for a double cover of the conformal group of the hyperplane.
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\FirstPageHeading
\ShortArticleName
Knapp–Stein Type Intertwining Operators for Symmetric Pairs II
\ArticleName
Knapp–Stein Type Intertwining Operators
for Symmetric Pairs II. – The Translation Principle
and Intertwining Operators for Spinors
\Author
Jan FRAHM and Bent ØRSTED \AuthorNameForHeadingJ. Frahm and B. Ørsted \AddressDepartment of Mathematics, Aarhus University,
Ny Munkegade 118, 8000 Aarhus C, Denmark \Email[email protected], [email protected]
\ArticleDates
Received May 17, 2019, in final form October 29, 2019; Published online November 02, 2019
\Abstract
For a symmetric pair of reductive groups we extend to a large class of generalized principal series representations our previous construction of meromorphic families of symmetry breaking operators. These operators intertwine between a possibly vector-valued principal series of and one for and are given explicitly in terms of their integral kernels. As an application we give a complete classification of symmetry breaking operators from spinors on a Euclidean space to spinors on a hyperplane, intertwining for a double cover of the conformal group of the hyperplane.
\Keywords
Knapp–Stein intertwiners; intertwining operators; symmetry breaking operators; symmetric pairs; principal series; translation principle
\Classification
22E45; 47G10
1 Introduction
In the study of representations of real reductive Lie groups, intertwining operators play a decisive 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 . More recently, intertwining operators have been studied and used in the framework of branching problems, i.e., the restriction of a representation to a subgroup and its decomposition. Here one is interested in operators from a representation of to a representation of , intertwining for the subgroup. Such operators are also called symmetry breaking operators, a term coined by T. Kobayashi in his program for branching problems (see, e.g., [7]).
Such symmetry breaking operators have been studied in great detail in the special case of the conformal groups corresponding to a Euclidean space and a hyperplane by Kobayashi–Speh [10]. In this case some of the operators turn out to be differential operators, namely exactly the operators found by A. Juhl [5] in connection with his study of -curvature and holography in conformal geometry. Further connections to elliptic boundary value problems [15] and automorphic forms [12] indicate the broad spectrum of applications that these operators provide.
In our recent work with Y. Oshima [14] we generalized the construction of symmetry breaking operators by Kobayashi–Speh to a large class of symmetric pairs and spherical principal series representations, proving meromorphic dependence on the parameters in general and generic uniqueness in some special cases. In this paper we shall further extend our construction to the case of vector-valued principal series representations; we establish many of the properties, now for arbitrary vector bundles, in particular the meromorphic dependence on the parameters. The main argument is a version of a well-known translation principle, namely by tensoring with a finite-dimensional representation of the group. As an application and illustration we give all details for the case of spinors on a Euclidean space carrying a representation of the spin cover of the conformal group; here we invoke a variation of the method of finding the eigenvalues on -types of Knapp–Stein operators. The result is a complete classification of symmetry breaking operators from spinors on the Euclidean space to spinors on a hyperplane.
Let us now explain our results in more detail.
1.1 The translation principle
Let be a real reductive group and a reductive subgroup. For parabolic subgroups and we form the generalized principal series representations (smooth normalized parabolic induction)
[TABLE]
where and are finite-dimensional representations of and , and , , the complexified duals of the Lie algebras of and . Consider the space
[TABLE]
of continuous -intertwining operators. Realizing \textup{Ind}_{P}^{G}\big{(}\xi\otimes e^{\lambda}\otimes\mathbf{1}\big{)} and \textup{Ind}_{P_{H}}^{H}\big{(}\eta\otimes e^{\nu}\otimes\mathbf{1}\big{)} on the spaces of smooth sections of the vector bundles
[TABLE]
where and are the half sums of positive roots, one can identify continuous -intertwining operators with their distribution kernels. More precisely, Kobayashi–Speh [10] showed that taking distribution kernels is a linear isomorphism
[TABLE]
where is the dual bundle of and the -representation defining (see Section 2.2 for the precise definition).
Now suppose is an irreducible finite-dimensional representation of . Then the restriction of to contains a unique irreducible subrepresentation (see Lemma 3.1). With respect to the Langlands decomposition this subrepresentation is of the form for some irreducible finite-dimensional representation of and . Let denote the dual of .
Theorem A** (see Theorem 3.3 and Proposition 3.4).**
For every -equivariant quotient with as -representations, there is a unique linear map
[TABLE]
with the property that for every intertwining operator with distribution kernel , the distribution kernel of is given by , the multiplication of with the smooth section \varphi\in C^{\infty}\big{(}G/P,G\times_{P}(E^{\prime})^{\vee}\big{)}\otimes E^{\prime\prime} defined by
[TABLE]
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 classify intertwining operators (see, e.g., Theorem 4.10).
We note that although is a non-trivial analytic section, the map might be trivial for certain parameters. However, being a multiplication operator, behaves nicely when applied to holomorphic/meromorphic families of intertwining operators (see Remark 3.5 for details).
1.2 Knapp–Stein type intertwining operators
Now assume that is a symmetric pair, i.e., is an open subgroup of the fixed points of an involution of . For simplicity we further assume that is in the Harish-Chandra class. Let be a -stable parabolic subgroup, then is a parabolic subgroup of and we write for its Langlands decomposition.
In our previous paper [14] with Y. Oshima we constructed meromorphic families of intertwining operators between the spherical principal series representations \textup{Ind}_{P}^{G}\big{(}\mathbf{1}\otimes e^{\lambda}\otimes\mathbf{1}\big{)} and \textup{Ind}_{P_{H}}^{H}\big{(}\mathbf{1}\otimes e^{\nu}\otimes\mathbf{1}\big{)}. We now generalize this construction and obtain intertwining operators between vector-valued principal series representations using the translation principle.
Assume that and its opposite parabolic are conjugate via the Weyl group, i.e., , where is a representative of the longest Weyl group element . Then for any finite-dimensional representation of and we consider the function
[TABLE]
where and are the densely defined projections of onto the - and -component. For the trivial representation, these are the kernel functions constructed in [14]. Since is only defined on an open dense subset, it may happen that the factor a\big{(}\tilde{w}_{0}^{-1}g^{-1}\sigma(g)\big{)} is not defined for any , whence we additionally assume that the domain of definition for a\big{(}\tilde{w}_{0}^{-1}g^{-1}\sigma(g)\big{)} is not empty. In [14] we showed that in this case is defined on an open dense subset in , and also gave a criterion to check this.
Under the above assumptions, the translation principle combined with our previous results from [14] yields:
Corollary B** (see Corollary 3.9).**
Assume that the finite-dimensional representation of is extendible to see Section 3.4 for the precise definition. Then the functions extend to a meromorphic family of distributions
[TABLE]
where
[TABLE]
Therefore, they give rise to a meromorphic family of intertwining operators
[TABLE]
As in [14, Corollary B] this construction gives in particular lower bounds on multiplicities:
[TABLE]
for all parameters of the form (1.1).
1.3 Symmetry breaking operators for rank one orthogonal groups
We illustrate the translation principle in two examples, the first one being scalar-valued principal series representations for the symmetric pair . The parabolic subgroups and satisfy
[TABLE]
We identify such that , and denote by sgn the non-trivial character of . Then for and we consider intertwining operators between
[TABLE]
Tensoring with characters of , it is easy to see that
[TABLE]
For the pair , the restriction of a distribution kernel to the open dense Bruhat cell in , which is isomorphic to , defines an isomorphism onto a subspace of \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)} (see Kobayashi–Speh [10, Theorem 3.16] or Theorem 2.2 for details). Let \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+}, resp. \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{-}, denote the space of distribution kernels defining intertwining operators in with , resp. .
The space \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+} was classified by Kobayashi–Speh [10], and we briefly describe this classification, borrowing their notation. First, they construct a meromorphic family of distributions K^{\mathbb{A},+}_{\lambda,\nu}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+}, , given by
[TABLE]
Then they show that every distribution in \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+} is given by or a regularization of it. By a detailed analysis of the meromorphic nature, the poles and all possible residues of the family they obtain a complete description of \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+} for all . More precisely, let
[TABLE]
then the corresponding statement for symmetry breaking operators is: For we have
[TABLE]
and every intertwining operator is given by the distribution kernel or a regularization of it.
We apply the translation principle to the kernels to obtain a meromorphic family K_{\lambda,\nu}^{\mathbb{A},-}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{-} given by
[TABLE]
In Theorem 4.7 we derive from the poles and residues of all poles and all possible residues of the family and use them to classify intertwining operators. For the statement let
[TABLE]
Theorem C** (see Theorems 4.7 and 4.10).**
For we have
[TABLE]
and every intertwining operator is given by the distribution kernel or a regularization of it.
We remark that for , , there exists a residue of which is supported at the origin in , inducing a differential intertwining operator. In this setting and , and the differential intertwining operators form the family of odd order conformally invariant differential operators C^{\infty}\big{(}S^{n},\mathcal{V}_{\lambda}\big{)}\to C^{\infty}\big{(}S^{n-1},\mathcal{W}_{\nu}\big{)} studied previously by Juhl [5] (see also [9]). The even order Juhl operators were already obtained as residues of the family by Kobayashi–Speh [10].
1.4 Symmetry breaking operators for rank one pin groups
The second (and more involved) illustration of the translation principle is for the symmetric pair \big{(}\widetilde{G},\widetilde{H}\big{)}=(\textup{Pin}(n+1,1),\textup{Pin}(n,1)). Here denotes a certain double cover of the group (see Appendix A for details) so that we have compatible double covers and with and as in the previous section. The preimages of the parabolic subgroups and under the covering maps are and with , , and as in the previous section and
[TABLE]
The fundamental representations of the Lie algebra of are the exterior power representations and the spin representations. Symmetry breaking operators for the exterior power representations have been investigated in detail by Kobayashi–Speh [11] (see also Fischmann–Juhl–Somberg [4] and Kobayashi–Kubo–Pevzner [8] for the case of differential symmetry breaking operators). Here we focus on the fundamental spin representations.
The group can be realized inside the Clifford algebra with generators (see Appendix A for details). The fundamental spin representations of are the restrictions of irreducible representations of the complex Clifford algebra to , and therefore we do not distinguish between the representations of and their restrictions to . For even the Clifford algebra has a unique irreducible representation, and for odd it has two inequivalent irreducible representations. Let be an irreducible representation of and an irreducible representation of . If sgn denotes the non-trivial representation of , we have for all and principal series representations
[TABLE]
and we study intertwining operators in the space
[TABLE]
As above, taking distribution kernels and restricting them to the open dense Bruhat cell in identifies the space of intertwining operators with a subspace
[TABLE]
where the sign represents and the sign represents .
The translation principle applied to the distribution kernels K_{\lambda,\nu}^{\mathbb{A},\pm}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{\pm} yields meromorphic families of distribution kernels P\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1})\big{)}_{\lambda,\nu}^{\pm} given by
[TABLE]
where is the value of the representation of the Clifford algebra at the vector and . (Note that the spin representation of occurs in the restriction with multiplicity one, where denotes the determinant character.)
To state our result on the classification of intertwining operators, let
[TABLE]
Theorem D** (see Theorems 5.3, 5.4 and 6.5).**
For we have
[TABLE]
and every intertwining operator is given by the distribution kernel or a regularization of it. 2.
For we have
[TABLE]
and every intertwining operator is given by the distribution kernel or a regularization of it.
In Theorems 5.3 and 5.4 we determine all poles and residues of the meromorphic families , and hence give an explicit description of the distribution kernels of all intertwining operators between spinor-valued principal series representations.
We remark that for , resp. , , there exists a residue of , resp. , which is supported at the origin in , inducing a differential intertwining operator. These families of spinor-valued differential intertwining operators were previously obtained by Kobayashi–Ørsted–Somberg–Souček [9], and it was conjectured that these are all differential intertwining operators in this setting. Our classification confirms this conjecture (see Remark 5.6).
In contrast to the proof of Theorem C which only uses the translation principle, we employ the method developed in [13] for the proof of Theorem D. This method describes intertwining operators between the underlying Harish-Chandra modules of principal series representations in terms of their action on the different -types. The explicit knowledge of the action on -types also allows us to determine the dimensions of intertwining operators between the irreducible constituents of and at reducibility points.
The representation is reducible if and only if \lambda=\pm\big{(}\rho+\frac{1}{2}+i\big{)}, . More precisely, for the representation has a finite-dimensional irreducible subrepresentation and the quotient is irreducible. The composition factors at can be described in terms of and by tensoring with the determinant character (see Lemma 6.6 for the precise statement). We use the analogous notation for the composition factors and of at , .
Theorem E** (see Theorem 6.7).**
For and the multiplicities are given by
[TABLE]
[TABLE]
We remark that the multiplicities are the same as in the case of spherical principal series (see [10, Theorem 1.2]).
1.5 Relation to conformal geometry
Let be a connected oriented Riemannian manifold of dimension with a spin structure. Let denote the conformal group of , i.e., the group of diffeomorphisms such that there exists a conformal factor with for all and . Write for the character of which takes the value on orientation preserving diffeomorphisms and on orientation reversing diffeomorphisms. Let denote the spin bundle and its smooth sections, also called spinors. For a spinor and a diffeomorphism the pullback can in general not be defined unambiguously, but there exists a double covering and a smooth action of on which resolves this ambiguity. Abusing notation, we lift the orientation character and the conformal factor to the double cover . This gives rise to a family of representations of on depending on two parameters and given by
[TABLE]
For with the Euclidean metric the group is essentially and the definition of agrees with previous definition since .
For an oriented submanifold we may consider the group which acts conformally on . Fixing a spin structure on we denote by the space of spinors and by (, ) the corresponding representations of on .
In this context it is natural to ask for a construction and classification of (differential) operators such that for all . The analogous question for differential forms was previously discussed by Kobayashi–Kubo–Pevzner [8] and Kobayashi–Speh [11]. In the model case (X,Y)=\big{(}S^{n},S^{n-1}\big{)} a complete classification for differential forms was obtained in [4, 8, 11], and our results in Theorem D can be viewed as the analogous classification for spinors. We also refer to [2] for the case .
1.6 Structure of the paper
In Section 2 we fix the notation for (generalized) principal series representations of real reductive groups, explain how to describe intertwining operators between them in terms of invariant distributions, and recall the construction of Knapp–Stein type intertwining operators for symmetric pairs from our joint work with Y. Oshima [14]. Section 3 explains the idea of the translation principle in detail and contains the proofs of Theorem A and Corollary B. We then apply the translation principle in two different situations. Firstly, in Section 4 we construct and classify intertwining operators between principal series representations of induced from one-dimensional representations (see Theorems 4.7 and 4.10 for a detailed version of Theorem C). Secondly, in Section 5 we construct intertwining operators between principal series representations of \big{(}\widetilde{G},\widetilde{H}\big{)}=(\textup{Pin}(n+1,1),\textup{Pin}(n,1)) induced from spin-representations (see Theorems 5.3 and 5.4). To also obtain a classification in the second situation, we employ in Section 6 the method developed in [13], yielding the classification results in Theorems 6.5 and 6.7. Together with Theorems 5.3 and 5.4 this proves Theorems D and E. Finally, Appendix A contains some elementary material about Clifford algebras and spin representations needed in Sections 5 and 6.
Notation. , , .
2 Preliminaries
We recall the basic facts about (generalized) principal series representations of real reductive groups, symmetry breaking operators, and their construction for symmetric pairs. More details can be found in [10, 14].
2.1 Generalized principal series representations
Let be a real reductive group and a parabolic subgroup with Langlands decomposition . We write , , and for the Lie algebras of , , and . Then and . Let denote the nilradical opposite to and the corresponding closed connected subgroup of . Then is the parabolic subgroup opposite to .
The multiplication map is a diffeomorphism onto an open dense subset of , and for we define , , and by .
We consider representations of which are parabolically induced from finite-dimensional representations of . Let be a finite-dimensional representation of , and denote by the trivial representation of , then is a finite-dimensional representation of . Write for half the sum of all positive roots and let . We define the generalized principal series representation \textup{Ind}^{G}_{P}\big{(}\xi\otimes e^{\lambda}\otimes\mathbf{1}\big{)} as the left-regular representation on the space
[TABLE]
If we write
[TABLE]
for the homogeneous vector bundle associated to the representation of , then can be identified with the space of smooth sections of .
2.2 Distribution sections of vector bundles
Let , where is the contragredient representation of , and write for the dual bundle of . We define the space of distribution sections of the bundle as the (topological) dual of :
[TABLE]
Note that since is compact, smooth sections on are automatically compactly supported. Then embeds -equivariantly into by , where
[TABLE]
and denotes the normalized Haar measure on .
Let be another finite-dimensional representation of and the corresponding vector bundle over . Then every smooth section defines a continuous linear multiplication operator
[TABLE]
which is dual to the composition
[TABLE]
of the pointwise tensor product and the push-forward by the contraction map .
2.3 Symmetry breaking operators
Now let be a reductive subgroup of and a parabolic subgroup of . Similarly, we define for a finite-dimensional representation of and as the left-regular representation of on , where , . Then identifies with the space of smooth sections of the vector bundle .
In this paper we study continuous -intertwining operators
[TABLE]
By the Schwartz kernel theorem such maps are identified with -invariant distribution sections of some vector bundle over . Using the isomorphism which is induced by , , invariant distributions on reduce to invariant distributions on (see [10, Section 3.2]):
Proposition 2.1** ([10, Proposition 3.2]).**
We have natural isomorphisms of vector spaces
[TABLE]
Under the isomorphism an -intertwining operator A\colon\textup{Ind}^{G}_{P}\big{(}\xi\otimes e^{\lambda}\otimes\mathbf{1}\big{)}\to\textup{Ind}^{H}_{P_{H}}\big{(}\eta\otimes e^{\nu}\otimes\mathbf{1}\big{)} maps to the distribution kernel such that
[TABLE]
where is the contraction map, and the integral has to be understood in the distribution sense.
Sometimes it is convenient to work on the open dense Bruhat cell in , because it is isomorphic to the vector space . Under certain conditions on the parabolic subgroups and the restriction of the invariant distribution sections in Proposition 2.1 to the open dense Bruhat cell is injective:
Theorem 2.2** ([10, Theorem 3.16]).**
Assume that
[TABLE]
If additionally then the restriction to defines a linear isomorphism
[TABLE]
Here the action of on is the obvious action induced by the actions of on , and , and the action of is induced by the infinitesimal action on viewed as subset of the generalized flag variety .
2.4 Symmetric pairs and Knapp–Stein type intertwining operators
For symmetric pairs we provided in [14] explicit expressions of invariant distributions defining symmetry breaking operators, which we briefly recall.
Let be an involution of and let be an open subgroup of , the fixed points of . Then forms a symmetric pair. We make the following two additional assumptions:
[TABLE]
Then (G) implies , where is a representative of the longest Weyl group element . Further, (H) implies that is a parabolic subgroup of .
Recall the -projection from Section 2.1 which is defined on the open dense subset . For sufficiently positive we define
[TABLE]
Since the second factor might not be defined for any , we make the additional assumption
[TABLE]
In [14, Proposition 2.5] we showed that in this case the domain of definition for is already open and dense in , and gave a criterion to check this.
In [14] we studied the meromorphic continuation of in the parameters , and proved that they give rise to intertwining operators between spherical principal series:
Theorem 2.3** ([14, Theorems 3.1 and 3.3]).**
Under the assumptions (G), (H) and (D), the functions extend to a meromorphic family of distributions
[TABLE]
where
[TABLE]
Therefore, they give rise to a meromorphic family of intertwining operators
[TABLE]
3 The translation principle
We describe a technique, called the translation principle, which allows to obtain new symmetry breaking operators from existing ones by tensoring with finite-dimensional representations of .
3.1 General technique
Fix a principal series representation \textup{Ind}_{P}^{G}\big{(}\xi\otimes e^{\lambda}\otimes\mathbf{1}\big{)} of and let be a finite-dimensional representation of , then there is a natural -equivariant isomorphism
[TABLE]
When we view both sides as -valued functions on , this isomorphism is given by
[TABLE]
Now, for any -stable subspace we have a natural injective map
[TABLE]
Suppose that acts trivially on and acts by a fixed character , , then the -action on can be written as . The above map becomes
[TABLE]
Assuming irreducibility of and , there is essentially only one choice of such a -stable subspace :
Lemma 3.1**.**
For every irreducible finite-dimensional representation of the restriction contains a unique irreducible subrepresentation . Moreover, is generated by the highest weight space of , and acts on by , where is an irreducible representation of , and the trivial representation of .
Proof.
It suffices to treat the case of connected since meets all connected components of . We first fix some notation. Let be a Cartan subalgebra, then is a Cartan subalgebra of and is a Cartan subalgebra of . Choose any system of positive roots such that the non-zero restrictions of positive roots to are the roots of . Then the non-zero restrictions of positive roots in to form a positive system of roots . We consider highest weights with respect to these positive systems.
Now let be any irreducible subrepresentation for . Since is reductive, decomposes into the direct sum of irreducible -representations of highest weight . Since and commute, acts by a character on . Now, let such that is maximal among the , . Then it is easy to see that is trivial on . Hence, is stable under . Since was assumed to be irreducible for , we have and hence acts trivially on .
To show that is unique, we simply observe that a highest weight vector for the action of on is automatically a highest weight vector for the action of on which is unique (up to scalar multiples). Hence is the -subrepresentation of generated by the highest weight space. ∎
Further, in the case of minimal parabolic subgroups, essentially every irreducible finite-dimensional representation of extends to :
Lemma 3.2** ([16, Theorem 2.1]).**
Assume that is a linear connected reductive Lie group and is minimal parabolic. Then every irreducible finite-dimensional representation of is conjugate via the Weyl group to a representation that occurs as a direct summand in an irreducible finite-dimensional representation of and on which acts trivially.
Similarly, for a fixed principal series representation we have an isomorphism
[TABLE]
We also take a -quotient space on which acts trivially and acts by a character . Note that such a quotient always exists since the contragredient representation of possesses a -stable subspace on which acts trivially by Lemma 3.1. However, in contrast to Lemma 3.1, there might be several possibilities for since might not be irreducible. Denoting the -action on by , we get a map
[TABLE]
Now suppose that an -intertwining operator
[TABLE]
is given, and form the tensor product
[TABLE]
Then we obtain an -intertwining operator
[TABLE]
by composing the maps (3.2), (3.1), (3.5), (3.3) and (3.4), namely
[TABLE]
This proves:
Theorem 3.3**.**
Let be a finite-dimensional -representation, a -stable subspace with and a -equivariant quotient with . Then (3.6) defines a linear map
[TABLE]
for all finite-dimensional representations of and of and all , .
3.2 Integral kernels
Recall that -intertwining operators are given by distribution kernels (see Proposition 2.1). Let us see how the integral kernel behaves under the translation principle. Suppose that
[TABLE]
is given by a distribution kernel in the sense that
[TABLE]
where denotes the contraction map and the integral is meant in the distribution sense (see Section 2.3 for details). Write for the inclusion map and for the quotient map, and let and denote the corresponding homogeneous vector bundles over and . Let , then is given by
[TABLE]
This implies:
Proposition 3.4**.**
The integral kernel \Phi\big{(}K^{A}\big{)}=K^{\Phi(A)} of is given by
[TABLE]
where is given by
[TABLE]
Remark 3.5**.**
Since the operator is given by multiplication with the fixed smooth function , and the multiplication map
[TABLE]
is continuous, the operator maps holomorphic families of distributions to holomorphic families. More precisely, if depends holomorphically on , then depends holomorphically on . More generally, if depends meromorphically on with poles in the set , then depends meromorphically on and its poles are contained in . However, it may of course happen that has a pole at whereas is regular at since the multiplication map can have a non-trivial kernel (see, e.g., Remark 4.9). This also implies that a holomorphic/meromorphic family , which does not vanish identically, might be mapped to for all (see, e.g., Remark 4.5). If, however, the family has generically full support, i.e., for generic , then cannot be identically zero for all . In fact, is an analytic function which is non-zero due to the irreducibility of , so it has full support and hence
[TABLE]
Remark 3.6**.**
Since is a matrix coefficient of a finite-dimensional repesentation, it is obviously smooth in . In view of Theorem 2.2 one can in some cases study the distribution kernels by their restriction to . On the function is actually a polynomial. In fact, the nilpotency of implies that there exists such that for all and . This shows that is a polynomial of degree at most .
3.3 Reformulation using
We can also use the opposite parabolic subgroup instead of in the above procedure. Assume there exists an element such that and . Then we have a -equivariant isomorphism
[TABLE]
Here, denotes the representation of on given by \big{(}\tilde{w}_{0}^{-1}\xi\big{)}(m)=\xi\big{(}\tilde{w}_{0}m\tilde{w}_{0}^{-1}\big{)}. Suppose that an -intertwining operator
[TABLE]
is given. Composing with the above ismorphism we have
[TABLE]
Then in a similar way, using instead of , we obtain an -intertwining operator
[TABLE]
for every -stable subspace with and every -stable quotient . Composing with the map , we get
[TABLE]
Moreover, if is given by the distribution kernel , then we similarly see that has distribution kernel \Psi\big{(}K^{A}\big{)}=K^{\Psi(A)} given by
[TABLE]
Now assume additionally that with and , then and can be chosen compatibly. More precisely, let be the unique irreducible -subrepresentation (see Lemma 3.1), then is the lowest -weight space and . Write for the direct sum of all other -weight spaces, then and the projection is -equivariant. Hence we can take :
Corollary 3.7**.**
Assume that and that and . Let be an irreducible finite-dimensional -representation and the unique irreducible -subrepresentation with . Then for all finite-dimensional representations of and of and all , we obtain a linear map
[TABLE]
which maps an intertwining operator with distribution kernel to the intertwining operator with distribution kernel given by
[TABLE]
where is given by
[TABLE]
Proof.
It remains to show the formula for . Write
[TABLE]
then we have
[TABLE]
Since acts trivially on we have for , and similarly for . Hence
[TABLE]
Remark 3.8**.**
We note that the function resembles the integral kernel of the standard Knapp–Stein intertwining operator
[TABLE]
More precisely, it is shown in [6, Chapter VII, Section 7] that
[TABLE]
3.4 Application to Knapp–Stein type intertwining operators
We now specialize to the setting of Section 2.4. In this case, all assumptions of Corollary 3.7 are satisfied and we can apply it to the family of intertwining operators
[TABLE]
and given by (2.1), with distribution kernels
[TABLE]
We call an irreducible finite-dimensional representation of extendible if there exists a finite-dimensional irreducible representation of such that
[TABLE]
Corollary 3.9**.**
Assume conditions (G), (H) and (D). Then for every extendible irreducible finite-dimensional -representation the functions
[TABLE]
extend to a meromorphic family of distributions
[TABLE]
for and given by (2.1) and \mathcal{V}_{\lambda}=G\times_{P}\big{(}\tilde{w}_{0}\xi\otimes e^{\lambda+\rho}\otimes\mathbf{1}\big{)}, . Therefore, they give rise to a meromorphic family of intertwining operators
[TABLE]
Proof.
Let be an irreducible finite-dimensional representation of with as -representations, then is -stable of the form . Now the statement follows from Theorem 2.3 and Corollary 3.7. ∎
Remark 3.10**.**
For one can choose a representative of that centralizes . Hence, for all irreducible finite-dimensional representations of . By Lemma 3.2 either or is extendible, so that in this case all irreducible finite-dimensional -representations are extendible.
4 Example:
We apply the translation principle to the symmetry breaking operators between spherical principal series of rank one orthogonal groups studied by Kobayashi–Speh [10] and obtain new symmetry breaking operators between non-spherical scalar-valued principal series.
4.1 Parabolic subgroups and the symmetric pair
Let , , realized as the subgroup of preserving the indefinite bilinear form
[TABLE]
We choose the (minimal) parabolic subgroup such that with
[TABLE]
and for with . Then
[TABLE]
Let and identify with , then . We further identify by
[TABLE]
and use this identification to parametrize by
[TABLE]
The group acts on by the adjoint action as follows:
[TABLE]
Further, is conjugate to its opposite parabolic by
[TABLE]
Let us identify by , so that .
Lemma 4.1**.**
For :
[TABLE]
and in this case for we have
[TABLE]
Now let be the involution of given by conjugation with the matrix . Then and forms a symmetric pair. It is easy to see that the pair satisfies the assumptions in Theorem 2.2 and we write . Then
[TABLE]
Further, under the identification the involution acts by
[TABLE]
and therefore the subalgebra is given by the standard embedding of into as the first coordinates.
4.2 Principal series representations and symmetry breaking operators
Denote by the character of given by for and . Abusing notation we also write sgn for the corresponding character of . For and we define the scalar principal series representations (smooth normalized parabolic induction)
[TABLE]
We consider the space of symmetry breaking operators between and . By Theorem 2.2 every such operator is given by a distribution kernel satisfying certain invariance conditions for the action of and . Since acts by , it is clear that only depends on the parity of . Identifying as above we write \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+}, resp. \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+}, for the space \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}^{M_{H}A_{H},\mathfrak{n}_{H}} with , resp. . Then, taking distribution kernels is a linear isomorphism
[TABLE]
4.3 Construction of symmetry breaking operators
In this section we describe all intertwining operators in for all , . We note that the notation is essentially due to Kobayashi–Speh [10].
4.3.1 Spherical principal series
For the representations and are spherical, i.e. possess a vector invariant under a maximal compact subgroup. In this setting, Section 2.4 provides a meromorphic family of intertwining operators given by the distribution kernels
[TABLE]
By analyzing the poles and residues of explicitly, Kobayashi–Speh [10] completely determine the space \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+} for all . (Note that our parameters are normalized such that and are unitary for . Therefore, in our notation one has to replace by to obtain Kobayashi–Speh’s notation, see [10].) To summarize their results let
[TABLE]
For with we further define
[TABLE]
and for with we put
[TABLE]
Note that both and depend holomorphically on (or ).
Theorem 4.2**.**
The renormalized distribution
[TABLE]
depends holomorphically on and vanishes only for . More precisely,
For we have . 2.
For with , , we have
[TABLE]
In particular, . 3.
For with , , we have
[TABLE]
In particular, . 4.
For , , restricting the function to two different complex hyperplanes in through and renormalizing gives two holomorphic families of distributions:
[TABLE]
Their special values at satisfy
[TABLE]
more precisely, for odd and
[TABLE]
Proof.
This is a summary of the results of [10]. ∎
These results can be used to describe the space \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+}:
Theorem 4.3**.**
We have
[TABLE]
Proof.
See [10, Theorem 1.9] and also [13, Theorem 4.9]. ∎
4.3.2 Scalar principal series
We use the translation principle to describe \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{-} in terms of \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+}.
Let be the defining representation of on . Then
[TABLE]
is the maximal subspace on which acts trivially, and . For the -quotient space we choose
[TABLE]
then . Now let us compute the -valued function on . For , , we have
[TABLE]
Hence, by Theorem 3.3, we get a linear map
[TABLE]
Applying this map to the meromorphic family of distributions we obtain another meromorphic family
[TABLE]
To normalize so that it becomes holomorphic in we first consider the kernel of the map
[TABLE]
Lemma 4.4**.**
We have
[TABLE]
Note that for (\lambda,\nu)=\big{(}{-}\frac{1}{2},0\big{)} we have .
Proof.
If K\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+} such that then clearly . Hence,
[TABLE]
for some distributions u_{m}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n-1}\big{)}. Since we must have for , so that . By the classification in Theorems 4.2 and 4.3 the only possibilities for such in the space \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{+} are
[TABLE]
with (i.e., ) and
[TABLE]
with (i.e., ). ∎
Remark 4.5**.**
Lemma 4.4 implies that the families
[TABLE]
which depend holomorphically on and are not identically zero, are mapped to identically zero families by (4.1), i.e., for all . However, the family which depends holomorphically on has generically full support, i.e., for generic , so it follows from Remark 3.5 that also for generic . In particular, the holomorphic family is not identically zero.
For the full classification of \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{-} we also need to understand the kernel of the map
[TABLE]
Lemma 4.6**.**
We have
[TABLE]
Proof.
As in the previous proof any with must be of the form . Now, if K\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{-} then by the invariance under we have and since this implies
[TABLE]
On the other hand, invariance of under implies for all , in particular for , hence
[TABLE]
This shows that and the proof is complete. ∎
To state the analogue of Theorem 4.2 for let
[TABLE]
For with we further define
[TABLE]
and for with we put
[TABLE]
Theorem 4.7**.**
The renormalized distribution
[TABLE]
depends holomorphically on and vanishes only for . More precisely,
For we have . 2.
For with , , we have
[TABLE]
In particular, . 3.
For with , , we have
[TABLE]
In particular, . 4.
For , , restricting the function to two different complex hyperplanes in through and renormalizing gives two holomorphic families of distributions:
[TABLE]
Their special values at satisfy
[TABLE]
more precisely, for odd and
[TABLE]
Proof.
We first apply the map
[TABLE]
to the holomorphic family and obtain a holomorphic family of distributions
[TABLE]
By Theorem 4.2 and Lemma 4.4 we have if or . We can therefore renormalize the kernels
[TABLE]
This shows that depends holomorphically on . Next, we apply the map
[TABLE]
to and find
[TABLE]
By Lemma 4.6 the map (4.2) is injective, hence if and only if , which is only the case for , i.e., .
Let . Then and hence \operatorname{supp}\big{(}x_{n}\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}\big{)}=\operatorname{supp}\big{(}\widetilde{K}^{\mathbb{A},+}_{\lambda+1,\nu}\big{)}=\mathbb{R}^{n}. This implies \operatorname{supp}\big{(}\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}\big{)}=\mathbb{R}^{n}. 2. 2.
Let with , then it is easy to see, using , that
[TABLE]
Hence,
[TABLE] 3. 3.
For the same method as in (2) applies to show (3). Here we use that for . 4. 4.
Now let . The first statement about follows immediately from (3). For the second statement note that and we can use Theorem 4.2(4). Restricting to we have
[TABLE]
The denominator is equal to \tfrac{1}{2}\big{(}\lambda+\nu-\tfrac{1}{2}\big{)}=-\frac{1}{2}(n+i+j)=-(k+1)\neq 0, whence {\mathchoice{\widetilde{\hbox{\set@color\displaystyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\textstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptscriptstyle\widetilde{K}}}}}^{\mathbb{A},-}_{\lambda,\nu} depends holomorphically on . Moreover, for even we have \operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color\displaystyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\textstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptscriptstyle\widetilde{K}}}}}^{\mathbb{A},+}_{\lambda-1,\nu}=\mathbb{R}^{n} so that \operatorname{supp}\big{(}x_{n}{\mathchoice{\widetilde{\hbox{\set@color\displaystyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\textstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptscriptstyle\widetilde{K}}}}}^{\mathbb{A},+}_{\lambda-1,\nu}\big{)}=\mathbb{R}^{n} and hence \operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color\displaystyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\textstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptscriptstyle\widetilde{K}}}}}^{\mathbb{A},-}_{\lambda,\nu}=\mathbb{R}^{n}. For odd it follows from that
[TABLE]
Remark 4.8**.**
The differential intertwining operators belonging to the distribution kernels are the even order conformally covariant differential operators found by Juhl [5], sometimes also referred to as Juhl operators. Kobayashi–Speh showed that they are obtained as residue families of the non-local intertwining operators with kernels . Theorem 4.7 proves that also the odd order Juhl operators with integral kernels can be obtained in this way. Further, the translation principle explains the following relation between even and odd order Juhl operators:
[TABLE]
Remark 4.9**.**
Theorem 4.2 implies that the unnormalized meromorphic family has poles for . Multiplication with gives the meromorphic family which, by Theorem 4.7, has poles for . This shows that for and the family has a pole, while does not.
Using the translation principle we can finally determine the space \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{-} completely:
Theorem 4.10**.**
We have
[TABLE]
Proof.
By Lemma 4.6 the map
[TABLE]
is injective and hence
[TABLE]
For we have 0\neq\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{-}, and hence the left hand side of (4.3) is . For we have {\mathchoice{\widetilde{\hbox{\set@color\displaystyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\textstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptscriptstyle\widetilde{K}}}}}^{\mathbb{A},-}_{\lambda,\nu},\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{-} and these distributions are linearly independent since
[TABLE]
This shows that the left hand side of (4.3) is . Now note that if and only if , and therefore (4.3) is actually an equality for all , thanks to Theorem 4.3. This finishes the proof. ∎
5 Example:
We explicitly construct intertwining operators between principal series representations of and induced from fundamental spin representations of and using the translation principle.
5.1 Parabolic subgroups and the symmetric pair
We consider the two-fold covering
[TABLE]
realized inside the Clifford algebra of . For details on the definition and properties of the pin groups and Clifford algebras we refer the reader to Appendix A.1. In what follows we will write for any subgroup .
We choose the same parabolic subgroup as in Section 4.1, then , where the embedding of into is the restriction of the embedding of Clifford algebras induced by
[TABLE]
We note that commutes with and hence . Both and split in the cover, so that and , and we identify and with . Then is the Langlands decomposition of the parabolic subgroup of .
For the Weyl group element we choose the representative
[TABLE]
with the sign yet to be determined. Since , the element corresponds to the representative chosen in Section 4.1 for the group . Hence, by Lemma 4.1 we find
[TABLE]
The matrix is the orthogonal reflection in at the hyperplane orthogonal to , and the corresponding elements in the cover are . Therefore, we may choose the sign in (5.1) so that under the identification we have
[TABLE]
Further we note that
[TABLE]
where denotes the canonical automorphism of the Clifford algebra and the determinant character of .
The symmetric pair as introduced in Section 4.1 takes in the cover the form \big{(}\widetilde{G},\widetilde{H}\big{)}=(\textup{Pin}(n+1,1),\textup{Pin}(n,1)). We have
[TABLE]
where the embedding of into is induced by the embedding
[TABLE]
Further, is a parabolic subgroup of with .
5.2 Principal series representations and symmetry breaking operators
For the fundamental spin representations of we use the notation introduced in Appendix A.2. The group has for even one fundamental spin representation, and for odd two fundamental spin representations. For even we have for the fundamental spin representation , and for odd and any fundamental spin representation of the representations and are non-equivalent.
Denote by the non-trivial character of the two-element group . Then for and a fundamental spin representation of we consider the representation of . Note that
[TABLE]
For a fundamental spin representation of , and we form the principal series representations
[TABLE]
Similarly, for a fundamental spin representation of , and we form principal series representations of :
[TABLE]
The main result of this section is the construction of symmetry breaking operators
[TABLE]
In Section 6 we then show that this construction actually gives a full classification of symmetry breaking operators between spinor-valued principal series.
By Theorem 2.2 every intertwining operator in (5.3) is uniquely determined by its distribution kernel K\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1})\big{)}. As explained in Section 4.2, the space of distribution kernels describing intertwining operators in (5.3) only depends on and the parity of . We therefore denote by
[TABLE]
the corresponding space of distribution kernels of intertwining operators, where the sign describes kernels for and the sign describes kernels for .
5.3 Construction of symmetry breaking operators
We extend the representation of to a representation of the Clifford algebra and write for short (see Appendix A.2). Consider the representation of . By Lemma 3.2 there exists a representation of containing a subspace invariant under such that or for some . By possibly replacing by its twist by the Cartan involution we may assume that . It is easy to see that . By (5.2) we have
[TABLE]
Further, using Lemma 4.1 we have the following expression for the translation kernel:
[TABLE]
Then Corollary 3.7 gives linear maps
[TABLE]
which are on the level of integral kernels given by
[TABLE]
To obtain intertwining operators into note that occurs in the restriction with multiplicity one, and fix a projection . Then we have a linear map
[TABLE]
which is on the level of integral kernels given by
[TABLE]
By Theorems 4.3 and 4.10 all symmetry breaking operators in are obtained from the meromorphic family of kernels K^{\mathbb{A},\pm}_{\lambda,\nu}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{\pm}. We therefore consider the -valued kernels
[TABLE]
For odd the restriction of the -representation to is isomorphic to , so the map is an isomorphism. Therefore, we can as well study the -valued distributions
[TABLE]
For even the restriction of the -representation to is isomorphic to the direct sum of and . The next result shows that the poles of P\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1})\big{)} and \not{K}_{\lambda,\nu}^{\mathbb{A},\pm}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n};\textup{End}_{\mathbb{C}}(\SS)\big{)} agree and that the residues of agree with the composition of the residues of with . Note that
[TABLE]
where .
Proposition 5.1**.**
Assume is even and let u\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n};\textup{End}_{\mathbb{C}}(\SS_{n})\big{)} be a distribution of the form
[TABLE]
with u_{i}\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}. Then , in particular if and only if .
Proof.
We have
[TABLE]
We claim that the operators are linearly independent, then the statement follows.
Decompose into irreducible -representations , where is an isomorphism and . Now note that for we have
[TABLE]
and therefore vanishes on . On the other hand, maps to , so that vanishes on . Now assume that
[TABLE]
Restricting to this implies
[TABLE]
and hence since are linearly independent. Then also and the proof is complete. ∎
To find the right normalization making holomorphically dependent on we first study the kernel in \mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)}_{\lambda,\nu}^{\pm} of the map (5.4). Note that , and the operators are linearly independent. Hence, for any distribution u\in\mathcal{D}^{\prime}\big{(}\mathbb{R}^{n}\big{)} we have
[TABLE]
and
[TABLE]
Lemma 5.2**.**
For we have
[TABLE]
Proof.
It is easy to see that multiples of are the only distributions with for all . Then the claim follows from Theorems 4.2, 4.3, 4.7 and 4.10. ∎
To state the analogues of Theorems 4.2 and 4.7 for we write and for the operators
[TABLE]
5.3.1 The distributions
Let
[TABLE]
For with we further define
[TABLE]
and for with we put
[TABLE]
Note that both and depend holomorphically on (or ).
Theorem 5.3**.**
The renormalized distribution
[TABLE]
depends holomorphically on and vanishes only for . More precisely,
For we have . 2.
For with , , we have
[TABLE]
In particular, . 3.
For with , , we have
[TABLE]
In particular, . 4.
For \big{(}{-}\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j\big{)}\in\not{L}_{\textup{even}}, , restricting the function to two different complex hyperplanes in through \big{(}{-}\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j\big{)} and renormalizing gives two holomorphic families of distributions:
[TABLE]
Their special values at (\lambda,\nu)=\big{(}{-}\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j\big{)} satisfy
[TABLE]
more precisely, for odd and
[TABLE]
Proof.
By Theorem 4.7 the distribution
[TABLE]
depends holomorphically on , and by Lemma 5.2 it vanishes if and only if \big{(}\lambda-\frac{1}{2},\nu+\frac{1}{2}\big{)}\in L_{\textup{odd}}, i.e., . The rest is similar to the proof of Theorem 5.4 which we carry out in detail, some arguments are even easier since in this case we do not need to renormalize the kernel and we have
[TABLE]
5.3.2 The distributions
Let
[TABLE]
For with we further define
[TABLE]
and for with we put
[TABLE]
Note that both and depend holomorphically on (or ).
Theorem 5.4**.**
The renormalized distribution
[TABLE]
depends holomorphically on and vanishes only for . More precisely,
For we have . 2.
For with , , we have
[TABLE]
In particular, . 3.
For with , , we have
[TABLE]
In particular, . 4.
For \big{(}{-}\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j\big{)}\in\not{L}_{\textup{odd}}, , restricting the function to two different complex hyperplanes in through \big{(}{-}\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j\big{)} and renormalizing gives two holomorphic families of distributions:
[TABLE]
Their special values at (\lambda,\nu)=\big{(}{-}\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j\big{)} satisfy
[TABLE]
more precisely, for odd and :
[TABLE]
Proof.
By Theorem 4.2 the distribution
[TABLE]
depends holomorphically on , and by Lemma 5.2 it vanishes if and only if or \big{(}\lambda-\frac{1}{2},\nu+\frac{1}{2}\big{)}\in L_{\textup{even}}. Renormalizing shows that
[TABLE]
depends holomorphically on . To prove the remaining statements, note that
[TABLE]
and hence
[TABLE]
1. Let , then and hence . Now (5.6) implies .
2. Let with , then \big{(}\lambda-\frac{1}{2},\nu+\frac{1}{2}\big{)}\in\,\,\backslash\kern-8.00003pt{\backslash}\;^{+} and by Theorem 4.2(2) and using we have
[TABLE]
3. Let with , then \big{(}\lambda-\frac{1}{2},\nu+\frac{1}{2}\big{)}\in\,\,/\kern-8.00003pt{/}\;^{+} with \big{(}\lambda-\frac{1}{2})-(\nu+\frac{1}{2}\big{)}=-\frac{1}{2}-2(\ell+1) and by Theorem 4.2(3) and using \big{[}\Delta_{\mathbb{R}^{n-1}}^{j},x_{i}\big{]}=2j\Delta^{j-1}\frac{\partial}{\partial x_{i}} we have
[TABLE]
4. Now let \big{(}{-}\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j\big{)}\in\not{L}_{\textup{odd}}. The first statement about follows immediately from (3). For the second statement note that and we can use Theorem 4.2(4). Restricting to we have
[TABLE]
which clearly depends holomorphically on . Moreover, \operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color\displaystyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\textstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptscriptstyle\widetilde{K}}}}}^{\mathbb{A},+}_{-\rho-(i+1),-\rho_{H}-j}=\mathbb{R}^{n}, resp. , so that \operatorname{supp}x_{i}{\mathchoice{\widetilde{\hbox{\set@color\displaystyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\textstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptstyle\widetilde{K}}}}{\widetilde{\hbox{\set@color\scriptscriptstyle\widetilde{K}}}}}^{\mathbb{A},+}_{-\rho-(i+1),-\rho_{H}-j}=\mathbb{R}^{n}, resp. , for , and hence \operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color\displaystyle\widetilde{\not{K}}}}}{\widetilde{\hbox{\set@color\textstyle\widetilde{\not{K}}}}}{\widetilde{\hbox{\set@color\scriptstyle\widetilde{\not{K}}}}}{\widetilde{\hbox{\set@color\scriptscriptstyle\widetilde{\not{K}}}}}}^{\mathbb{A},-}_{-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j}=\mathbb{R}^{n}, resp. , by (5.5). For odd and we further have by Theorem 4.2(4)
[TABLE]
Remark 5.5**.**
The identity
[TABLE]
that was used in the previous proof can be explained by again applying the translation principle, now with the dual representation , and then projecting onto the trivial constituent in which is spanned by .
Remark 5.6**.**
The intertwining differential operators C^{\infty}\big{(}\mathbb{R}^{n};\SS\big{)}\to C^{\infty}\big{(}\mathbb{R}^{n-1},\SS\big{)} with integral kernel have been obtained before [9, Theorem 5.7] and it was conjectured that these operators exhaust the space of all intertwining differential operators. Theorem 6.5 confirms this conjecture.
6 The compact picture of symmetry breaking operators
between spinors
We use the method developed in [13] to show that the symmetry breaking operators found in Section 5 between spinor-valued principal series span the space of all symmetry breaking operators. The notation will be as in the previous section.
6.1 Reduction to the pair \boldsymbol{\big{(}\widetilde{G}_{1},\widetilde{H}_{1}\big{)}}
To simplify computations we first reduce the study of symmetry breaking operators for the pair \big{(}\widetilde{G},\widetilde{H}\big{)}=(\textup{Pin}(n+1,1),\textup{Pin}(n,1)) to the pair \big{(}\widetilde{G}_{1},\widetilde{H}_{1}\big{)}=(\textup{Pin}(n+1,1)_{1},\textup{Pin}(n,1)_{1}), where
[TABLE]
We refer the reader to Appendix A.1 for the definition of . Note that is a parabolic subgroup of with . Similarly, is a parabolic subgroup of with . Further, and , and hence
[TABLE]
Note that the restrictions are independent of and .
Lemma 6.1**.**
For and we have
[TABLE]
Proof.
The proof of the two identities is analogous to [8, Theorem 2.10] and uses and . ∎
In the remaining part of this section we determine using the technique developed in [13].
6.2 Harish-Chandra modules
Let denote the Cartan involution of given by conjugation with the matrix . Then lifts to a Cartan involution of which restricts to Cartan involutions of , and . This gives the following maximal compact subgroups:
[TABLE]
The embedding is given by the embedding induced from the standard embedding .
The method in [13] actually constructs and classifies intertwining operators between the underlying Harish-Chandra modules of and which we denote by and . As vector space is the space of all -finite vectors in , or equivalently the sum of all irreducible -subrepresentations of . Hence, clearly carries a -action. It is further invariant under the action of the Lie algebra of and hence carries the structure of a \big{(}\mathfrak{g},\widetilde{K}_{1}\big{)}-module.
Since is dense in , we obtain an injective map
[TABLE]
Using the method in [13] we determine in this section the dimension of the space of intertwining operators between the Harish-Chandra modules. This gives an upper bound for the dimension of the space of intertwining operators between the representations and . From the explicit construction of intertwining operators in Section 5 we also have a lower bound, and it turns out that these bounds agree, so that the injective map in fact is a bijection.
6.3 -types
We study the representations and in the compact picture, i.e. on sections of spin bundles over and . More precisely,
[TABLE]
By Frobenius Reciprocity and using the classical branching rules for spin groups we easily obtain
[TABLE]
where denote the higher spin representations of (see Appendix A.3). By Appendix A.4
[TABLE]
We use the notation and for even , and , for odd , and write if . Denote by the subspace of C^{\infty}\big{(}\widetilde{K}_{1}\times_{\widetilde{M}_{1}}\zeta_{n}\big{)} which is isomorphic to the -representation if and if , and similar for the corresponding subspace of C^{\infty}\big{(}\widetilde{K}_{H,1}\times_{\widetilde{M}_{H,1}}\zeta_{n-1}\big{)}. In this notation
[TABLE]
Further,
[TABLE]
where . We fix a non-zero -intertwining operator
[TABLE]
for each pair with .
6.4 Scalar identities for symmetry breaking operators
Write for the eigenspace decomposition of under and for the one of . We identify via
[TABLE]
Consider the map \omega\colon\mathfrak{s}_{\mathbb{C}}\to C^{\infty}\big{(}\widetilde{K}_{1}/\widetilde{M}_{1}\big{)}=C^{\infty}(S^{n}) given by
[TABLE]
Multiplication by defines an operator M(\omega(Y))\colon C^{\infty}\big{(}\widetilde{K}_{1}\times_{\widetilde{M}_{1}}\zeta_{n}\big{)}\to C^{\infty}\big{(}\widetilde{K}_{1}\times_{\widetilde{M}_{1}}\zeta_{n}\big{)}. From the weights of it is easy to see that maps to if and only if
[TABLE]
Write if this is the case.
We use the same notation for and and write for multiplication by the function , , . Further write if , and , .
For define
[TABLE]
Then both and are -intertwining operators . Since occurs in with multiplicity at most one, we have
[TABLE]
for some constant .
Now, every -intertwining operator is uniquely determined by its restriction to the subspaces on which it is given by
[TABLE]
for some scalars . In [13] it is proven that is -intertwining if and only if for all and we have
[TABLE]
Here the constants and are essentially the eigenvalues of the Casimir element on the -types and . From [1, Section 3.a] it follows that
[TABLE]
and similarly for . Hence, the only missing constants in (6.3) are the numbers which we now compute after choosing explicit operators .
6.5 Explicit embeddings of -types
We realize the -types explicitly inside .
Assume first that is even, then the -representation is equivalent to the restriction of the -representation to . Hence
[TABLE]
The underlying Harish-Chandra module is in this picture given by restrictions of -valued polynomials on to the sphere . Denote by \mathcal{M}_{i}\big{(}\mathbb{R}^{n+1},\SS_{n+1}\big{)} the space of monogenic polynomials of degree (see Appendix A.3 for details). By the Fischer decomposition (A.2) we have
[TABLE]
where we identify a homogeneous polynomial with its restriction to the unit sphere . Then \mathcal{M}_{i}\big{(}\mathbb{R}^{n+1},\SS_{n+1}\big{)} carries the representation and \underline{x}\mathcal{M}_{i}\big{(}\mathbb{R}^{n+1},\SS_{n+1}\big{)} carries the representation . In the notation of Section 6.3 this means
[TABLE]
Now let be odd, then the restriction of the -representation to decomposes into , where and . Hence
[TABLE]
Again, the underlying Harish-Chandra module is given by
[TABLE]
but in this case \mathcal{M}_{i}\big{(}\mathbb{R}^{n+1},\SS_{n+1}\big{)} and \underline{x}\mathcal{M}_{i}\big{(}\mathbb{R}^{n+1},\SS_{n+1}\big{)} are isomorphic -representations, more precisely they both are equivalent to . A short calculation using Lemma A.2 shows that the -types belonging to are given by
[TABLE]
where , being the map defined in (A.1) that intertwines with . In the notation of Section 6.3 this reads
[TABLE]
Using the same identifications for the -types we now fix for each pair a -intertwining operator
[TABLE]
as follows: For even set
[TABLE]
and for odd we put
[TABLE]
Then is a -equivariant isomorphism and we can define the -intertwining operator by . For this particular choice of operators we can now compute the proportionality constants defined in (6.1).
Lemma 6.2**.**
Let .
For even and we have
[TABLE] 2.
For odd and we have
[TABLE]
Proof.
Since the defining equation (6.1) for is equivalent to
[TABLE]
Let first be even and , . Then for , , and , , we have
[TABLE]
Hence,
[TABLE]
and
[TABLE]
Note that by Lemma A.1(2). On the other hand,
[TABLE]
and
[TABLE]
Then Lemma A.5 gives the constants for . The case and is treated similarly. The verification for odd is left to the reader. ∎
6.6 Multiplicities for symmetry breaking operators
between spinor-valued principal series
Inserting the explicit constants determined in Lemma 6.2 into the scalar identities (6.3) we obtain an explicit characterization of symmetry breaking operators in terms of scalar identities. We assume that is even for the statement of these identities, the case of odd is similar.
Corollary 6.3**.**
Assume is even. Then a -intertwining operator is \big{(}\mathfrak{h},\widetilde{K}_{H,1}\big{)}-intertwining if and only if the scalars defined by (6.2) satisfy the following three relations:
[TABLE]
As previously carried out in [13, Section 4.3] for similar relations, we solve this system to obtain the following result about multiplicities of symmetry breaking operators:
Theorem 6.4**.**
[TABLE]
Proof.
We assume is even, the case of odd is treated similarly. It is more convenient to work with the scalars
[TABLE]
then the identities in Corollary 6.3 become
[TABLE]
Note that each identity only involves scalars with one choice of sign , so that the system of equations degenerates into two systems of equations, one for and one for . Fixing a sign we have to find the dimension of the space of tuples satisfying the identities (6.4)–(6.6). Let us first visualize the --type picture in a diagram:
i$$j
Here the dot at position represents the scalar . Now, each identity is a linear relation between certain neighboring dots in this diagram, visualized as
Note that the only coefficients in the identities that can possibly vanish are those involving and . Identity (6.4) at gives
[TABLE]
It is easy to see that the dimension of the space of diagonal sequences satisfying (6.7) is equal to if and equal to if (see [13, Lemma 4.4] for the same argument in a similar setting).
Step 1. Now let us first assume that , in particular . Then identity (6.5) can be used to define in terms of and since the coefficient \big{(}\lambda+\rho+\frac{1}{2}+i\big{)} of never vanishes. This shows that each diagonal sequence uniquely determines the numbers for all and hence, for both signs the dimension of the space of sequences \big{(}s_{i,j}^{\pm}\big{)} satisfying (6.4)–(6.6) is .
Step 2. Now assume , , but . Then the extension argument from Step 1 using (6.5) still works for and hence any diagonal sequence uniquely determines the numbers for . Next, we can repeatedly use (6.4) with to define in terms of , and . Note that the coefficient \big{(}\nu+\rho_{H}+\frac{1}{2}+j\big{)} of never vanishes since and . Therefore, also in this case, for both signs the space of sequences satisfying (6.4)–(6.6) is one-dimensional.
Step 3. Finally assume (\lambda,\nu)=\big{(}{-}\rho-\frac{1}{2}-k,-\rho_{H}-\frac{1}{2}-\ell\big{)}\in\not{L}, . We only discuss the case where , the case is treated similarly. We divide the –-type picture into four regions accoding to whether or and or :
\dashline1(-2,9.5)(30,9.5) \dashline1(17.5,0)(17.5,28) *$$*[math][math][math]*$$*$$i$$j$$k$$k+1$$\ell$$\ell+1
Then from the diagonal identity (6.7) it follows that the diagonal entries in the upper left region are [math] whereas the diagonal entries in the upper right and the lower left region can be chosen independently (as indicated by the stars and zeros). This shows that the space of diagonal sequences satisfying (6.7) is two-dimensional. Using (6.5) as before shows that in fact all scalars in the upper left region have to be [math]. Now we study what happens near the intersection of the two dashed lines. For instance, we have two identities that relate and , namely (6.5) for :
[TABLE]
and (6.6) for :
[TABLE]
Let us first consider the positive sign scalars . In this case the coefficient of in (6.8) is
[TABLE]
so that (6.8) is a multiple of (6.9). Hence, the two identities (6.8) and (6.9) are dependent and do not force and to be [math]. As above one can uniquely extend any diagonal sequence to the whole lower left region with , . For the negative sign scalars the opposite is true: the relations (6.8) and (6.9) are independent and hence . Extending the zeros in the other direction forces all scalars in the lower left region to be [math]. The same happens for the upper right region, where (6.4) for and (6.5) for are two dependent resp. independent relations for and , resp. and . This shows that the space of scalars in the upper left, lower left and upper right region satisfying (6.4)–(6.6) is two-dimensional for and trivial for . It remains to extend such tuples to the lower right region. Assuming that we already have defined in the first three regions, there are two linear relations for and , namely (6.5) for :
[TABLE]
and (6.6) for :
[TABLE]
Here we write for the right hand side which is a linear combination of scalars from the other three regions which we already specified. Now, for the negative sign scalars the coefficient of in (6.10) is equal to
[TABLE]
and the two relations are dependent. This means that every choice of can be uniquely extended to , , using the extension methods from (1) and (2) (with if or ). Hence, the dimension of the space of tuples satisfying (6.4)–(6.6) is . For the positive sign scalars, it is easy to see that the relations (6.10) and (6.11) are independent, and therefore and are uniquely determined by the chosen scalars in the other three regions. Together with the extension techniques outlined in Steps 1 and 2 this implies that the dimension of the space of tuples satisfying (6.4)–(6.6) is . This proves the claim. ∎
6.7 Classification of symmetry breaking operators
between spinor-valued principal series
Combining Theorems 5.3, 5.4 and Proposition 5.1 with Lemma 6.1 and Theorem 6.4 we obtain a full classification of symmetry breaking operators between spinor-valued principal series representations:
Theorem 6.5**.**
We have
[TABLE]
6.8 Symmetry breaking operators at reducibility points
We study symmetry breaking operators between irreducible constituents of and at reducibility points. For this we first describe the irreducible constituents.
Lemma 6.6**.**
The representation is reducible if and only if . 2.
For , , the representation has a unique irreducible subrepresentation which is finite-dimensional, and the quotient is irreducible. 3.
For , , the representation has a unique irreducible subrepresentation isomorphic to an irreducible quotient isomorphic to , where is the determinant character.
Proof.
We first describe the -type decomposition of . The maximal compact subgroup of is a semidirect product of with the two-element group which acts on via the canonical automorphism of the Clifford algebra (see Appendix A.1 for details). As remarked in Section 6.3, the restriction of to decomposes into a multiplicity-free direct sum of higher spin representations. For even, it is the direct sum of , , and it is easy to see that extends uniquely to an irreducible representation of . For odd, the restriction of to is the direct sum of , , and there are two inequivalent ways of extending to an irreducible -representation. For fixed let denote the extension which occurs in . Then, for both even and odd we have
[TABLE]
The Lie algebra action of maps into the direct sum of , and . From [1] and the computations in Section 6.4 it follows that one can reach if and only if , and one can reach if and only if . Then statements (1) and (2) follow with
[TABLE]
To prove (3) observe that the standard intertwining operators in this situation (see, e.g., Remark 3.8)
[TABLE]
map irreducible quotients to irreducible subrepresentations. Since the character of extends to we have
[TABLE]
and the claim follows by specializing to \lambda=\pm\big{(}\rho+\frac{1}{2}+i\big{)}. ∎
Note that the representations and depend on the chosen spin-representation of that is induced from. However, as in the case of the full principal series, the multiplicities of intertwining operators turn out to be independent of .
Denote by and the corresponding composition factors of at \nu=\pm\big{(}\rho_{H}+\frac{1}{2}+j\big{)}, .
Theorem 6.7**.**
For the representations and the multiplicities are given by
[TABLE]
[TABLE]
Proof.
We only treat the case of even , for odd similar arguments can be used. Write
[TABLE]
for the decomposition into irreducible representations of and . Then, as in the proof of Theorem 6.4, an intertwining operator is given by a sequence of scalars, describing the action of the operator between and . Analyzing the action of it is easy to see that intertwining operators are described by the scalars if and by the scalars if . We can therefore use the identities (6.4), (6.5) and (6.6) for classification.
Next, observe that all intertwining operators between the irreducible constituents can be obtained from intertwining operators between the full principal series. For instance, intertwining operators are intertwining operators for and which vanish on the finite-dimensional subrepresentation of (hence factor to the quotient ) and whose image is contained in the irreducible subrepresentation . By replacing the spin representation of , that is induced from, by we may as well consider intertwining operators . Then the multiplicity is the dimension of the space of sequences satisfying (6.4), (6.5) and (6.6) (with for and for ) such that whenever or . By Theorem 6.5 there is up to scalar multiples a unique intertwiner , and the arguments in the proof of Theorem 6.4 show that for this intertwiner we have whenever or if and only if and . The other multiplicities are computed similarly. ∎
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 we let \mathbb{R}^{p,q}=\big{(}\mathbb{R}^{p+q},Q_{p,q}\big{)}, where is the following quadratic form on :
[TABLE]
Abusing notation, we also write for the associated symmetric bilinear form on .
We define the Clifford algebra to be the unital -algebra generated by subject to the relation
[TABLE]
For the standard basis vectors this implies
[TABLE]
For we also write for short. The Clifford algebra has a natural grading into even and odd elements
[TABLE]
The map of which acts by on and by on is an algebra involution called the canonical automorphism.
Denote by and the complexifications of and . Abusing notation, we will also use for the extension of the symmetric -bilinear form on to a symmetric -bilinear form on . Note that as -algebras.
We define the groups and by
[TABLE]
Further, put and . Then is connected and , where . For the group has four connected components
[TABLE]
where the first (resp. ) means that the number of ’s with in the product of an element is even (resp. odd), and the second (resp. ) the same for the number of ’s with . Then clearly , so has two connected components.
For all the group is a double cover of the indefinite orthogonal group , the covering map being
[TABLE]
The restriction of to induces a double covering
[TABLE]
A.2 Clifford modules and fundamental spin representations
We now describe the irreducible representations of . For even there is only one irreducible representation, and for odd there are two. To construct these we choose the maximal isotropic subspaces
[TABLE]
of , where w_{i}=\frac{1}{2}\big{(}e_{2i-1}+\sqrt{-1}e_{2i}\big{)}, w_{i}^{\prime}=\frac{1}{2}\big{(}e_{2i-1}-\sqrt{-1}e_{2i}\big{)}. Then
[TABLE]
We define an irreducible representation of on by
[TABLE]
for , , and in case is odd additionally
[TABLE]
Then for even is the unique irreducible representation of and , where is the canonical automorphism of . More precisely, the map
[TABLE]
intertwines and . For odd, and are inequivalent and we obtain two irreducible inequivalent representations of . For we write
[TABLE]
whenever the representation is clear from the context.
The restriction of any irreducible complex representation of the Clifford algebra to defines an irreducible representation of . These representations are called fundamental spin representations and will also be denoted by for even and for odd. Note that for even and for odd , where is the one-dimensional representation of given by the determinant character .
For even the restriction of to decomposes into the direct sum of two irreducible representations of according to the decomposition
[TABLE]
They have highest weights \big{(}\frac{1}{2},\ldots,\frac{1}{2},\pm\frac{1}{2}\big{)} in the standard notation. For odd the restrictions of and to define equivalent irreducible representations of whose highest weight is \big{(}\frac{1}{2},\ldots,\frac{1}{2}\big{)}.
A.3 Higher spin representations on monogenic polynomials
Let . For even the direct sum of the two irreducible -representations with highest weight \big{(}i+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2},\pm\frac{1}{2}\big{)} extends uniquely to an irreducible representation of which we denote by . Note that . For odd the irreducible -representation with highest weight \big{(}i+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2}\big{)} has two inequivalent extensions to which we denote by .
We realize the representations as monogenic polynomials on . For each the group acts on the space \operatorname{Pol}_{i}\big{(}\mathbb{R}^{n};\SS_{n}\big{)} of -valued homogeneous polynomials on of degree by
[TABLE]
We consider the Dirac operator on \operatorname{Pol}\big{(}\mathbb{R}^{n};\SS_{n}\big{)}=\bigoplus_{i=0}^{\infty}\operatorname{Pol}_{i}\big{(}\mathbb{R}^{n};\SS_{n}\big{)} given by
[TABLE]
Then the space
[TABLE]
of homogeneous monogenic polynomials of degree is invariant under the action of and defines an irreducible representation of . For even, , the isomorphism being
[TABLE]
For odd, and define inequivalent irreducible representations \big{(}\zeta_{n,i}^{\pm},\mathcal{M}_{i}\big{(}\mathbb{R}^{n};\SS_{n}\big{)}\big{)} of .
A.4 Branching laws
We give the explicit branching laws for the restriction of the -representations to .
A.4.1 Fundamental spin representations
We use the explicit realizations of the representations given in Section A.2. Then for even , and for odd . Recall the map from (A.1).
For even, the restriction of \big{(}\zeta_{n+1}^{\pm},\SS_{n+1}\big{)} to stays irreducible and is isomorphic to :
Lemma A.1**.**
Assume is even.
The explicit isomorphisms are given by
[TABLE] 2.
.
Proof.
This follows immediately from the definition of and and the fact that intertwines with . ∎
For odd, the restriction of to decomposes into the direct sum of and :
Lemma A.2**.**
Assume is odd.
The explicit embeddings are given by
[TABLE]
In particular, the image of the embedding is equal to
[TABLE] 2.
. In particular, the map is the canonical projection .
Proof.
It is easy to see that for we have
[TABLE]
then the claims follow. ∎
For the representations of this implies
[TABLE]
A.4.2 Higher spin representations
Using classical branching laws for the pair of Lie algebras, it is easy to see that the branching laws for the higher spin representations are
[TABLE]
To make this branching explicit in the realizations on monogenic polynomials, we use the classical Gegenbauer polynomials given by (see, e.g., [3, Chapter 10.9, equation (18)])
[TABLE]
The polynomial satisfies the differential equation (see [3, Chapter 10.9, equation (14)])
[TABLE]
Lemma A.3**.**
Let be a fundamental spin representation of and assume occurs in the restriction of to . If we identify with a subspace of , then for every the map
[TABLE]
is -intertwining, where .
Proof.
We first show the intertwining property. Let , then and . Further,
[TABLE]
Then the intertwining property follows. It remains to show that is monogenic, and for this we abbreviate
[TABLE]
Then
[TABLE]
Applying the Dirac operator to yields, after a short computation:
[TABLE]
where . This vanishes if and only if
[TABLE]
Multiplying the first equation with and subtracting it from the second one yields
[TABLE]
If we now insert this into the first equation we obtain
[TABLE]
which is the Gegenbauer differential equation with solution , the classical Gegenbauer polynomial. Finally, using (see [3, Chapter 10.9, equation (23)])
[TABLE]
and renormalizing and shows that indeed and the proof is complete. ∎
A.5 Multiplication with coordinates
By the Fischer decomposition we have
[TABLE]
Note that . We now study how the product of a monogenic polynomial with a coordinate function decomposes according to this decomposition.
Lemma A.4**.**
Let \phi\in\mathcal{M}_{i}\big{(}\mathbb{R}^{n};\SS_{n}\big{)} and . Then
[TABLE]
More precisely, with
[TABLE]
Proof.
This follows from the following identities which are easily verified:
[TABLE]
Now, recall the intertwining map I_{j\to i}\colon\mathcal{M}_{j}\big{(}\mathbb{R}^{n};\SS_{n}\big{)}\to\mathcal{M}_{i}\big{(}\mathbb{R}^{n+1},\SS_{n+1}\big{)} from Lemma A.3. The next result relates the decompositions of the polynomials and for .
Lemma A.5**.**
For and we have
[TABLE]
Note that we identify with a subspace of , so that is also a subspace of which is invariant under . More precisely, if acts on by , then it acts on by .
Proof.
This a lengthy but elementary computation involving several identities for Gegenbauer polynomials. We provide these identities and leave the computation to the reader. First, we have
[TABLE]
(A.3) is [3, Chapter 10.9, equation (23)], (A.4) follows from [3, Chapter 10.9, equation (24)] and (A.3), (A.5) is a consequence of [3, Chapter 10.9, equation (25)] and (A.3), (A.6) is [3, Chapter 10.9, equation (35)] and (A.7) is [3, Chapter 10.9, equation (36)]. Moreover, (A.5) combined with (A.6) implies
[TABLE]
[TABLE]
and (A.4) together with (A.8) implies
[TABLE]
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