# Knapp–Stein type intertwining operators for symmetric pairs II. – The
translation principle and intertwining operators for spinors

Jan Möllers  Department Mathematik, FAU Erlangen–Nürnberg, Cauerstr. 11, 91058
Erlangen, Germany<EMAIL_ADDRESS>and  Bent Ørsted  Institut for
Matematiske Fag, Aarhus Universitet, Ny Munkegade 118, 8000 Aarhus C, Denmark
<EMAIL_ADDRESS>
(Date: February 7, 2017)

###### Abstract.

For a symmetric pair $(G,H)$ 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 $G$ and one
for $H$ 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.

###### Key words and phrases:

Knapp–Stein intertwiners, intertwining operators, symmetry breaking operators,
symmetric pairs, principal series, translation principle.

###### 2010 Mathematics Subject Classification:

Primary 22E45; Secondary 47G10.

## 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 $G$. 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 $H\subseteq G$ and its
decomposition. 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, a term 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 of the 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, namely exactly the operators found by A. Juhl
[4] in connection with his study of Q-curvature and holography in conformal
geometry. Further connections to elliptic boundary value problems [13] and
automorphic forms [11] indicate the broad spectrum of applications that these
operators provide.

In our recent work with Y. Oshima [12] we generalized the construction of
symmetry breaking operators by Kobayashi–Speh to a large class of symmetric
pairs $(G,H)$ 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 $K$-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.

### The translation principle

Let $G$ be a real reductive group and $H\subseteq G$ a reductive subgroup. For
parabolic subgroups $P=MAN\subseteq G$ and $P_{H}=M_{H}A_{H}N_{H}\subseteq H$
we form the generalized principal series representations (smooth normalized
parabolic induction)

$\textup{Ind}_{P}^{G}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\qquad\mbox{and}\qquad\textup{Ind}_{P_{H}}^{H}(\eta\otimes
e^{\nu}\otimes\mathbf{1}),$

where $\xi$ and $\eta$ are finite-dimensional representations of $M$ and
$M_{H}$, and $\lambda\in\mathfrak{a}_{\mathbb{C}}^{*}$,
$\nu\in\mathfrak{a}_{H,\mathbb{C}}^{*}$, the complexified duals of the Lie
algebras of $A$ and $A_{H}$. Consider the space

$\textup{Hom}_{H}(\textup{Ind}_{P}^{G}(\xi\otimes
e^{\lambda}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}(\eta\otimes
e^{\nu}\otimes\mathbf{1}))$

of continuous $H$-intertwining operators. Realizing
$\textup{Ind}_{P}^{G}(\xi\otimes e^{\lambda}\otimes\mathbf{1})$ and
$\textup{Ind}_{P_{H}}^{H}(\eta\otimes e^{\nu}\otimes\mathbf{1})$ on the spaces
of smooth sections of the vector bundles

$\mathcal{V}_{\lambda}=G\times_{P}(\xi\otimes
e^{\lambda+\rho}\otimes\mathbf{1})\to
G/P\qquad\mbox{and}\qquad\mathcal{W}_{\nu}=H\times_{P_{H}}(\eta\otimes
e^{\nu+\rho_{H}}\otimes\mathbf{1})\to H/P_{H},$

where $\rho\in\mathfrak{a}^{*}$ and $\rho_{H}\in\mathfrak{a}_{H}^{*}$ 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 isomorphism

$\textup{Hom}_{H}(\textup{Ind}_{P}^{G}(\xi\otimes
e^{\lambda}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}(\eta\otimes
e^{\nu}\otimes\mathbf{1}))\stackrel{{\scriptstyle\sim}}{{\to}}(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})},\quad A\mapsto K^{A},$

where $\mathcal{V}_{\lambda}^{*}$ is the dual bundle of
$\mathcal{V}_{\lambda}$ and $W_{\nu}$ the $P_{H}$-representation defining
$\mathcal{W}_{\nu}$ (see Section 1.2 for the precise definition).

Now suppose $(\tau,E)$ is an irreducible finite-dimensional representation of
$G$. Then the restriction $\tau|_{P}$ of $\tau$ to $P$ contains a unique
irreducible subrepresentation $i:E^{\prime}\hookrightarrow E$ (see Lemma 2.1).
With respect to the Langlands decomposition $P=MAN$ this subrepresentation is
of the form $\tau^{\prime}\otimes e^{\mu^{\prime}}\otimes\mathbf{1}$ for some
irreducible finite-dimensional representation $\tau$ of $M$ and
$\mu^{\prime}\in\mathfrak{a}^{*}$. Let
$(E^{\prime})^{\vee}=\textup{Hom}_{\mathbb{C}}(E^{\prime},\mathbb{C})$ denote
the dual of $E^{\prime}$.

######  Theorem A (see Theorem 2.3 and Proposition 2.4).

For every $P_{H}$-equivariant quotient $p:E|_{P_{H}}\twoheadrightarrow
E^{\prime\prime}$ with $E^{\prime\prime}\simeq\tau^{\prime\prime}\otimes
e^{\mu^{\prime\prime}}\otimes\mathbf{1}$ as $P_{H}$-representations, there is
a unique linear map

$\Phi:\textup{Hom}_{H}(\textup{Ind}_{P}^{G}(\xi\otimes
e^{\lambda}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}(\eta\otimes
e^{\nu}\otimes\mathbf{1}))\\\
\to\textup{Hom}_{H}(\textup{Ind}_{P}^{G}((\xi\otimes\tau^{\prime})\otimes
e^{\lambda+\mu^{\prime}}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}((\eta\otimes\tau^{\prime\prime})\otimes
e^{\nu+\mu^{\prime\prime}}\otimes\mathbf{1}))$

with the property that for every intertwining operator $A$ with distribution
kernel $K^{A}$, the distribution kernel $K^{\Phi(A)}$ of $\Phi(A)$ is given by
$K^{\Phi(A)}=\varphi\otimes K^{A}$, the multiplication of $K^{A}$ with the
smooth section $\varphi\in
C^{\infty}(G/P,G\times_{P}(E^{\prime})^{\vee})\otimes E^{\prime\prime}$
defined by

$\varphi(g)=p\circ\tau(g)\circ
i\in\textup{Hom}_{\mathbb{C}}(E^{\prime},E^{\prime\prime})\simeq(E^{\prime})^{\vee}\otimes
E^{\prime\prime},\qquad g\in G.$

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 classify intertwining operators (see e.g. Theorem
3.8).

### Knapp–Stein type intertwining operators

Now assume that $(G,H)$ is a symmetric pair, i.e. $H$ is an open subgroup of
the fixed points $G^{\sigma}$ of an involution $\sigma$ of $G$. For simplicity
we further assume that $G$ is in the Harish-Chandra class. Let $P=MAN\subseteq
G$ be a $\sigma$-stable parabolic subgroup, then $P_{H}=P\cap H$ 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 intertwining operators between the spherical principal series
representations $\textup{Ind}_{P}^{G}(\mathbf{1}\otimes
e^{\lambda}\otimes\mathbf{1})$ and $\textup{Ind}_{P_{H}}^{H}(\mathbf{1}\otimes
e^{\nu}\otimes\mathbf{1})$. We now generalize this construction and obtain
intertwining operators between vector-valued principal series representations
using the translation principle.

Assume that $P$ and its opposite parabolic $\overline{P}$ are conjugate via
the Weyl group, i.e. $P=\tilde{w}_{0}\overline{P}\tilde{w}_{0}^{-1}$, where
$\tilde{w}_{0}$ is a representative of the longest Weyl group element $w_{0}$.
Then for any finite-dimensional representation $\xi$ of $M$ and
$\alpha,\beta\in\mathfrak{a}_{\mathbb{C}}^{*}$ we consider the function

$K_{\xi,\alpha,\beta}(g)=\xi(m(\tilde{w}_{0}^{-1}g^{-1}))^{-1}a(\tilde{w}_{0}^{-1}g^{-1})^{\alpha}a(\tilde{w}_{0}^{-1}g^{-1}\sigma(g))^{\beta},\qquad
g\in G,$

where $m(g)$ and $a(g)$ are the densely defined projections of
$g\in\overline{N}MAN$ onto the $M$\- and $A$-component. For $\xi=\mathbf{1}$
the trivial representation, these are the kernel functions constructed in
[12]. Since $a(g)$ is only defined on an open dense subset, it may happen that
the factor $a(\tilde{w}_{0}^{-1}g^{-1}\sigma(g))$ is not defined for any $g\in
G$, whence we additionally assume that the domain of definition for
$a(\tilde{w}_{0}^{-1}g^{-1}\sigma(g))$ is not empty. In [12] we showed that in
this case $K_{\xi,\alpha,\beta}(g)$ is defined 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 results from [12] yields:

######  Corollary B (see Corollary 2.9).

Assume that the finite-dimensional representation $\xi$ of $M$ is extendible
to $G$ (see Section 2.4 for the precise definition). Then the functions
$K_{\xi,\alpha,\beta}(g)$ extend to a meromorphic family of distributions

$K_{\xi,\alpha,\beta}\in(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})},\qquad\alpha,\beta\in\mathfrak{a}_{\mathbb{C}}^{*},$

where

(0.1) $\lambda=-w_{0}\alpha+\sigma\beta-
w_{0}\beta+\rho,\qquad\nu=-\alpha|_{\mathfrak{a}_{H,\mathbb{C}}}-\rho_{H}.$

Therefore, they give rise to a meromorphic family of intertwining operators

$A(\xi,\alpha,\beta)\in\textup{Hom}_{H}(\textup{Ind}_{P}^{G}(\tilde{w}_{0}\xi\otimes
e^{\lambda}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}(\xi|_{M_{H}}\otimes
e^{\nu}\otimes\mathbf{1})).$

As in [12, Corollary B] this construction gives in particular lower bounds on
multiplicities:

$\dim\textup{Hom}_{H}(\textup{Ind}_{P}^{G}(\tilde{w}_{0}\xi\otimes
e^{\lambda}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}(\xi|_{M_{H}}\otimes
e^{\nu}\otimes\mathbf{1}))\geq 1$

for all parameters $(\lambda,\nu)$ of the form (0.1).

### 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
$(G,H)=(\textup{O}(n+1,1),\textup{O}(n,1))$. The parabolic subgroups $P$ and
$P_{H}$ satisfy

$M_{H}\simeq\textup{O}(n-1)\times\textup{O}(1)\subseteq\textup{O}(n)\times\textup{O}(1)\simeq
M,\qquad A_{H}=A\simeq\mathbb{R}_{+},\qquad
N_{H}\simeq\mathbb{R}^{n-1}\subseteq\mathbb{R}^{n}\simeq N.$

We identify $\mathfrak{a}_{\mathbb{C}}^{*}\simeq\mathbb{C}$ such that
$\rho=\frac{n}{2}$, and denote by sgn the non-trivial character of
$O(1)\subseteq M_{H}\subseteq M$. Then for
$\delta,\varepsilon\in\mathbb{Z}/2\mathbb{Z}$ and $\lambda,\nu\in\mathbb{C}$
we consider intertwining operators between

$\pi_{\lambda,\delta}=\textup{Ind}_{P}^{G}(\textup{sgn}^{\delta}\otimes
e^{\lambda}\otimes\mathbf{1})\qquad\mbox{and}\qquad\tau_{\nu,\varepsilon}=\textup{Ind}_{P_{H}}^{H}(\textup{sgn}^{\varepsilon}\otimes
e^{\nu}\otimes\mathbf{1}).$

Tensoring with characters of $G$, it is easy to see that

$\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})\simeq\textup{Hom}_{H}(\pi_{\lambda,1-\delta}|_{H},\tau_{\nu,1-\varepsilon}).$

For the pair $(G,H)$, the restriction of a distribution kernel
$K\in(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})}$ to the open dense Bruhat cell in $G/P$, which is
isomorphic to $\overline{N}\simeq\mathbb{R}^{n}$, defines an isomorphism onto
a subspace of $\mathcal{D}^{\prime}(\mathbb{R}^{n})$ (see Kobayashi–Speh [9,
Theorem 3.16] or Theorem 1.2 for details). Let
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ resp.
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$ denote the space of
distribution kernels defining intertwining operators in
$\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})$ with
$\delta+\varepsilon\equiv 0(2)$ resp. $\delta+\varepsilon\equiv 1(2)$.

The space $\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ was
classified by Kobayashi–Speh [9], and we briefly describe this classification,
borrowing their notation. First, they construct a meromorphic family of
distributions
$K^{\mathbb{A},+}_{\lambda,\nu}\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$,
$(\lambda,\nu)\in\mathbb{C}^{2}$, given by

$K^{\mathbb{A},+}_{\lambda,\nu}(x^{\prime},x_{n})=|x_{n}|^{\lambda+\nu-\frac{1}{2}}(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n-1}{2}},\qquad(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}=\mathbb{R}^{n}.$

Then they show that every distribution in
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ is given by
$K_{\lambda,\nu}^{\mathbb{A},+}$ or a regularization of it. By a detailed
analysis of the meromorphic nature, the poles and all possible residues of the
family $K^{\mathbb{A},+}_{\lambda,\nu}$ they obtain a complete description of
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ for all
$(\lambda,\nu)\in\mathbb{C}^{2}$. More precisely, let

$L_{\textup{even}}=\\{(-\rho-i,-\rho_{H}-j):i,j\in\mathbb{N},i-j\in
2\mathbb{N}\\},$

then the corresponding statement for symmetry breaking operators is: For
$\delta+\varepsilon\equiv 0(2)$ we have

$\dim\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})=\dim\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}=\begin{cases}2&\mbox{for
$(\lambda,\nu)\in L_{\textup{even}}$,}\\\ 1&\mbox{for
$(\lambda,\nu)\in\mathbb{C}^{2}-L_{\textup{even}}$,}\end{cases}$

and every intertwining operator is given by the distribution kernel
$K^{\mathbb{A},+}_{\lambda,\nu}$ or a regularization of it.

We apply the translation principle to the kernels
$K^{\mathbb{A},+}_{\lambda,\nu}$ to obtain a meromorphic family
$K_{\lambda,\nu}^{\mathbb{A},-}\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$
given by

$K^{\mathbb{A},-}_{\lambda,\nu}(x)=x_{n}\cdot
K^{\mathbb{A},+}_{\lambda-1,\nu}(x)=\textup{sgn}(x_{n})|x_{n}|^{\lambda+\nu-\frac{1}{2}}(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n-1}{2}},\qquad
x\in\mathbb{R}^{n}.$

In Theorem 3.6 we derive from the poles and residues of
$K^{\mathbb{A},+}_{\lambda,\nu}$ all poles and all possible residues of the
family $K^{\mathbb{A},-}_{\lambda,\nu}$ and use them to classify intertwining
operators. For the statement let

$L_{\textup{odd}}=\\{(-\rho-i,-\rho_{H}-j):i,j\in\mathbb{N},i-j\in
2\mathbb{N}+1\\}.$

######  Theorem C (see Theorems 3.6 and 3.8).

For $\delta+\varepsilon\equiv 1(2)$ we have

$\dim\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})=\begin{cases}2&\mbox{for
$(\lambda,\nu)\in L_{\textup{odd}}$,}\\\ 1&\mbox{for
$(\lambda,\nu)\in\mathbb{C}^{2}-L_{\textup{odd}}$,}\end{cases}$

and every intertwining operator is given by the distribution kernel
$K^{\mathbb{A},-}_{\lambda,\nu}$ or a regularization of it.

We remark that for $\lambda-\nu=-\frac{3}{2}-2\ell$, $\ell\in\mathbb{N}$,
there exists a residue of $K^{\mathbb{A},-}_{\lambda,\nu}$ which is supported
at the origin in $\mathbb{R}^{n}$, inducing a differential intertwining
operator. In this setting $G/P\simeq S^{n}$ and $H/P_{H}\simeq S^{n-1}$, and
the differential intertwining operators form the family of odd order
conformally invariant differential operators
$C^{\infty}(S^{n},\mathcal{V}_{\lambda})\to
C^{\infty}(S^{n-1},\mathcal{W}_{\nu})$ studied previously by Juhl [4] (see
also [8]). The even order Juhl operators were already obtained as residues of
the family $K^{\mathbb{A},+}_{\lambda,\nu}$ by Kobayashi–Speh [9].

### Symmetry breaking operators for rank one pin groups

The second illustration of the translation principle is for the symmetric pair
$(\widetilde{G},\widetilde{H})=(\textup{Pin}(n+1,1),\textup{Pin}(n,1))$. Here
$\textup{Pin}(p,q)$ denotes a certain double cover of the group
$\textup{O}(p,q)$ (see Appendix A for details) so that we have compatible
double covers $\widetilde{G}\to G$ and $\widetilde{H}\to H$ with $G$ and $H$
as in the previous section. The preimages of the parabolic subgroups $P$ and
$P_{H}$ under the covering maps are $\widetilde{P}\simeq\widetilde{M}AN$ and
$\widetilde{P}_{H}\simeq\widetilde{M}_{H}A_{H}N_{H}$ with $A$, $N$, $A_{H}$
and $N_{H}$ as in the previous section and

$\widetilde{M}_{H}\simeq\textup{Pin}(n-1)\times\textup{O}(1)\subseteq\textup{Pin}(n)\times\textup{O}(1)\simeq\widetilde{M}.$

The fundamental representations of the Lie algebra $\mathfrak{so}(n)$ of $M$
are the exterior power representations $\wedge^{k}\mathbb{R}^{n}$ and the spin
representations. Symmetry breaking operators for the exterior power
representations are recently being investigated in detail by Kobayashi–Speh
(see also Fischmann–Juhl–Somberg [3] and Kobayashi–Kubo–Pevzner [7] for the
case of differential symmetry breaking operators). Here we focus on the
fundamental spin representations.

The group $\textup{Pin}(n)$ can be realized inside the Clifford algebra
$\operatorname{Cl}(n)$ with $n$ generators (see Appendix A for details). The
fundamental spin representations of $\textup{Pin}(n)$ are the restrictions of
irreducible representations of the complex Clifford algebra
$\operatorname{Cl}(n;\mathbb{C})=\operatorname{Cl}(n)\otimes_{\mathbb{R}}\mathbb{C}$
to $\textup{Pin}(n)$, and therefore we do not distinguish between the
representations of $\operatorname{Cl}(n;\mathbb{C})$ and their restrictions to
$\textup{Pin}(n)$. For even $n$ the the Clifford algebra
$\operatorname{Cl}(n;\mathbb{C})$ has a unique irreducible representation, and
for odd $n$ it has two inequivalent irreducible representations. Let
$(\zeta_{n},\SS_{n})$ be an irreducible representation of
$\operatorname{Cl}(n;\mathbb{C})$ and $(\zeta_{n-1},\SS_{n-1})$ an irreducible
representation of $\operatorname{Cl}(n-1;\mathbb{C})$. If sgn denotes the non-
trivial representation of $\textup{O}(1)$, we have for all
$\lambda,\nu\in\mathbb{C}$ and $\delta,\varepsilon\in\mathbb{Z}/2\mathbb{Z}$
principal series representations

$\not{\pi}_{\lambda,\delta}=\textup{Ind}_{\widetilde{P}}^{\widetilde{G}}((\zeta_{n}\otimes\textup{sgn}^{\delta})\otimes
e^{\lambda}\otimes\mathbf{1})\qquad\mbox{and}\qquad\not{\tau}_{\nu,\varepsilon}=\textup{Ind}_{\widetilde{P}_{H}}^{\widetilde{H}}((\zeta_{n-1}\otimes\textup{sgn}^{\varepsilon})\otimes
e^{\nu}\otimes\mathbf{1}),$

and we study intertwining operators in the space

$\textup{Hom}_{\widetilde{H}}(\not{\pi}_{\lambda,\delta}|_{\widetilde{H}},\not{\tau}_{\nu,\varepsilon}).$

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

$\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}))_{\lambda,\nu}^{\pm}\subseteq\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}))\simeq\mathcal{D}^{\prime}(\mathbb{R}^{n})\otimes\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}),$

where the sign $+$ represents $\delta+\varepsilon\equiv 0(2)$ and the sign $-$
represents $\delta+\varepsilon\equiv 1(2)$.

The translation principle applied to the distribution kernels
$K_{\lambda,\nu}^{\mathbb{A},\pm}\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{\pm}$
yields meromorphic families of distribution kernels
$P\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}\in\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}))_{\lambda,\nu}^{\pm}$
given by

$P\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}(x)=(P\zeta_{n}(x))\cdot
K_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}^{\mathbb{A},\mp}(x),$

where $\zeta_{n}(x)\in\textup{End}_{\mathbb{C}}(\SS_{n})$ is the value of the
representation $\zeta_{n}$ of the Clifford algebra
$\operatorname{Cl}(n;\mathbb{C})$ at the vector
$x\in\mathbb{R}^{n}\subseteq\operatorname{Cl}(n)\subseteq\operatorname{Cl}(n,\mathbb{C})$
and $0\neq
P\in\textup{Hom}_{\textup{Pin}(n-1)}([\zeta_{n}\otimes\det]|_{\textup{Pin}(n-1)},\zeta_{n-1})$.
(Note that the spin representation $\zeta_{n-1}$ of $\textup{Pin}(n-1)$ occurs
in the restriction $[\zeta_{n}\otimes\det]|_{\textup{Pin}(n-1)}$ with
multiplicity one, where $\det:\textup{Pin}(n)\to\textup{O}(n)\to\\{\pm 1\\}$
denotes the determinant character.)

To state our result on the classification of intertwining operators, let

$\displaystyle\not{L}_{\textup{even}}$
$\displaystyle=\\{(-\rho-\tfrac{1}{2}-i,-\rho_{H}-\tfrac{1}{2}-j):i,j\in\mathbb{N},i-j\in
2\mathbb{N}\\},$ $\displaystyle\not{L}_{\textup{odd}}$
$\displaystyle=\\{(-\rho-\tfrac{1}{2}-i,-\rho_{H}-\tfrac{1}{2}-j):i,j\in\mathbb{N},i-j\in
2\mathbb{N}+1\\}.$

######  Theorem D (see Theorems 4.3, 4.4 and 5.5).

  1. (1)

For $\delta+\varepsilon\equiv 0(2)$ we have

$\dim\textup{Hom}_{\widetilde{H}}(\not{\pi}_{\lambda,\delta}|_{\widetilde{H}},\not{\tau}_{\nu,\varepsilon})=\begin{cases}2&\mbox{for
$(\lambda,\nu)\in\not{L}_{\textup{even}}$,}\\\ 1&\mbox{for
$(\lambda,\nu)\in\mathbb{C}^{2}-\not{L}_{\textup{even}}$,}\end{cases}$

and every intertwining operator is given by the distribution kernel
$P\not{K}_{\lambda,\nu}^{\mathbb{A},+}$ or a regularization of it.

  2. (2)

For $\delta+\varepsilon\equiv 1(2)$ we have

$\dim\textup{Hom}_{\widetilde{H}}(\not{\pi}_{\lambda,\delta}|_{\widetilde{H}},\not{\tau}_{\nu,\varepsilon})=\begin{cases}2&\mbox{for
$(\lambda,\nu)\in\not{L}_{\textup{odd}}$,}\\\ 1&\mbox{for
$(\lambda,\nu)\in\mathbb{C}^{2}-\not{L}_{\textup{odd}}$,}\end{cases}$

and every intertwining operator is given by the distribution kernel
$P\not{K}_{\lambda,\nu}^{\mathbb{A},-}$ or a regularization of it.

In Theorems 4.3 and 4.4 we determine all poles and residues of the meromorphic
families $P\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}$, and hence give an explicit
description of the distribution kernels of all intertwining operators between
spinor-valued principal series representations.

We remark that for $\lambda-\nu=-\frac{1}{2}-2\ell$ resp.
$\lambda-\nu=-\frac{3}{2}-2\ell$, $\ell\in\mathbb{N}$, there exists a residue
of $P\not{K}_{\lambda,\nu}^{\mathbb{A},+}$ resp.
$P\not{K}_{\lambda,\nu}^{\mathbb{A},-}$ which is supported at the origin in
$\mathbb{R}^{n}$, inducing a differential intertwining operator. These
families of spinor-valued differential intertwining operators were previously
obtained by Kobayashi–Ørsted–Somberg–Souček [8], and it was conjectured that
these are all differential intertwining operators in this setting. Our
classification confirms this conjecture (see Remark 4.6).

In contrast to the proof of Theorem C which only uses the translation
principle, we employ the method developed in [10] 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 $K$-types. The explicit knowledge of the action on $K$-types
also allows us to determine the dimensions of intertwining operators between
the irreducible constituents of $\not{\pi}_{\lambda,\delta}$ and
$\not{\tau}_{\nu,\varepsilon}$ at reducibility points.

The representation $\not{\pi}_{\lambda,\delta}$ is reducible if and only if
$\lambda=\pm(\rho+\frac{1}{2}+i)$, $i\in\mathbb{N}$. More precisely, for
$\lambda=-\rho-\frac{1}{2}-i$ the representation $\not{\pi}_{\lambda,\delta}$
has a finite-dimensional irreducible subrepresentation
$\mathcal{F}_{\delta}(i)$ and the quotient
$\mathcal{T}_{\delta}(i)=\not{\pi}_{\lambda,\delta}/\mathcal{F}_{\delta}(i)$
is irreducible. The composition factors at $\lambda=\rho+\frac{1}{2}+i$ can be
described in terms of $\mathcal{F}_{\delta}(i)$ and $\mathcal{T}_{\delta}(i)$
by tensoring with the determinant character (see Lemma 5.6 for the precise
statement). We use the analogous notation for the composition factors
$\mathcal{F}^{\prime}_{\varepsilon}(j)$ and
$\mathcal{T}^{\prime}_{\varepsilon}(j)$ of $\not{\tau}_{\nu,\varepsilon}$ at
$\nu=-\rho_{H}-\frac{1}{2}-j$, $j\in\mathbb{N}$.

######  Theorem E (see Theorem 5.7).

For $\pi\in\\{\mathcal{T}_{\delta}(i),\mathcal{F}_{\delta}(i)\\}$ and
$\tau\in\\{\mathcal{T}^{\prime}_{\varepsilon}(j),\mathcal{F}^{\prime}_{\varepsilon}(j)\\}$
the multiplicities
$\dim\textup{Hom}_{\widetilde{H}}(\pi|_{\widetilde{H}},\tau)$ are given by

$\pi$$\tau$ | $\mathcal{F}^{\prime}_{\varepsilon}(j)$ | $\mathcal{T}^{\prime}_{\varepsilon}(j)$
---|---|---
$\mathcal{F}_{\delta}(i)$ | $1$ | $0$
$\mathcal{T}_{\delta}(i)$ | $0$ | $1$
for $0\leq j\leq i$,
$i+j\equiv\delta+\varepsilon(2)$,
$\pi$$\tau$ | $\mathcal{F}^{\prime}_{\varepsilon}(j)$ | $\mathcal{T}^{\prime}_{\varepsilon}(j)$
---|---|---
$\mathcal{F}_{\delta}(i)$ | $0$ | $0$
$\mathcal{T}_{\delta}(i)$ | $1$ | $0$
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 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 [12]. Section 2 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 3 we construct and classify intertwining
operators between principal series representations of
$(G,H)=(\textup{O}(n+1,1),\textup{O}(n,1))$ induced from one-dimensional
representations (see Theorems 3.6 and 3.8 for a detailed version of Theorem
C). Secondly, in Section 4 we construct intertwining operators between
principal series representations of
$(\widetilde{G},\widetilde{H})=(\textup{Pin}(n+1,1),\textup{Pin}(n,1))$
induced from spin-representations (see Theorems 4.3 and 4.4). To also obtain a
classification in the second situation, we employ in Section 5 the method
developed in [10], yielding the classification results in Theorems 5.5 and
5.7. Together with Theorems 4.3 and 4.4 this proves Theorems D and E. Finally,
Appendix A contains some elementary material about Clifford algebras and spin
representations needed in Sections 4 and 5.

### Notation

$\mathbb{N}=\\{0,1,2,\ldots\\}$,
$(\lambda)_{n}=\lambda(\lambda+1)\cdots(\lambda+n-1)$, $A-B=\\{a\in A:a\notin
B\\}$.

##  1\. 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 [9, 12].

###  1.1. Generalized principal series representations

Let $G$ be a real reductive group and $P\subseteq G$ a parabolic subgroup with
Langlands decomposition $P=MAN$. We write $\mathfrak{g}$, $\mathfrak{m}$,
$\mathfrak{a}$ and $\mathfrak{n}$ for the Lie algebras of $G$, $M$, $A$ and
$N$. Then $A=\exp(\mathfrak{a})$ and $N=\exp(\mathfrak{n})$. Let
$\overline{\mathfrak{n}}$ denote the nilradical opposite to $\mathfrak{n}$ and
$\overline{N}=\exp(\overline{\mathfrak{n}})$ the corresponding closed
connected subgroup of $G$. Then $\overline{P}=MA\overline{N}\subseteq G$ is
the parabolic subgroup opposite to $P$.

The multiplication map $\overline{N}\times M\times A\times N\to G$ is a
diffeomorphism onto an open dense subset of $G$, and for $g\in\overline{N}MAN$
we define $\overline{n}(g)\in\overline{N}$, $m(g)\in M$, $a(g)\in A$ and
$n(g)\in N$ by $g=\overline{n}(g)m(g)a(g)n(g)$.

We consider representations of $G$ which are parabolically induced from
finite-dimensional representations of $P$. Let $(\xi,V)$ be a finite-
dimensional representation of $M$, $\lambda\in\mathfrak{a}_{\mathbb{C}}^{*}$
and denote by $\mathbf{1}$ the trivial representation of $N$, then $\xi\otimes
e^{\lambda}\otimes\mathbf{1}$ is a finite-dimensional representation of
$P=MAN$. Write
$\rho=\frac{1}{2}\operatorname{tr}\textup{ad}_{\mathfrak{n}}\in\mathfrak{a}^{*}$
for half the sum of all positive roots and let $V_{\lambda}=\xi\otimes
e^{\lambda+\rho}\otimes\mathbf{1}$. We define the generalized principal series
representation $\textup{Ind}^{G}_{P}(\xi\otimes e^{\lambda}\otimes\mathbf{1})$
as the left-regular representation on the space

$C^{\infty}(G,V_{\lambda})^{P}=\\{f\in
C^{\infty}(G,V):f(gman)=a^{-\lambda-\rho}\xi(m)^{-1}f(g)\,\forall\,g\in
G,man\in MAN\\}.$

If we write

$\mathcal{V}_{\lambda}=G\times_{P}V_{\lambda}\to G/P$

for the homogeneous vector bundle associated to the representation
$V_{\lambda}$ of $P$, then $C^{\infty}(G,V_{\lambda})^{P}$ can be identified
with the space $C^{\infty}(G/P,\mathcal{V}_{\lambda})$ of smooth sections of
$\mathcal{V}_{\lambda}$.

###  1.2. Distribution sections of vector bundles

Let $V_{\lambda}^{*}=\xi^{\vee}\otimes e^{-\lambda+\rho}\otimes\mathbf{1}$,
where $\xi^{\vee}$ is the contragredient representation of $\xi$, and write
$\mathcal{V}_{\lambda}^{*}=G\times_{P}V_{\lambda}^{*}$ for the dual bundle of
$\mathcal{V}_{\lambda}$. We define the space
$\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda})$ of distribution sections of
the bundle $\mathcal{V}_{\lambda}$ as the (topological) dual of
$C^{\infty}(G/P,\mathcal{V}_{\lambda}^{*})$:

$\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda})=C^{\infty}(G/P,\mathcal{V}_{\lambda}^{*})^{\prime}.$

Note that since $G/P$ is compact, smooth sections on $G/P$ are automatically
compactly supported. Then $C^{\infty}(G/P,\mathcal{V}_{\lambda})\simeq
C^{\infty}(G,V_{\lambda})^{P}$ embeds $G$-equivariantly into
$\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda})$ by $f\mapsto T_{f}$, where

$\langle T_{f},\varphi\rangle=\int_{K}\langle
f(k),\varphi(k)\rangle\,dk\qquad\forall\varphi\in
C^{\infty}(G,V_{\lambda}^{*})^{P},$

and $dk$ denotes the normalized Haar measure on $K$.

Let $(\tau,E)$ be another finite-dimensional representation of $P$ and
$\mathcal{E}=G\times_{P}E$ the corresponding vector bundle over $G/P$. Then
every smooth section $f\in C^{\infty}(G/P,\mathcal{E})$ defines a continuous
linear multiplication operator

$\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda})\to\mathcal{D}^{\prime}(G/P,\mathcal{E}\otimes\mathcal{V}_{\lambda}),\quad
u\mapsto f\otimes u,$

which is dual to the composition

$C^{\infty}(G,E^{\vee}\otimes V_{\lambda}^{*})^{P}\stackrel{{\scriptstyle
f\otimes}}{{\to}}C^{\infty}(G,E\otimes E^{\vee}\otimes
V_{\lambda}^{*})^{P}\stackrel{{\scriptstyle
c_{*}}}{{\to}}C^{\infty}(G,V_{\lambda}^{*})^{P}$

of the pointwise tensor product $f\otimes$ and the push-forward by the
contraction map $c:E\otimes E^{\vee}\otimes V_{\lambda}^{*}\to
V_{\lambda}^{*}$.

###  1.3. Symmetry breaking operators

Now let $H\subseteq G$ be a reductive subgroup of $G$ and
$P_{H}=M_{H}A_{H}N_{H}\subseteq H$ a parabolic subgroup of $H$. Similarly, we
define $\textup{Ind}^{H}_{P_{H}}(\eta\otimes e^{\nu}\otimes\mathbf{1})$ for a
finite-dimensional representation $(\eta,W)$ of $M_{H}$ and
$\nu\in\mathfrak{a}_{H,\mathbb{C}}^{*}$ as the left-regular representation of
$H$ on $C^{\infty}(H,W_{\nu})^{P_{H}}$, where $W_{\nu}=\eta\otimes
e^{\nu+\rho_{H}}\otimes\mathbf{1}$,
$\rho_{H}=\frac{1}{2}\operatorname{tr}\textup{ad}_{\mathfrak{n}_{H}}$. Then
$C^{\infty}(H,W_{\nu})^{P_{H}}$ identifies with the space
$C^{\infty}(H/P_{H},\mathcal{W}_{\nu})$ of smooth sections of the vector
bundle $\mathcal{W}_{\nu}=H\times_{P_{H}}W_{\nu}\to H/P_{H}$.

In this paper we study continuous $H$-intertwining operators

$\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1}).$

By the Schwartz Kernel Theorem such maps are identified with $H$-invariant
distribution sections of some vector bundle over $G/P\times H/P_{H}$. Using
the isomorphism $\Delta(H)\backslash(G\times H)\simeq G$ which is induced by
$G\times H\to G,\,(g,h)\mapsto g^{-1}h$, invariant distributions on $G/P\times
H/P_{H}$ reduce to invariant distributions on $G/P$ (see [9, Section 3.2]):

######  Proposition 1.1 ([9, Proposition 3.2]).

We have natural isomorphisms of vector spaces

$\displaystyle\textup{Hom}_{H}(\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1}),\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1}))$ $\displaystyle\simeq\mathcal{D}^{\prime}(G/P\times
H/P_{H},\mathcal{V}_{\lambda}^{*}\otimes\mathcal{W}_{\nu})^{\Delta(H)}$
$\displaystyle\simeq(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})}.$

Under the isomorphism an $H$-intertwining operator
$A:\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1})$ maps to the distribution kernel
$K^{A}\in(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})}$ such that

$Af(h)=\int_{G/P}c(f(x)\otimes K^{A}(h^{-1}x))\,dx,\qquad f\in
C^{\infty}(G/P,\mathcal{V}_{\lambda}),$

where $c:V_{\lambda}\otimes V_{\lambda}^{*}\otimes W_{\nu}\to W_{\nu}$ 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 $G/P$,
because it is isomorphic to the vector space $\overline{\mathfrak{n}}$. 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 dense
Bruhat cell is injective:

######  Theorem 1.2 ([9, Theorem 3.16]).

Assume that

$P_{H}=P\cap H,\quad M_{H}=M\cap H,\quad A_{H}=A\cap H,\quad N_{H}=N\cap H.$

If additionally $G=P_{H}\overline{N}P$ then the restriction to
$\overline{\mathfrak{n}}\simeq\overline{N}\hookrightarrow G/P$ defines a
linear isomorphism

$(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})}\to\mathcal{D}^{\prime}(\overline{\mathfrak{n}},V_{\lambda}^{*}\otimes
W_{\nu})^{M_{H}A_{H},\mathfrak{n}_{H}}.$

Here the action of $M_{H}A_{H}$ on
$\mathcal{D}^{\prime}(\overline{\mathfrak{n}},V_{\lambda}^{*}\otimes W_{\nu})$
is the obvious action induced by the actions of $M_{H}A_{H}$ on
$\overline{\mathfrak{n}}$, $V_{\lambda}^{*}$ and $W_{\nu}$, and the action of
$\mathfrak{n}_{H}$ is induced by the infinitesimal action on
$\overline{\mathfrak{n}}$ viewed as subset of the generalized flag variety
$G/P$.

###  1.4. Symmetric pairs and Knapp–Stein type intertwining operators

For symmetric pairs $(G,H)$ we provided in [12] explicit expressions of
invariant distributions defining symmetry breaking operators, which we briefly
recall.

Let $\sigma$ be an involution of $G$ and let $H$ be an open subgroup of
$G^{\sigma}$, the fixed points of $\sigma$. Then $(G,H)$ forms a symmetric
pair. We make the following two additional assumptions:

(G) $P$ and $\overline{P}$ are conjugate via the Weyl group (H)
$\displaystyle\mbox{$P$ is $\sigma$-stable}.$

Then (G) implies $P=\tilde{w}_{0}\overline{P}\tilde{w}_{0}^{-1}$, where
$\tilde{w}_{0}$ is a representative of the longest Weyl group element $w_{0}$.
Further, (H) implies that $P_{H}=P\cap H$ is a parabolic subgroup of $H$.

Recall the $a$-projection $g\mapsto a(g)\in A$ from Section 1.1 which is
defined on the open dense subset $\overline{N}MAN\subseteq G$. For
$\alpha,\beta\in\mathfrak{a}_{\mathbb{C}}^{*}$ sufficiently positive we define

$K_{\alpha,\beta}(g)=a(\tilde{w}_{0}^{-1}g^{-1})^{\alpha}a(\tilde{w}_{0}^{-1}g^{-1}\sigma(g))^{\beta},\qquad
g\in G.$

Since the second factor might not be defined for any $g\in G$, we make the
additional assumption

(D) The domain of definition for $K_{\alpha,\beta}$ is non-empty.

In [12, Proposition 2.5] we showed that in this case the domain of definition
for $K_{\alpha,\beta}$ is already open and dense in $G$, and gave a criterion
to check this.

In [12] we studied the meromorphic continuation of $K_{\alpha,\beta}$ in the
parameters $\alpha,\beta\in\mathfrak{a}_{\mathbb{C}}^{*}$, and proved 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), the functions $K_{\alpha,\beta}$
extend to a meromorphic family of distributions

$K_{\alpha,\beta}\in(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})},$

where

(1.1) $\lambda=-w_{0}\alpha+\sigma\beta-
w_{0}\beta+\rho,\qquad\nu=-\alpha|_{\mathfrak{a}_{H,\mathbb{C}}}-\rho_{H}.$

Therefore, they give rise to a meromorphic family of intertwining operators

$A(\alpha,\beta):\textup{Ind}_{P}^{G}(\mathbf{1}\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}_{P_{H}}^{H}(\mathbf{1}\otimes
e^{\nu}\otimes\mathbf{1}).$

##  2\. 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 $G$.

###  2.1. General technique

Fix a principal series representation $\textup{Ind}_{P}^{G}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})$ of $G$ and let $(\tau,E)$ be a finite-
dimensional representation of $G$, then there is a natural $G$-equivariant
isomorphism

(2.1) $\iota_{\tau}:\textup{Ind}^{G}_{P}((\xi\otimes
e^{\lambda}\otimes\mathbf{1})\otimes
E|_{P})\stackrel{{\scriptstyle\sim}}{{\to}}\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\otimes E.$

When we view both sides as $(V\otimes E)$-valued functions on $G$, this
isomorphism is given by

$(\iota_{\tau}f)(g)=(\textup{id}\otimes\tau(g))f(g),\qquad g\in G.$

Now, for any $P$-stable subspace $E^{\prime}\subset E$ we have a natural
injective map

$\displaystyle\textup{Ind}^{G}_{P}((\xi\otimes
e^{\lambda}\otimes\mathbf{1})\otimes
E^{\prime})\hookrightarrow\textup{Ind}^{G}_{P}((\xi\otimes
e^{\lambda}\otimes\mathbf{1})\otimes E|_{P}).$

Suppose that $N$ acts trivially on $E^{\prime}$ and $A$ acts by a fixed
character $e^{\mu^{\prime}}$, $\mu\in\mathfrak{a}^{*}$, then the $P$-action on
$E^{\prime}$ can be written as $\tau^{\prime}\otimes
e^{\mu^{\prime}}\otimes\mathbf{1}$. The above map becomes

(2.2) $\textup{Ind}^{G}_{P}((\xi\otimes\tau^{\prime})\otimes
e^{\lambda+\mu^{\prime}}\otimes\mathbf{1})\hookrightarrow\textup{Ind}^{G}_{P}((\xi\otimes
e^{\lambda}\otimes\mathbf{1})\otimes E|_{P}).$

Assuming irreducibility of $\tau$ and $\tau^{\prime}$, there is essentially
only one choice of such a $P$-stable subspace $E^{\prime}\subseteq E$:

######  Lemma 2.1.

For every irreducible finite-dimensional representation $(\tau,E)$ of $G$ the
restriction $\tau|_{P}$ contains a unique irreducible subrepresentation
$E^{\prime}$. Moreover, $E^{\prime}$ is generated by the highest weight space
of $E$, and $P=MAN$ acts on $E^{\prime}$ by $\tau^{\prime}\otimes
e^{\mu^{\prime}}\otimes\mathbf{1}$, where $\tau^{\prime}$ is an irreducible
representation of $M$, $\mu^{\prime}\in\mathfrak{a}^{*}$ and $\mathbf{1}$ the
trivial representation of $N$.

###### Proof.

It suffices to treat the case of $G$ connected since $M$ meets all connected
components of $G$. We first fix some notation. Let
$\mathfrak{t}\subseteq\mathfrak{m}$ be a Cartan subalgebra, then
$\mathfrak{c}=\mathfrak{t}\oplus\mathfrak{a}$ is a Cartan subalgebra of
$\mathfrak{g}$ and $\mathfrak{c}_{\mathbb{C}}$ is a Cartan subalgebra of
$\mathfrak{g}_{\mathbb{C}}$. Choose any system of positive roots
$\Sigma^{+}(\mathfrak{g}_{\mathbb{C}},\mathfrak{c}_{\mathbb{C}})\subseteq\Sigma(\mathfrak{g}_{\mathbb{C}},\mathfrak{c}_{\mathbb{C}})$
such that the non-zero restrictions of positive roots to $\mathfrak{a}$ are
the roots of $\mathfrak{n}$. Then the non-zero restrictions of positive roots
in $\mathfrak{m}$ to $\mathfrak{t}_{\mathbb{C}}$ form a positive system of
roots
$\Sigma^{+}(\mathfrak{m}_{\mathbb{C}},\mathfrak{t}_{\mathbb{C}})\subseteq\Sigma(\mathfrak{m}_{\mathbb{C}},\mathfrak{t}_{\mathbb{C}})$.
We consider highest weights with respect to these positive systems.
Now let $E^{\prime}\subseteq E$ be any irreducible subrepresentation for
$\tau|_{P}$. Since $M$ is reductive, $E^{\prime}$ decomposes into the direct
sum $E^{\prime}=E_{1}^{\prime}\oplus\cdots\oplus E_{m}^{\prime}$ of
irreducible $M$-representations $E_{i}^{\prime}$ of highest weight
$\lambda_{i}\in\mathfrak{t}_{\mathbb{C}}^{*}$. Since $M$ and $A$ commute, $A$
acts by a character $\mu_{i}\in\mathfrak{a}^{*}$ on $E_{i}^{\prime}$. Now, let
$1\leq i\leq m$ such that $\lambda_{i}+\mu_{i}$ is maximal among the
$\lambda_{j}+\mu_{j}$, $1\leq j\leq m$. Then it is easy to see that
$\tau|_{N}$ is trivial on $E_{i}^{\prime}$. Hence, $E_{i}^{\prime}\subseteq E$
is stable under $P$. Since $E^{\prime}$ was assumed to be irreducible for $P$,
we have $E^{\prime}=E_{i}^{\prime}$ and hence $N$ acts trivially on
$E^{\prime}$.
To show that $E^{\prime}$ is unique, we simply observe that a highest weight
vector for the action of $M$ on $E^{\prime}$ is automatically a highest weight
vector for the action of $G$ on $E$ which is unique (up to scalar multiples).
Hence $E^{\prime}$ is the $P$-subrepresentation of $E$ generated by the
highest weight space. ∎

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 $P\subseteq G$
is minimal parabolic. Then every irreducible finite-dimensional representation
$(\tau^{\prime},E^{\prime})$ of $M$ is conjugate via the Weyl group to a
representation that occurs as a direct summand in an irreducible finite-
dimensional representation $(\tau,E)$ of $G$ and on which $N$ acts trivially.

Similarly, for a fixed principal series representation
$\textup{Ind}_{P_{H}}^{H}(\eta\otimes e^{\nu}\otimes\mathbf{1})$ we have an
isomorphism

(2.3) $\textup{Ind}_{P_{H}}^{H}(\eta\otimes e^{\nu}\otimes\mathbf{1})\otimes
E\stackrel{{\scriptstyle\sim}}{{\to}}\textup{Ind}^{H}_{P_{H}}((\eta\otimes
e^{\nu}\otimes\mathbf{1})\otimes E|_{P_{H}}).$

We also take a $P_{H}$-quotient space $E\twoheadrightarrow E^{\prime\prime}$
on which $N_{H}$ acts trivially and $A_{H}$ acts by a character
$e^{\mu^{\prime\prime}}$. Note that such a quotient always exists since the
contragredient representation $E^{\vee}$ 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 $E^{\prime\prime}$.
Denoting the $P_{H}$-action on $E^{\prime\prime}$ by
$\tau^{\prime\prime}\otimes e^{\mu^{\prime\prime}}\otimes\mathbf{1}$, we get a
map

(2.4) $\textup{Ind}^{H}_{P_{H}}((\eta\otimes e^{\nu}\otimes\mathbf{1})\otimes
E|_{P_{H}})\twoheadrightarrow\textup{Ind}^{H}_{P_{H}}((\eta\otimes\tau^{\prime\prime})\otimes
e^{\nu+\mu^{\prime\prime}}\otimes\mathbf{1}).$

Now suppose that an $H$-intertwining operator

$\displaystyle A:\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1})$

is given, and form the tensor product

(2.5) $A\otimes\textup{id}_{E}:\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\otimes E\to\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1})\otimes E.$

Then we obtain an $H$-intertwining operator

$\displaystyle\Phi(A):\textup{Ind}^{G}_{P}((\xi\otimes\tau^{\prime})\otimes
e^{\lambda+\mu^{\prime}}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}((\eta\otimes\tau^{\prime\prime})\otimes
e^{\nu+\mu^{\prime\prime}}\otimes\mathbf{1})$

by composing the maps (2.2), (2.1), (2.5), (2.3) and (2.4), namely

$\displaystyle\textup{Ind}^{G}_{P}((\xi\otimes\tau^{\prime})\otimes
e^{\lambda+\mu^{\prime}}\otimes\mathbf{1})\hookrightarrow{}$
$\displaystyle\textup{Ind}^{G}_{P}((\xi\otimes
e^{\lambda}\otimes\mathbf{1})\otimes E|_{P})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}{}$
$\displaystyle\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\otimes E$ (2.6) $\displaystyle\to{}$
$\displaystyle\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1})\otimes E$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}{}$
$\displaystyle\textup{Ind}^{H}_{P_{H}}((\eta\otimes
e^{\nu}\otimes\mathbf{1})\otimes E|_{P_{H}})$
$\displaystyle\twoheadrightarrow{}$
$\displaystyle\textup{Ind}^{H}_{P_{H}}((\eta\otimes\tau^{\prime\prime})\otimes
e^{\nu+\mu^{\prime\prime}}\otimes\mathbf{1}).$

This proves:

######  Theorem 2.3.

Let $(\tau,E)$ be a finite-dimensional $G$-representation,
$E^{\prime}\subseteq E$ a $P$-stable subspace with
$E^{\prime}|_{P}=\tau^{\prime}\otimes e^{\mu^{\prime}}\otimes\mathbf{1}$ and
$E\twoheadrightarrow E^{\prime\prime}$ a $P_{H}$-equivariant quotient with
$E^{\prime\prime}|_{P_{H}}=\tau^{\prime\prime}\otimes
e^{\mu^{\prime\prime}}\otimes\mathbf{1}$. Then (2.6) defines a linear map

$\Phi:\textup{Hom}_{H}(\textup{Ind}_{P}^{G}(\xi\otimes
e^{\lambda}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}(\eta\otimes
e^{\nu}\otimes\mathbf{1}))\\\
\to\textup{Hom}_{H}(\textup{Ind}_{P}^{G}((\xi\otimes\tau^{\prime})\otimes
e^{\lambda+\mu^{\prime}}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}((\eta\otimes\tau^{\prime\prime})\otimes
e^{\nu+\mu^{\prime\prime}}\otimes\mathbf{1}))$

for all finite-dimensional representations $\xi$ of $M$ and $\eta$ of $M_{H}$
and all $\lambda\in\mathfrak{a}_{\mathbb{C}}^{*}$,
$\nu\in\mathfrak{a}_{H,\mathbb{C}}^{*}$.

###  2.2. Integral kernels

Recall that $H$-intertwining operators are given by distribution kernels (see
Proposition 1.1). Let us see how the integral kernel behaves under the
translation principle. Suppose that

$\displaystyle A:\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1})$

is given by a the distribution kernel
$K^{A}\in(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})}$ in the sense that

$Af(h)=\int_{G/P}c(f(x)\otimes K^{A}(h^{-1}x))\,dx,\qquad f\in
C^{\infty}(G/P,\mathcal{V}_{\lambda}),$

where $c$ denotes the contraction map $V_{\lambda}\otimes
V_{\lambda}^{*}\otimes W_{\nu}\to W_{\nu}$ and the integral is meant in the
distribution sense (see Section 1.3 for details). Write $i:E^{\prime}\to E$
for the inclusion map and $p:E\to E^{\prime\prime}$ for the quotient map, and
let $\mathcal{E}^{\prime}=G\times_{P}E^{\prime}$ and
$\mathcal{E}^{\prime\prime}=H\times_{P_{H}}E^{\prime\prime}$ denote the
corresponding homogeneous vector bundles over $G/P$ and $H/P_{H}$. Let $u\in
C^{\infty}(G/P,\mathcal{E}^{\prime}\otimes\mathcal{V}_{\lambda})$, then
$\Phi(A)u\in
C^{\infty}(H/P_{H},\mathcal{E}^{\prime\prime}\otimes\mathcal{W}_{\nu})$ is
given by

$\displaystyle\Phi(A)f(h)$ $\displaystyle\hskip
14.22636pt=\Bigg{(}\Big{(}\big{(}p\circ\tau(h^{-1})\big{)}\otimes\textup{id}_{W_{\nu}}\Big{)}\int_{G/P}(\textup{id}_{E}\otimes
c)\bigg{(}\Big{(}(\tau(x)\circ
i)\otimes\textup{id}_{V_{\lambda}}\Big{)}\big{(}f(x)\big{)}\otimes
K^{A}(h^{-1}x)\bigg{)}\,dx\Bigg{)}$ $\displaystyle\hskip
14.22636pt=\int_{G/P}c\Big{(}\big{(}p\circ\tau(h^{-1}x)\circ
i\big{)}f(x)\otimes K(h^{-1}x)\Big{)}\,dx.$

This implies:

######  Proposition 2.4.

The integral kernel $\Phi(K^{A})=K^{\Phi(A)}$ of $\Phi(A)$ is given by

$\displaystyle\Phi(K^{A})=\varphi\otimes K^{A},$

where $\varphi\in C^{\infty}(G/P,(\mathcal{E}^{\prime})^{\vee})\otimes
E^{\prime\prime}\simeq C^{\infty}(G,(E^{\prime})^{\vee}\otimes
E^{\prime\prime})^{P}$ is given by

$\varphi(g)=p\circ\tau(g)\circ
i\in\textup{Hom}_{\mathbb{C}}(E^{\prime},E^{\prime\prime})\simeq(E^{\prime})^{\vee}\otimes
E^{\prime\prime}.$

######  Remark 2.5.

Since the operator $\Phi$ is given by multiplication with the fixed smooth
function $\varphi$, and the multiplication map

$\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu}\to\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*}\otimes(\mathcal{E}^{\prime})^{\vee})\otimes(W_{\nu}\otimes
E^{\prime\prime}),\quad K\mapsto\varphi\otimes K$

is continuous, the operator $\Phi$ maps holomorphic families of distributions
to holomorphic families. More precisely, if
$K_{z}\in(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})}$ depends holomorphically on
$z\in\Omega\subseteq\mathbb{C}^{n}$, then $\Phi(K_{z})$ depends
holomorphically on $z\in\Omega$. More generally, if $K_{z}$ depends
meromorphically on $z\in\Omega$ with poles in the set $\Sigma\subseteq\Omega$,
then $\Phi(K_{z})$ depends meromorphically on $z\in\Omega$ and its poles are
contained in $\Sigma$. However, it may of course happen that $K_{z}$ has a
pole at $z=z_{0}$ whereas $\Phi(K_{z})$ is regular at $z=z_{0}$ since the
multiplication by $\varphi$ can have a non-trivial kernel.

######  Remark 2.6.

Since $\varphi(g)$ is a matrix coefficient of a finite-dimensional
repesentation, it is obviously smooth in $g\in G$. In view of Theorem 1.2 one
can in some cases study the distribution kernels by their restriction to
$\overline{\mathfrak{n}}\simeq\overline{N}\hookrightarrow G/P$. On
$\overline{\mathfrak{n}}$ the function $\varphi(g)$ is actually a polynomial.
In fact, the nilpotency of $\overline{\mathfrak{n}}$ implies that there exists
$N\in\mathbb{N}$ such that $\tau(X_{1})\cdots\tau(X_{n})=0$ for all
$X_{1},\ldots,X_{n}\in\overline{\mathfrak{n}}$ and $n\geq N$. This shows that
$\varphi|_{\overline{\mathfrak{n}}}$ is a polynomial of degree at most $N$.

###  2.3. Reformulation using $\overline{P}$

We can also use the opposite parabolic subgroup $\overline{P}$ instead of $P$
in the above procedure. Assume there exists an element $\tilde{w}_{0}\in K$
such that $\tilde{w}_{0}\overline{P}\tilde{w}_{0}^{-1}=P$ and
$\tilde{w}_{0}A\tilde{w}_{0}^{-1}=A$. Then we have a $G$-equivariant
isomorphism

$\displaystyle\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\simeq\textup{Ind}^{G}_{\overline{P}}(\tilde{w}_{0}^{-1}\xi\otimes
e^{w_{0}^{-1}\lambda}\otimes\mathbf{1}),\quad f\mapsto
f(\,\cdot\,\tilde{w}_{0}^{-1}).$

Here, $\tilde{w}_{0}^{-1}\xi$ denotes the representation of $M$ on $V$ given
by $(\tilde{w}_{0}^{-1}\xi)(m)=\xi(\tilde{w}_{0}m\tilde{w}_{0}^{-1})$. Suppose
that an $H$-intertwining operator

$\displaystyle A:\textup{Ind}^{G}_{P}(\xi\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1})$

is given. Composing with the above ismorphism we have

$\displaystyle\textup{Ind}^{G}_{\overline{P}}(\tilde{w}_{0}^{-1}\xi\otimes
e^{w_{0}^{-1}\lambda}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}(\eta\otimes
e^{\nu}\otimes\mathbf{1})$

Then in a similar way, using $\overline{P}$ instead of $P$, we obtain an
$H$-intertwining operator

$\displaystyle\textup{Ind}^{G}_{\overline{P}}((\tilde{w}_{0}^{-1}\xi\otimes\tau^{\prime})\otimes
e^{w_{0}^{-1}\lambda+\mu^{\prime}}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}((\eta\otimes\tau^{\prime\prime})\otimes
e^{\nu+\mu^{\prime\prime}}\otimes\mathbf{1})$

for every $\overline{P}$-stable subspace $i:E^{\prime}\hookrightarrow E$ with
$E^{\prime}\simeq\tau^{\prime}\otimes e^{\mu^{\prime}}\otimes\mathbf{1}$ and
every $P_{H}$-stable quotient $p:E\to E^{\prime\prime}$. Composing with the
map $f\mapsto f(\,\cdot\,\tilde{w}_{0})$, we get

$\displaystyle\Psi(A):\textup{Ind}^{G}_{P}((\xi\otimes\tilde{w}_{0}\tau^{\prime})\otimes
e^{\lambda+w_{0}\mu^{\prime}}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}((\eta\otimes\tau^{\prime\prime})\otimes
e^{\nu+\mu^{\prime\prime}}\otimes\mathbf{1}).$

Moreover, if $A$ is given by the distribution kernel
$K^{A}\in(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})}$, then we similarly see that $\Psi(A)$ has
distribution kernel $\Psi(K^{A})=K^{\Psi(A)}$ given by

$\displaystyle\Psi(K^{A})=(p\circ\tau(\,\cdot\,\tilde{w}_{0})\circ i)\otimes
K^{A}.$

Now assume additionally that $P_{H}\subseteq P$ with $M_{H}A_{H}\subseteq MA$
and $N_{H}\subseteq N$, then $E^{\prime}$ and $E^{\prime\prime}$ can be chosen
compatibly. More precisely, let $E^{\prime}\subseteq E$ be the unique
irreducible $\overline{P}$-subrepresentation (see Lemma 2.1), then
$E^{\prime}$ is the lowest $\mathfrak{a}$-weight space and
$E^{\prime}\simeq\tau^{\prime}\otimes e^{\mu^{\prime}}\otimes\mathbf{1}$.
Write $E_{1}$ for the direct sum of all other $\mathfrak{a}$-weight spaces,
then $E=E^{\prime}\oplus E_{1}$ and the projection $E\twoheadrightarrow
E^{\prime}$ is $P_{H}$-equivariant. Hence we can take
$E^{\prime\prime}=E^{\prime}$:

######  Corollary 2.7.

Assume that $\tilde{w}_{0}\overline{P}\tilde{w}_{0}^{-1}=P$ and that
$M_{H}A_{H}\subseteq MA$ and $N_{H}\subseteq N$. Let $(\tau,E)$ be an
irreducible finite-dimensional $G$-representation and $E^{\prime}\subseteq E$
the unique irreducible $\overline{P}$-subrepresentation with
$E^{\prime}=\tau^{\prime}\otimes e^{\mu^{\prime}}\otimes\mathbf{1}$. Then for
all finite-dimensional representations $\xi$ of $M$ and $\eta$ of $M_{H}$ and
all $\lambda\in\mathfrak{a}_{\mathbb{C}}^{*}$,
$\nu\in\mathfrak{a}_{H,\mathbb{C}}^{*}$ we obtain a linear map

$\Psi:\textup{Hom}_{H}(\textup{Ind}_{P}^{G}(\xi\otimes
e^{\lambda}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}(\eta\otimes
e^{\nu}\otimes\mathbf{1}))\\\
\to\textup{Hom}_{H}(\textup{Ind}_{P}^{G}((\xi\otimes\tilde{w}_{0}\tau^{\prime})\otimes
e^{\lambda+w_{0}\mu^{\prime}}\otimes\mathbf{1}),\textup{Ind}_{P_{H}}^{H}((\eta\otimes\tau^{\prime}|_{M_{H}})\otimes
e^{\nu+\mu^{\prime}|_{\mathfrak{a}_{H}}}\otimes\mathbf{1}))$

which maps an intertwining operator $A$ with distribution kernel $K^{A}$ to
the intertwining operator $\Psi(A)$ with distribution kernel
$\Psi(K^{A})=K^{\Psi(A)}$ given by

$\Psi(K^{A})=\psi\otimes K^{A},$

where $\psi\in C^{\infty}(G/P,(\mathcal{E}^{\prime})^{\vee})\otimes
E^{\prime}\simeq C^{\infty}(G,(E^{\prime})^{\vee}\otimes E^{\prime})^{P}$ is
given by

$\psi(g)=\tau^{\prime}(m(\tilde{w}_{0}^{-1}g^{-1}))^{-1}\cdot
a(\tilde{w}_{0}^{-1}g^{-1})^{-\mu^{\prime}}\in\textup{End}_{\mathbb{C}}(E^{\prime})\simeq(E^{\prime})^{\vee}\otimes
E^{\prime}.$

###### Proof.

It remains to show the formula for $\psi(g)$. Write

$\tilde{w}_{0}^{-1}g^{-1}=\overline{n}(\tilde{w}_{0}^{-1}g^{-1})m(\tilde{w}_{0}^{-1}g^{-1})a(\tilde{w}_{0}^{-1}g^{-1})n(\tilde{w}_{0}^{-1}g^{-1})\in\overline{N}MAN,$

then we have

$g\tilde{w}_{0}=n(\tilde{w}_{0}^{-1}g^{-1})^{-1}m(\tilde{w}_{0}^{-1}g^{-1})^{-1}a(\tilde{w}_{0}^{-1}g^{-1})^{-1}\overline{n}(\tilde{w}_{0}^{-1}g^{-1})^{-1}\in
NMA\overline{N}.$

Since $\overline{N}$ acts trivially on $E^{\prime}$ we have
$\tau(\overline{n})\circ i=i$ for $\overline{n}\in\overline{N}$, and similarly
$p\circ\tau(n)=p$ for $n\in N_{H}\subseteq N$. Hence

$p\circ\tau(g\tilde{w}_{0})\circ
i=\tau^{\prime}(m(\tilde{w}_{0}^{-1}g^{-1})^{-1})\cdot
a(\tilde{w}_{0}^{-1}g^{-1})^{-\mu^{\prime}}.\qed$

######  Remark 2.8.

We note that the function $\psi(g)$ resembles the integral kernel of the
standard Knapp–Stein intertwining operator

$A(\lambda):\textup{Ind}_{P}^{G}(\tau^{\prime}\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}_{P}^{G}(\tilde{w}_{0}\tau^{\prime}\otimes
e^{w_{0}\lambda}\otimes\mathbf{1}),\quad
Af(g)=\int_{\overline{N}}f(g\tilde{w}_{0}\overline{n})\,d\overline{n}.$

More precisely, it is shown in [5, Chapter VII, §7] that

$Af(g)=\int_{\overline{N}}\tau^{\prime}(m(\tilde{w}_{0}^{-1}\overline{n}))a(\tilde{w}_{0}^{-1}\overline{n})^{\lambda-\rho}f(g\overline{n})\,d\overline{n}.$

###  2.4. Application to Knapp–Stein type intertwining operators

We now specialize to the setting of Section 1.4. In this case, all assumptions
of Corollary 2.7 are satisfied and we can apply it to the family
$A(\alpha,\beta)$ of intertwining operators

$\displaystyle A(\alpha,\beta):\textup{Ind}^{G}_{P}(\mathbf{1}\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}(\mathbf{1}\otimes
e^{\nu}\otimes\mathbf{1}),$

$\lambda$ and $\nu$ given by (1.1), with distribution kernels

$\displaystyle
K_{\alpha,\beta}(g)=a(\tilde{w}_{0}^{-1}g^{-1})^{\alpha}a(\tilde{w}_{0}^{-1}g^{-1}\sigma(g))^{\beta},\qquad
g\in G.$

We call an irreducible finite-dimensional representation $\xi$ of $M$
extendible if there exists a finite-dimensional irreducible representation
$(\tau,E)$ of $G$ such that

$E^{\overline{N}}=\\{v\in
E:\tau(\overline{n})v=v\,\forall\,\overline{n}\in\overline{N}\\}\simeq\xi\qquad\mbox{as
$M$-representations.}$

######  Corollary 2.9.

Assume conditions (G), (H) and (D). Then for every extendible irreducible
finite-dimensional $M$-representation $\xi$ the functions

$\displaystyle
K_{\xi,\alpha,\beta}(g)=\xi(m(\tilde{w}_{0}^{-1}g^{-1}))^{-1}a(\tilde{w}_{0}^{-1}g^{-1})^{\alpha}a(\tilde{w}_{0}^{-1}g^{-1}\sigma(g))^{\beta},\qquad
g\in G,$

extend to a meromorphic family of distributions

$K_{\xi,\alpha,\beta}\in(\mathcal{D}^{\prime}(G/P,\mathcal{V}_{\lambda}^{*})\otimes
W_{\nu})^{\Delta(P_{H})},$

for $\lambda$ and $\nu$ given by (1.1) and
$\mathcal{V}_{\lambda}=G\times_{P}(\tilde{w}_{0}\xi\otimes
e^{\lambda+\rho}\otimes\mathbf{1})$, $W_{\nu}=\xi|_{M_{H}}\otimes
e^{\nu+\rho_{H}}\otimes\mathbf{1}$. Therefore, they give rise to a meromorphic
family of intertwining operators

$A(\xi,\alpha,\beta):\textup{Ind}^{G}_{P}(\tilde{w}_{0}\xi\otimes
e^{\lambda}\otimes\mathbf{1})\to\textup{Ind}^{H}_{P_{H}}(\xi\otimes
e^{\nu}\otimes\mathbf{1}).$

###### Proof.

Let $(\tau,E)$ be an irreducible finite-dimensional representation of $G$ with
$E^{\overline{N}}\simeq\xi$ as $M$-representations, then
$E^{\prime}=E^{\overline{N}}\subseteq E$ is $\overline{P}$-stable of the form
$E^{\prime}\simeq\xi\otimes e^{\mu}\otimes\mathbf{1}$. Now the statement
follows from Theorem 1.3 and Corollary 2.7. ∎

######  Remark 2.10.

For $(G,H)=(\textup{O}(n+1,1),\textup{O}(n,1))$ one can choose a
representative $\tilde{w}_{0}$ of $w_{0}$ that centralizes $M$. Hence,
$\tilde{w}_{0}\xi=\xi$ for all irreducible finite-dimensional representations
$\xi$ of $M$. By Lemma 2.2 either $\xi$ or $\tilde{w}_{0}\xi=\xi$ is
extendible, so that in this case all irreducible finite-dimensional
$M$-representations are extendible.

##  3\. Example: $(G,H)=(\textup{O}(n+1,1),\textup{O}(n,1))$

We apply the translation principle to the symmetry breaking operators between
spherical principal series of rank one orthogonal groups studied by
Kobayashi–Speh [9] and obtain new symmetry breaking operators between non-
spherical scalar-valued principal series.

###  3.1. Parabolic subgroups and the symmetric pair

Let $G=\textup{O}(n+1,1)$, $n\geq 2$, realized as the subgroup of
$\textup{GL}(n+2,\mathbb{R})$ preserving the indefinite bilinear form

$\mathbb{R}^{n+2}\to\mathbb{R},\quad x\mapsto
x_{1}^{2}+\cdots+x_{n+1}^{2}-x_{n+2}^{2}.$

We choose the (minimal) parabolic subgroup $P=P_{\min}=MAN\subseteq G$ such
that $\mathfrak{a}=\mathbb{R}H_{0}$ with

$H_{0}=\left(\begin{array}[]{ccc}0&&1\\\ &\mathbf{0}_{n}&\\\
1&&0\end{array}\right)$

and $\mathfrak{n}=\mathfrak{g}_{\alpha}$ for $\alpha\in\mathfrak{a}^{*}$ with
$\alpha(H_{0})=1$. Then

$M=\left\\{\left(\begin{array}[]{ccc}\varepsilon&&\\\ &m&\\\
&&\varepsilon\end{array}\right):m\in\textup{O}(n),\varepsilon=\pm
1\right\\}\simeq\textup{O}(n)\times\mathbb{Z}/2\mathbb{Z}.$

Let $m_{0}=\textup{diag}(-1,1,\ldots,1,-1)$ and identify $m\in\textup{O}(n)$
with $\textup{diag}(1,m,1)\in M$, then $M=\textup{O}(n)\cup
m_{0}\textup{O}(n)$. We further identify
$\overline{\mathfrak{n}}=\mathfrak{g}_{-\alpha}\cong\mathbb{R}^{n}$ by

$\mathbb{R}^{n}\to\overline{\mathfrak{n}},\,x\mapsto\left(\begin{array}[]{ccc}0&x^{\top}&0\\\
x&\mathbf{0}_{n}&x\\\ 0&-x^{\top}&0\end{array}\right)$

and use this identification to parametrize
$\overline{N}=\exp(\overline{\mathfrak{n}})$ by

$\mathbb{R}^{n}\to\overline{\mathfrak{n}}\xrightarrow{\exp}\overline{N},\quad
x\mapsto\overline{n}_{x}=\left(\begin{array}[]{ccc}1+|x|^{2}/2&x^{\top}&|x|^{2}/2\\\
x&\mathbf{1}_{n}&x\\\ -|x|^{2}/2&-x^{\top}&1-|x|^{2}/2\end{array}\right).$

The group $M=\textup{O}(n)\cup m_{0}\textup{O}(n)$ acts on
$x\in\overline{\mathfrak{n}}\cong\mathbb{R}^{n}$ by the adjoint action as
follows:

$\textup{Ad}(m)x=mx\quad(m\in\textup{O}(n)),\qquad\qquad\textup{Ad}(m_{0})x=-x.$

Further, $P$ is conjugate to its opposite parabolic
$\overline{P}=MA\overline{N}$ by

$\tilde{w}_{0}=\textup{diag}(1,\ldots,1,-1).$

Let us identify $\mathfrak{a}_{\mathbb{C}}^{*}\cong\mathbb{C}$ by
$\lambda\mapsto\lambda(H_{0})$, so that $\rho=\frac{n}{2}$.

######  Lemma 3.1.

For $x\in\mathbb{R}^{n}$:

$\tilde{w}_{0}^{-1}\overline{n}_{x}\in\overline{N}MAN\quad\Leftrightarrow\quad
x\in\mathbb{R}^{n}-\\{0\\},$

and in this case for $\lambda\in\mathfrak{a}_{\mathbb{C}}^{*}\simeq\mathbb{C}$
we have

$a(\tilde{w}_{0}^{-1}\overline{n}_{x})^{\lambda}=|x|^{2\lambda},\qquad
m(\tilde{w}_{0}^{-1}\overline{n}_{x})=\left(\begin{array}[]{ccc}1&&\\\
&\mathbf{1}_{n}-2\frac{xx^{\top}}{|x|^{2}}&\\\ &&1\end{array}\right).$

Now let $\sigma$ be the involution of $G$ given by conjugation with the matrix
$\textup{diag}(1,\ldots,1,-1,1)$. Then $H=G^{\sigma}\simeq\textup{O}(n,1)$ 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 $P_{H}=P\cap
H=M_{H}A_{H}N_{H}$. Then

$M_{H}=M\cap H=\textup{O}(n-1)\cup
m_{0}\textup{O}(n-1)\qquad\mbox{and}\qquad\mathfrak{a}_{H}=\mathfrak{a}=\mathbb{R}H_{0}.$

Further, under the identification
$\overline{\mathfrak{n}}\simeq\mathbb{R}^{n}$ the involution $\sigma$ acts by

$\sigma(x)=(x_{1},\ldots,x_{n-1},-x_{n}),\qquad x\in\mathbb{R}^{n},$

and therefore the subalgebra
$\overline{\mathfrak{n}}_{H}=\overline{\mathfrak{n}}^{\sigma}$ is given by the
standard embedding of $\mathbb{R}^{n-1}$ into $\mathbb{R}^{n}$ as the first
$n-1$ coordinates.

###  3.2. Principal series representations and symmetry breaking operators

Denote by $\textup{sgn}:M\to\\{\pm 1\\}$ the character of $M=\textup{O}(n)\cup
m_{0}\textup{O}(n)$ given by $\textup{sgn}(m)=1$ for $m\in\textup{O}(n)$ and
$\textup{sgn}(m_{0})=-1$. Abusing notation we also write sgn for the
corresponding character of $M_{H}$. For
$\delta,\varepsilon\in\mathbb{Z}/2\mathbb{Z}$ and
$\lambda,\nu\in\mathfrak{a}_{\mathbb{C}}^{*}\simeq\mathbb{C}$ we define the
scalar principal series representations (smooth normalized parabolic
induction)

$\pi_{\lambda,\delta}=\textup{Ind}_{P}^{G}(\textup{sgn}^{\delta}\otimes
e^{\lambda}\otimes\mathbf{1})\qquad\mbox{and}\qquad\tau_{\nu,\varepsilon}=\textup{Ind}_{P_{H}}^{H}(\textup{sgn}^{\varepsilon}\otimes
e^{\nu}\otimes\mathbf{1}).$

We consider the space

$\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})$

of symmetry breaking operators between $\pi_{\lambda,\delta}$ and
$\tau_{\nu,\varepsilon}$. By Theorem 1.2 every such operator is given by a
distribution kernel
$K\in\mathcal{D}^{\prime}(\overline{\mathfrak{n}})^{M_{H}A_{H},\mathfrak{n}_{H}}$
satisfying certain invariance conditions for the action of $M_{H}A_{H}$ and
$\mathfrak{n}_{H}$. Since $m_{0}\in M_{H}$ acts by
$K(x)\mapsto(-1)^{\delta+\varepsilon}K(-x)$, it is clear that
$\mathcal{D}^{\prime}(\overline{\mathfrak{n}})^{M_{H}A_{H},\mathfrak{n}_{H}}$
only depends on the parity of $\delta+\varepsilon$. Identfying
$\overline{\mathfrak{n}}\simeq\mathbb{R}^{n}$ as above we write
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ resp.
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ for the space
$\mathcal{D}^{\prime}(\mathbb{R}^{n})^{M_{H}A_{H},\mathfrak{n}_{H}}$ with
$\delta+\varepsilon\equiv 0(2)$ resp. $\delta+\varepsilon\equiv 1(2)$. Then,
taking distribution kernels is a linear isomorphism

$\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})\stackrel{{\scriptstyle\sim}}{{\to}}\begin{cases}\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}&\mbox{for
$\delta+\varepsilon\equiv 0(2)$,}\\\
\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}&\mbox{for
$\delta+\varepsilon\equiv 1(2)$.}\end{cases}$

###  3.3. Construction of symmetry breaking operators

In this section we describe all intertwining operators in
$\textup{Hom}_{H}(\pi_{\lambda,\delta},\tau_{\nu,\varepsilon})$ for all
$\lambda,\nu\in\mathbb{C}$, $\delta,\varepsilon\in\mathbb{Z}/2\mathbb{Z}$. We
note that the notation is essentially due to Kobayashi–Speh [9].

####  3.3.1. Spherical principal series

For $\delta=\varepsilon=0$ the representations $\pi_{\lambda,\delta}$ and
$\tau_{\nu,\varepsilon}$ are spherical, i.e. possess a vector invariant under
a maximal compact subgroup. In this setting, Section 1.4 provides a
meromorphic family of intertwining operators
$\mathbb{A}_{\lambda,\nu}\in\textup{Hom}_{H}(\pi_{\lambda,0}|_{H},\tau_{\nu,0})$
given by the distribution kernels

$K_{\lambda,\nu}^{\mathbb{A},+}(x)=|x_{n}|^{\lambda+\nu-\frac{1}{2}}(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n-1}{2}},\qquad
x\in\mathbb{R}^{n}.$

By analyzing the poles and residues of $K^{\mathbb{A},+}_{\lambda,\nu}$
explicitly, Kobayashi–Speh [9] completely determine the space
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ for all
$\lambda,\nu\in\mathbb{C}$. (Note that our parameters $(\lambda,\nu)$ are
normalized such that $\pi_{\lambda,\delta}$ and $\tau_{\nu,\varepsilon}$ are
unitary for $\lambda,\nu\in i\mathbb{R}$. Therefore, in our notation one has
to replace $(\lambda,\nu)$ by $(\lambda-\rho,\nu-\rho_{H})$ to obtain
Kobayashi–Speh’s notation, see [9].) To summarize their results let

$\displaystyle\backslash\backslash^{+}$
$\displaystyle=\\{(\lambda,\nu):\lambda+\nu=-\tfrac{1}{2}-2k,k\in\mathbb{N}\\},$
$\displaystyle//^{+}$
$\displaystyle=\\{(\lambda,\nu):\lambda-\nu=-\tfrac{1}{2}-2\ell,\ell\in\mathbb{N}\\},$
$\displaystyle L_{\textup{even}}$
$\displaystyle=\\{(\lambda,\nu)=(-\rho-i,-\rho_{H}-j):i,j\in\mathbb{N},i-j\in
2\mathbb{N}\\}\subseteq//^{+}.$

For $(\lambda,\nu)\in\backslash\backslash^{+}$ with
$\lambda+\nu=-\frac{1}{2}-2k$ we further define

$\displaystyle\widetilde{K}^{\mathbb{B},+}_{\lambda,\nu}(x)$
$\displaystyle=\frac{1}{\Gamma(\frac{1}{2}(\lambda-\nu+\frac{1}{2}))}(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n-1}{2}}\delta^{(2k)}(x_{n})$
$\displaystyle=\sum_{i=0}^{k}\frac{(-1)^{i}(2k)!(\nu+\frac{n-1}{2})_{i}}{i!(2k-2i)!\Gamma(\frac{1}{2}(\lambda-\nu+\frac{1}{2}))}|x^{\prime}|^{1-n-2\nu-2i}\delta^{(2k-2i)}(x_{n}),$
and for $(\lambda,\nu)\in//^{+}$ with $\lambda-\nu=-\frac{1}{2}-2\ell$ we put
$\displaystyle\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}(x)$
$\displaystyle=\sum_{j=0}^{\ell}\frac{2^{2\ell-2j}(\nu-\ell)_{\ell-j}}{j!(2\ell-2j)!}(\Delta_{\mathbb{R}^{n-1}}^{j}\partial_{n}^{2\ell-2j}\delta)(x).$

Note that both $\widetilde{K}^{\mathbb{B},+}_{\lambda,\nu}$ and
$\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}$ depend holomorphically on
$\nu\in\mathbb{C}$ (or $\lambda\in\mathbb{C}$).

######  Theorem 3.2.

The renormalized distribution

$\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}(x)=\frac{K^{\mathbb{A},+}_{\lambda,\nu}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu+\frac{1}{2}))\Gamma(\frac{1}{2}(\lambda-\nu+\frac{1}{2}))}$

depends holomorphically on $(\lambda,\nu)\in\mathbb{C}^{2}$ and vanishes only
for $(\lambda,\nu)\in L_{\textup{even}}$. More precisely,

  1. (1)

For $(\lambda,\nu)\in\mathbb{C}^{2}-(\backslash\backslash^{+}\cup//^{+})$ we
have
$\operatorname{supp}\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}=\mathbb{R}^{n}$.

  2. (2)

For $(\lambda,\nu)\in\backslash\backslash^{+}-//^{+}$ with
$\lambda+\nu=-\frac{1}{2}-2k$, $k\in\mathbb{N}$, we have

$\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}(x)=\frac{(-1)^{k}k!}{(2k)!}\widetilde{K}^{\mathbb{B},+}_{\lambda,\nu}(x).$

In particular,
$\operatorname{supp}\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}=\mathbb{R}^{n-1}$.

  3. (3)

For $(\lambda,\nu)\in//^{+}-L_{\textup{even}}$ with
$\lambda-\nu=-\frac{1}{2}-2\ell$, $\ell\in\mathbb{N}$, we have

$\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}(x)=\frac{(-1)^{\ell}\ell!\pi^{\frac{n-1}{2}}}{2^{2\ell}\Gamma(\nu+\frac{n-1}{2})}\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}(x).$

In particular,
$\operatorname{supp}\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}=\\{0\\}$.

  4. (4)

For $(-\rho-i,-\rho_{H}-j)\in L_{\textup{even}}$, $2\ell=i-j$, restricting the
function $(\lambda,\nu)\mapsto\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}$ to
two different complex hyperplanes in $\mathbb{C}^{2}$ through
$(-\rho-i,-\rho_{H}-j)$ and renormalizing gives two holomorphic families of
distributions:

$\displaystyle\qquad\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}(x)$
$\displaystyle=\frac{(-1)^{\ell}2^{2\ell}}{\ell!\pi^{\frac{n-1}{2}}}\Gamma(\nu+\rho_{H})\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}(x)$
$\displaystyle(\lambda-\nu=-\tfrac{1}{2}-2\ell),$
$\displaystyle\qquad{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},+}_{\lambda,\nu}(x)$
$\displaystyle=\Gamma(\tfrac{1}{2}(\lambda-\nu+\tfrac{1}{2}))\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}(x)$
$\displaystyle(\nu=-\rho_{H}-j).$

Their special values at $(-\rho-i,-\rho_{H}-j)$ satisfy

$\displaystyle\operatorname{supp}\widetilde{K}^{\mathbb{C},+}_{-\rho-i,-\rho_{H}-j}$
$\displaystyle=\\{0\\},$
$\displaystyle\operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},+}_{-\rho-i,-\rho_{H}-j}$
$\displaystyle=\begin{cases}\mathbb{R}^{n}&\mbox{for $n$ even,}\\\
\mathbb{R}^{n-1}&\mbox{for $n$ odd,}\end{cases}$

more precisely, for $n$ odd and $2k=i+j+n-1$

${\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},+}_{-\rho-i,-\rho_{H}-j}=\frac{(-1)^{k}k!}{(2k)!}(|x^{\prime}|^{2}+x_{n}^{2})^{j}\delta^{(2k)}(x_{n}).$

###### Proof.

This is a summary of the results of [9]. ∎

These results can be used to describe the space
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$:

######  Theorem 3.3.

We have

$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}=\begin{cases}\mathbb{C}\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}&\mbox{for
$(\lambda,\nu)\in\mathbb{C}^{2}-L_{\textup{even}}$,}\\\
\mathbb{C}{\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}\oplus\mathbb{C}\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}&\mbox{for
$(\lambda,\nu)\in L_{\textup{even}}$.}\end{cases}$

###### Proof.

See [9, Theorem 1.9] and also [10, Theorem 4.9]. ∎

####  3.3.2. Scalar principal series

We use the translation principle to describe
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$ in terms of
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$.

Let $(\tau,E)$ be the defining representation of $G=\textup{O}(n+1,1)$ on
$E=\mathbb{C}^{n+2}$. Then

$i:E^{\prime}=\mathbb{C}\hookrightarrow E,\quad
z\mapsto(z,0,\ldots,0,z)^{\top}$

is the maximal subspace on which $N$ acts trivially, and
$E^{\prime}|_{P}=\textup{sgn}\otimes e^{\alpha}\otimes\mathbf{1}$. For the
$P_{H}$-quotient space we choose

$p:E\twoheadrightarrow\mathbb{C},\quad(z_{1},\ldots,z_{n+2})\mapsto z_{n+1},$

then $E^{\prime\prime}|_{P_{H}}=\mathbf{1}\otimes\mathbf{1}\otimes\mathbf{1}$.
Now let us compute the $(E^{\prime})^{*}\otimes E^{\prime\prime}$-valued
function $p\circ\tau(g)\circ i$ on $\overline{N}$. For $g=\overline{n}_{x}$,
$x\in\mathbb{R}^{n}$, we have

$p\circ\tau(g)\circ i(1)^{\top}=p(1+|x|^{2},2x,1-|x|^{2})^{\top}=2x_{n}.$

Hence, by Theorem 2.3, we get a linear map

$x_{n}:\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{\pm}\to\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda+1,\nu}^{\mp},\quad
K(x)\mapsto x_{n}K(x).$

Applying this map to the meromorphic family of distributions
$K^{\mathbb{A},+}_{\lambda,\nu}(x)$ we obtain another meromorphic family

$K^{\mathbb{A},-}_{\lambda,\nu}(x)=x_{n}\cdot
K^{\mathbb{A},+}_{\lambda-1,\nu}=\textup{sgn}(x_{n})|x_{n}|^{\lambda+\nu-\frac{1}{2}}(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n-1}{2}},\qquad
x\in\mathbb{R}^{n}.$

To normalize $K^{\mathbb{A},-}_{\lambda,\nu}(x)$ so that it becomes
holomorphic in $(\lambda,\nu)\in\mathbb{C}^{2}$ we first consider the kernel
of the map

$x_{n}:\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}\to\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda+1,\nu}^{-}.$

######  Lemma 3.4.

$\\{K\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}:x_{n}K=0\\}=\begin{cases}\mathbb{C}\widetilde{K}^{\mathbb{B},+}_{\lambda,\nu}&\mbox{for
$\lambda+\nu=-\frac{1}{2}$,}\\\
\mathbb{C}\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}&\mbox{for
$\lambda-\nu=-\frac{1}{2}$,}\\\ \\{0\\}&\mbox{else.}\end{cases}$

Note that for $(\lambda,\nu)=(-\frac{1}{2},0)$ we have
$\widetilde{K}^{\mathbb{B},+}_{\lambda,\nu}(x)=\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}(x)=\delta(x)$.

###### Proof.

If $K\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ such that
$x_{n}K=0$ then clearly $\operatorname{supp}(K)\subseteq\mathbb{R}^{n-1}$.
Hence,

$K=\sum_{m=0}^{M}u_{m}(x^{\prime})\delta^{(m)}(x_{n})$

for some distributions $u_{m}\in\mathcal{D}^{\prime}(\mathbb{R}^{n-1})$. Since
$x_{n}\delta^{(m)}(x_{n})=-m\delta^{(m-1)}(x_{n})$ we must have $u_{m}=0$ for
$m>0$, so that $K(x)=u_{0}(x^{\prime})\delta(x_{n})$. By the classification in
Theorems 3.2 and 3.3 the only possibilities for such $K$ in the space
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}$ are

$K(x)=\widetilde{K}^{\mathbb{B},+}_{\lambda,\nu}(x)=\frac{|x^{\prime}|^{1-n-2\nu}\delta(x_{n})}{\Gamma(-\nu)}$

with $\lambda+\nu=-\frac{1}{2}$ (i.e. $k=0$) and

$K(x)=\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}(x)=\delta(x)$

with $\lambda-\nu=-\frac{1}{2}$ (i.e. $\ell=0$). ∎

For the full classification of
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$ we also need to
understand the kernel of the map

$x_{n}:\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}\to\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda+1,\nu}^{+}.$

######  Lemma 3.5.

$\\{K\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}:x_{n}K=0\\}=\\{0\\}.$

###### Proof.

As in the previous proof any $K$ with $x_{n}K=0$ must be of the form
$K(x)=u_{0}(x^{\prime})\delta(x_{n})$. Now, if
$K\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$ then by the
invariance under $m_{0}\in M_{H}$ we have $u(-x)=-u(x)$ and since
$\delta(-x_{n})=\delta(x_{n})$ this implies

$u_{0}(-x^{\prime})=-u_{0}(x^{\prime}).$

On the other hand, invariance of $K$ under $\textup{O}(n-1)\subseteq M_{H}$
implies $u_{0}(mx^{\prime})=u_{0}(x^{\prime})$ for all $m\in\textup{O}(n-1)$,
in particular for $m=-\mathbf{1}$, hence

$u_{0}(-x^{\prime})=u_{0}(x^{\prime}).$

This shows that $u_{0}=0$ and the proof is complete. ∎

To state the analogue of Theorem 3.2 for $K^{\mathbb{A},-}_{\lambda,\nu}(x)$
let

$\displaystyle\backslash\backslash^{-}$
$\displaystyle=\\{(\lambda,\nu):\lambda+\nu=-\tfrac{3}{2}-2k,k\in\mathbb{N}\\},$
$\displaystyle//^{-}$
$\displaystyle=\\{(\lambda,\nu):\lambda-\nu=-\tfrac{3}{2}-2\ell,\ell\in\mathbb{N}\\},$
$\displaystyle L_{\textup{odd}}$
$\displaystyle=\\{(\lambda,\nu)=(-\rho-i-1,-\rho_{H}-j):i,j\in\mathbb{N},i-j\in
2\mathbb{N}\\}\subseteq//^{-}.$

For $(\lambda,\nu)\in\backslash\backslash^{-}$ with
$\lambda+\nu=-\frac{3}{2}-2k$ we further define

$\displaystyle\widetilde{K}^{\mathbb{B},-}_{\lambda,\nu}(x)$
$\displaystyle=\frac{1}{\Gamma(\frac{1}{2}(\lambda-\nu+\frac{3}{2}))}(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n-1}{2}}\delta^{(2k+1)}(x_{n})$
$\displaystyle=\sum_{i=0}^{k}\frac{(-1)^{i}(2k+1)!(\nu+\frac{n-1}{2})_{i}}{i!(2k-2i+1)!\Gamma(\frac{1}{2}(\lambda-\nu+\frac{3}{2}))}|x^{\prime}|^{1-n-2\nu-2i}\delta^{(2k-2i+1)}(x_{n}),$
and for $(\lambda,\nu)\in//^{-}$ with $\lambda-\nu=-\frac{3}{2}-2\ell$ we put
$\displaystyle\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}(x)$
$\displaystyle=\sum_{j=0}^{\ell}\frac{2^{2\ell-2j}(\nu-\ell)_{\ell-j}}{j!(2\ell-2j+1)!}(\Delta_{\mathbb{R}^{n-1}}^{j}\partial_{n}^{2\ell-2j+1}\delta)(x).$

######  Theorem 3.6.

The renormalized distribution

$\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x)=\frac{K^{\mathbb{A},-}_{\lambda,\nu}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu+\frac{3}{2}))\Gamma(\frac{1}{2}(\lambda-\nu+\frac{3}{2}))}$

depends holomorphically on $(\lambda,\nu)\in\mathbb{C}^{2}$ and vanishes only
for $(\lambda,\nu)\in L_{\textup{odd}}$. More precisely,

  1. (1)

For $(\lambda,\nu)\in\mathbb{C}^{2}-(\backslash\backslash^{-}\cup//^{-})$ we
have
$\operatorname{supp}\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}=\mathbb{R}^{n}$.

  2. (2)

For $(\lambda,\nu)\in\backslash\backslash^{-}-//^{-}$ with
$\lambda+\nu=-\frac{3}{2}-2k$, $k\in\mathbb{N}$, we have

$\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x)=\frac{(-1)^{k+1}k!}{(2k+1)!}\widetilde{K}^{\mathbb{B},-}_{\lambda,\nu}(x).$

In particular,
$\operatorname{supp}\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}=\mathbb{R}^{n-1}$.

  3. (3)

For $(\lambda,\nu)\in//^{-}-L_{\textup{odd}}$ with
$\lambda-\nu=-\frac{3}{2}-2\ell$, $\ell\in\mathbb{N}$, we have

$\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x)=\frac{(-1)^{\ell+1}\ell!\pi^{\frac{n-1}{2}}}{2^{2\ell}\Gamma(\nu+\frac{n-1}{2})}\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}(x).$

In particular,
$\operatorname{supp}\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}=\\{0\\}$.

  4. (4)

For $(-\rho-i,-\rho_{H}-j)\in L_{\textup{odd}}$, $2\ell=i-j-1$, restricting
the function $(\lambda,\nu)\mapsto\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}$
to two different complex hyperplanes in $\mathbb{C}^{2}$ through
$(-\rho-i,-\rho_{H}-j)$ and renormalizing gives two holomorphic families of
distributions:

$\displaystyle\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}(x)$
$\displaystyle=\frac{(-1)^{\ell}2^{2\ell}}{\ell!\pi^{\frac{n-1}{2}}}\Gamma(\nu+\rho_{H})\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x),$
$\displaystyle(\lambda-\nu=-\tfrac{3}{2}-2\ell),$
$\displaystyle{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\Gamma(\tfrac{1}{2}(\lambda-\nu+\tfrac{3}{2}))\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x),$
$\displaystyle(\nu=-\rho_{H}-j).$

Their special values at $(-\rho-i,-\rho_{H}-j)$ satisfy

$\displaystyle\operatorname{supp}\widetilde{K}^{\mathbb{C},-}_{-\rho-i,-\rho_{H}-j}$
$\displaystyle=\\{0\\},$
$\displaystyle\operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},-}_{-\rho-i,-\rho_{H}-j}$
$\displaystyle=\begin{cases}\mathbb{R}^{n}&\mbox{for $n$ even,}\\\
\mathbb{R}^{n-1}&\mbox{for $n$ odd,}\end{cases}$

more precisely, for $n$ odd and $2k=n+i+j-2$

$\displaystyle{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\frac{(-1)^{k+1}k!}{(2k+1)!}(|x^{\prime}|^{2}+x_{n}^{2})^{j}\delta^{(2k+1)}(x_{n}),$

###### Proof.

We first apply the map

$x_{n}:\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda-1,\nu}^{+}\to\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$

to the holomorphic family $\widetilde{K}^{\mathbb{A},+}_{\lambda-1,\nu}$ and
obtain a holomorphic family of distributions

$x_{n}\widetilde{K}^{\mathbb{A},+}_{\lambda-1,\nu}(x)=\frac{K^{\mathbb{A},-}_{\lambda,\nu}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu-\frac{1}{2}))\Gamma(\frac{1}{2}(\lambda-\nu-\frac{1}{2}))}.$

By Theorem 3.2 and Lemma 3.4 we have
$x_{n}\widetilde{K}^{\mathbb{A},+}_{\lambda-1,\nu}=0$ if and only if
$\lambda+\nu=\frac{1}{2}$ or $\lambda-\nu=\frac{1}{2}$ or $(\lambda-1,\nu)\in
L_{\textup{even}}$. We can therefore renormalize the kernels

$\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x)=\frac{x_{n}\widetilde{K}^{\mathbb{A},+}_{\lambda-1,\nu}(x)}{\frac{1}{2}(\lambda+\nu-\frac{1}{2})\cdot\frac{1}{2}(\lambda-\nu-\frac{1}{2})}=\frac{K^{\mathbb{A},-}_{\lambda,\nu}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu+\frac{3}{2}))\Gamma(\frac{1}{2}(\lambda-\nu+\frac{3}{2}))}.$

This shows that $\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x)$ depends
holomorphically on $(\lambda,\nu)\in\mathbb{C}^{2}$. Next, we apply the map

(3.1)
$x_{n}:\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}\to\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda+1,\nu}^{+}$

to $\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x)$ and find

$x_{n}\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}=\widetilde{K}^{\mathbb{A},+}_{\lambda+1,\nu}.$

By Lemma 3.5 the map (3.1) is injective, hence
$\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}=0$ if and only if
$\widetilde{K}^{\mathbb{A},+}_{\lambda+1,\nu}=0$, which is only the case for
$(\lambda+1,\nu)\in L_{\textup{even}}$, i.e. $(\lambda,\nu)\in
L_{\textup{odd}}$.

  1. (1)

Let $(\lambda,\nu)\in\mathbb{C}^{2}-(\backslash\backslash^{-}\cup//^{-})$.
Then $(\lambda+1,\nu)\in\mathbb{C}^{2}-(\backslash\backslash^{+}\cup//^{+})$
and hence
$\operatorname{supp}(x_{n}\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu})=\operatorname{supp}(\widetilde{K}^{\mathbb{A},+}_{\lambda+1,\nu})=\mathbb{R}^{n}$.
This implies
$\operatorname{supp}(\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu})=\mathbb{R}^{n}$.

  2. (2)

Let $(\lambda,\nu)\in\backslash\backslash^{-}-//^{-}$ with
$\lambda+\nu=-\frac{3}{2}-2k$, then it is easy to see, using
$x_{n}\delta^{(m)}(x_{n})=-m\delta^{(m-1)}(x_{n})$, that

$x_{n}\widetilde{K}^{\mathbb{B},+}_{\lambda-1,\nu}=2(k+1)(\nu+k+1)\widetilde{K}^{\mathbb{B},-}_{\lambda,\nu}.$

Hence,

$\displaystyle\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}$
$\displaystyle=\frac{1}{\frac{1}{2}(\lambda+\nu-\frac{1}{2})\cdot\frac{1}{2}(\lambda-\nu-\frac{1}{2})}x_{n}\widetilde{K}^{\mathbb{A},+}_{\lambda-1,\nu}$
$\displaystyle=\frac{1}{(k+1)(\nu+k+1)}\frac{(-1)^{k+1}(k+1)!}{(2k+2)!}x_{n}\widetilde{K}^{\mathbb{B},+}_{\lambda-1,\nu}$
$\displaystyle=\frac{(-1)^{k+1}k!}{(2k+1)!}\widetilde{K}^{\mathbb{B},-}_{\lambda,\nu}.$

  3. (3)

For $(\lambda,\nu)\in//^{-}-L_{\textup{odd}}$ the same method as in (2)
applies to show (3). Here we use that
$x_{n}\widetilde{K}^{\mathbb{C},+}_{\lambda-1,\nu}=-4(\nu-\ell-1)\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}$
for $\lambda-\nu=-\frac{3}{2}-2\ell$.

  4. (4)

Now let $(-\rho-i,-\rho_{H}-j)\in L_{\textup{odd}}$. The first statement about
$\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}(x)$ follows immediately from (3).
For the second statement note that $(-\rho-(i+1),-\rho_{H}-j)\in
L_{\textup{even}}$ and we can use Theorem 3.2 (4). Restricting to
$\nu=-\rho_{H}-j$ we have

$\displaystyle{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\Gamma(\tfrac{1}{2}(\lambda-\nu+\tfrac{3}{2}))\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\frac{\Gamma(\tfrac{1}{2}(\lambda-\nu+\tfrac{3}{2}))x_{n}\widetilde{K}^{\mathbb{A},+}_{\lambda-1,\nu}(x)}{\frac{1}{2}(\lambda+\nu-\frac{1}{2})\cdot\frac{1}{2}(\lambda-\nu-\frac{1}{2})}=\frac{x_{n}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},+}_{\lambda-1,\nu}(x)}{\frac{1}{2}(\lambda+\nu-\frac{1}{2})}.$

The denominator is equal to
$\tfrac{1}{2}(\lambda+\nu-\tfrac{1}{2})=-\frac{1}{2}(n+i+j)=-(k+1)\neq 0$,
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 $\lambda\in\mathbb{C}$. Moreover, for even $n$ 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
$\operatorname{supp}(x_{n}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},+}_{\lambda-1,\nu})=\mathbb{R}^{n}$
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 $n$ it follows from $x_{n}\delta^{(m)}(x_{n})=-m\delta^{(m-1)}(x_{n})$
that

$\displaystyle{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=-\frac{1}{k+1}\cdot\frac{(-1)^{k+1}(k+1)!}{(2k+2)!}x_{n}(|x^{\prime}|^{2}+x_{n}^{2})^{j}\delta^{(2k+2)}(x_{n})$
$\displaystyle=\frac{(-1)^{k+1}k!}{(2k+1)!}(|x^{\prime}|^{2}+x_{n}^{2})^{j}\delta^{(2k+1)}(x_{n}).\qed$

######  Remark 3.7.

The differential intertwining operators belonging to the distribution kernels
$\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}$ are the even order conformally
covariant differential operators found by Juhl [4], 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
$\widetilde{K}^{\mathbb{A},+}_{\lambda,\nu}$. Theorem 3.6 proves that also the
odd order Juhl operators with integral kernels
$\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}$ can be obtained in this way.
Further, the translation principle explains the following relation between
even and odd order Juhl operators:

$x_{n}\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}=-2(\lambda+\nu+\tfrac{1}{2})\widetilde{K}^{\mathbb{C},-}_{\lambda+1,\nu}\qquad\mbox{and}\qquad
x_{n}\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}=-\widetilde{K}^{\mathbb{C},+}_{\lambda+1,\nu}.$

Using the translation principle we can finally determine the space
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$ completely:

######  Theorem 3.8.

We have

(3.2)
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}=\begin{cases}\mathbb{C}\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}&\mbox{for
$(\lambda,\nu)\in\mathbb{C}^{2}-L_{\textup{odd}}$,}\\\
\mathbb{C}{\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}\oplus\mathbb{C}\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}&\mbox{for
$(\lambda,\nu)\in L_{\textup{odd}}$.}\end{cases}$

###### Proof.

In the proof of Theorem 3.6 we showed that the map

$x_{n}:\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}\to\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda+1,\nu}^{+}$

is injective and hence

(3.3)
$\dim\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}\leq\dim\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda+1,\nu}^{+}.$

For $(\lambda,\nu)\in\mathbb{C}^{2}-L_{\textup{odd}}$ we have
$0\neq\widetilde{K}^{\mathbb{A},-}_{\lambda,\nu}\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$,
and hence the left hand side of (3.3) is $\geq 1$. For $(\lambda,\nu)\in
L_{\textup{odd}}$ 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}$}}}}^{\mathbb{A},-}_{\lambda,\nu},\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}$
and these distributions are linearly independent since

$\operatorname{supp}\widetilde{K}^{\mathbb{C},-}_{\lambda,\nu}=\\{0\\}\subsetneq\operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},-}_{\lambda,\nu}.$

This shows that the left hand side of (3.3) is $\geq 2$. Now note that
$(\lambda,\nu)\in L_{\textup{odd}}$ if and only if $(\lambda+1,\nu)\in
L_{\textup{even}}$, and therefore (3.3) is actually an equality for all
$(\lambda,\nu)\in\mathbb{C}^{2}$, thanks to Theorem 3.3. This finishes the
proof. ∎

##  4\. Example:
$(\widetilde{G},\widetilde{H})=(\textup{Pin}(n+1,1),\textup{Pin}(n,1))$

We explicitly construct intertwining operators between principal series
representations of $\widetilde{G}=\textup{Pin}(n+1,1)$ and
$\widetilde{H}=\textup{Pin}(n,1)$ induced from fundamental spin
representations of
$\widetilde{M}\simeq\textup{Pin}(n)\times\mathbb{Z}/2\mathbb{Z}$ and
$\widetilde{M}_{H}\simeq\textup{Pin}(n-1)\times\mathbb{Z}/2\mathbb{Z}$ using
the translation principle.

###  4.1. Parabolic subgroups and the symmetric pair

We consider the two-fold covering

$q:\widetilde{G}=\textup{Pin}(n+1,1)\to\textup{O}(n+1,1)=G,$

realized inside the Clifford algebra $\operatorname{Cl}(n+1,1)$ of
$\mathbb{R}^{n+1,1}$. 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 $\widetilde{S}=q^{-1}(S)$ for any subgroup $S\subseteq
G$.

We choose the same parabolic subgroup $P=MAN\subseteq G$ as in Section 3.1,
then $\widetilde{M}=\textup{Pin}(n)\cdot\\{1,m_{0}=e_{1}e_{n+2}\\}$, where the
embedding of $\textup{Pin}(n)$ into $\textup{Pin}(n+1,1)$ is the restriction
of the embedding of Clifford algebras
$\operatorname{Cl}(n)\hookrightarrow\operatorname{Cl}(n+1,1)$ induced by

$\mathbb{R}^{n}\hookrightarrow\mathbb{R}^{n+1,1},\quad e_{i}\mapsto
e_{i+1}\quad(1\leq i\leq n).$

We note that $m_{0}=e_{1}e_{n+2}$ commutes with $\textup{Pin}(n)$ and hence
$\widetilde{M}\simeq\textup{Pin}(n)\times\mathbb{Z}/2\mathbb{Z}$. Both $A$ and
$N$ split in the cover, so that $\widetilde{A}\simeq
A\times\mathbb{Z}/2\mathbb{Z}$ and $\widetilde{N}\simeq
N\times\mathbb{Z}/2\mathbb{Z}$, and we identify $A$ and $N$ with
$\widetilde{A}_{0},\widetilde{N}_{0}\subseteq\widetilde{G}$. Then
$\widetilde{P}=\widetilde{M}AN$ is the Langlands decomposition of the
parabolic subgroup $\widetilde{P}$ of $\widetilde{G}$.

For the Weyl group element $w_{0}$ we choose the representative

(4.1) $\tilde{w}_{0}=\pm e_{n+2}$

with the sign yet to be determined. Since
$q(\tilde{w}_{0})=\textup{diag}(1,\ldots,1,-1)$, the element $\tilde{w}_{0}$
corresponds to the representative chosen in Section 3.1 for the group $G$.
Hence, by Lemma 3.1 we find

$q(m(\tilde{w}_{0}^{-1}\overline{n}_{x}^{-1})^{-1})=\left(\begin{array}[]{ccc}1&&\\\
&\mathbf{1}_{n}-2\frac{xx^{\top}}{|x|^{2}}&\\\ &&1\end{array}\right),\qquad
x\in\mathbb{R}^{n}-\\{0\\}.$

The matrix $\mathbf{1}_{n}-2\frac{xx^{\top}}{|x|^{2}}\in\textup{O}(n)$ is the
orthogonal reflection in $\mathbb{R}^{n}$ at the hyperplane orthogonal to $x$,
and the corresponding elements in the cover
$\textup{Pin}(n)\subseteq\operatorname{Cl}(n)$ are $\pm\frac{x}{|x|}$.
Therefore, we may choose the sign in (4.1) so that under the identification
$\widetilde{M}\simeq\textup{Pin}(n)\times\mathbb{Z}/2\mathbb{Z}$ we have

$m(\tilde{w}_{0}^{-1}\overline{n}_{x}^{-1})^{-1}=-\tfrac{x}{|x|}\in\textup{Pin}(n)\subseteq\widetilde{M}.$

Further we note that

$\tilde{w}_{0}m\tilde{w}_{0}^{-1}=\alpha(m)=\det(m)m\quad(m\in\textup{Pin}(n)),\qquad\qquad\tilde{w}_{0}m_{0}\tilde{w}_{0}^{-1}=-m_{0},$

where $\alpha$ denotes the canonical automorphism of the Clifford algebra
$\operatorname{Cl}(n)$ and $\det:\textup{Pin}(n)\to\textup{O}(n)\to\\{\pm
1\\}$ the determinant character of $\textup{Pin}(n)$.

The symmetric pair $(G,H)=(\textup{O}(n+1,1),\textup{O}(n,1))$ as introduced
in Section 3.1 takes in the cover the form
$(\widetilde{G},\widetilde{H})=(\textup{Pin}(n+1,1),\textup{Pin}(n,1))$. We
have

$\widetilde{H}=\widetilde{G}\cap\operatorname{Cl}(n,1),$

where the embedding of $\operatorname{Cl}(n,1)$ into
$\operatorname{Cl}(n+1,1)$ is induced by the embedding

$\mathbb{R}^{n,1}\hookrightarrow\mathbb{R}^{n+1,1},\quad
x\mapsto(x_{1},\ldots,x_{n},0,x_{n+1}).$

Further,
$\widetilde{P}_{H}=\widetilde{P}\cap\widetilde{H}=\widetilde{M}_{H}AN_{H}$ is
a parabolic subgroup of $\widetilde{H}$ with
$\widetilde{M}_{H}=\textup{Pin}(n-1)\cdot\\{1,m_{0}\\}$.

###  4.2. Principal series representations and symmetry breaking operators

For the fundamental spin representations of $\textup{Pin}(n)$ we use the
notation introduced in Appendix A.2. The group $\textup{Pin}(n)$ has for even
$n$ one fundamental spin representation, and for odd $n$ two fundamental spin
representations. For $n$ even we have $\zeta_{n}\otimes\det\simeq\zeta_{n}$
for the fundamental spin representation $\zeta_{n}$, and for $n$ odd and
$\zeta_{n}$ any fundamental spin representation of $\textup{Pin}(n)$ the
representations $\zeta_{n}$ and $\zeta_{n}\otimes\det$ are non-equivalent.

Denote by $\textup{sgn}:\\{\mathbf{1},m_{0}\\}\to\\{\pm 1\\}$ the non-trivial
character of the two-element group $\\{\mathbf{1},m_{0}\\}$. Then for
$\delta\in\mathbb{Z}/2\mathbb{Z}$ and a fundamental spin representation
$\zeta_{n}$ of $\textup{Pin}(n)$ we consider the representation
$\zeta_{n}\otimes\textup{sgn}^{\delta}$ of
$\widetilde{M}=\textup{Pin}(n)\times\\{\mathbf{1},m_{0}\\}$. Note that

(4.2)
$\tilde{w}_{0}(\zeta_{n}\otimes\textup{sgn}^{\delta})=[\zeta_{n}\otimes\det]\otimes\textup{sgn}^{1-\delta}.$

For $(\zeta_{n},\SS_{n})$ a fundamental spin representation of
$\textup{Pin}(n)$, $\delta\in\mathbb{Z}/2\mathbb{Z}$ and
$\lambda\in\mathfrak{a}_{\mathbb{C}}^{*}\simeq\mathbb{C}$ we form the
principal series representations

$\not{\pi}_{\lambda,\delta}=\textup{Ind}_{\widetilde{P}}^{\widetilde{G}}((\zeta_{n}\otimes\textup{sgn}^{\delta})\otimes
e^{\lambda}\otimes\mathbf{1}).$

Similarly, for $(\zeta_{n-1},\SS_{n-1})$ a fundamental spin representation of
$\textup{Pin}(n-1)$, $\delta\in\mathbb{Z}/2\mathbb{Z}$ and
$\nu\in\mathfrak{a}_{H,\mathbb{C}}^{*}\simeq\mathbb{C}$ we form principal
series representations of $\widetilde{H}$:

$\not{\tau}_{\nu,\varepsilon}=\textup{Ind}_{\widetilde{P}_{H}}^{\widetilde{H}}((\zeta_{n-1}\otimes\textup{sgn}^{\varepsilon})\otimes
e^{\nu}\otimes\mathbf{1}).$

The main result of this section is the construction of symmetry breaking
operators

(4.3)
$A\in\textup{Hom}_{\widetilde{H}}(\not{\pi}_{\lambda,\delta}|_{\widetilde{H}},\not{\tau}_{\nu,\varepsilon}).$

In Section 5 we then show that this construction actually gives a full
classification of symmetry breaking operators between spinor-valued principal
series.

By Theorem 1.2 every intertwining operator in (4.3) is uniquely determined by
its distribution kernel
$K\in\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}))$.
As explained in Section 3.2, the space of distribution kernels describing
intertwining operators in (4.3) only depends on
$(\lambda,\nu)\in\mathbb{C}^{2}$ and the parity of $\delta+\varepsilon$. We
therefore denote by

$\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}))_{\lambda,\nu}^{\pm}\subseteq\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}))$

the corresponding space of distribution kernels of intertwining operators,
where the sign $+$ describes kernels for $\delta+\varepsilon\equiv 0(2)$ and
the sign $-$ describes kernels for $\delta+\varepsilon\equiv 1(2)$.

###  4.3. Construction of symmetry breaking operators

We extend the representation $(\zeta_{n},\SS_{n})$ of $\textup{Pin}(n)$ to a
representation of the Clifford algebra $\operatorname{Cl}(n;\mathbb{C})$ and
write $(\zeta,\SS)=(\zeta_{n},\SS_{n})$ for short (see Appendix A.2. Consider
the representation $\tau^{\prime}=[\zeta\otimes\det]\otimes\mathbf{1}$ of
$\widetilde{M}\simeq\textup{Pin}(n)\times\mathbb{Z}/2\mathbb{Z}$. By Lemma 2.2
there exists a representation $(\tau,E)$ of
$\widetilde{G}=\textup{Pin}(n+1,1)$ containing a subspace $E^{\prime}\subseteq
E$ invariant under $\widetilde{M}A\overline{N}$ such that
$E^{\prime}\simeq\tau^{\prime}\otimes e^{\mu^{\prime}}\otimes\mathbf{1}$ or
$E^{\prime}\simeq\tilde{w}_{0}\tau^{\prime}\otimes
e^{\mu^{\prime}}\otimes\mathbf{1}$ for some $\mu^{\prime}\in\mathfrak{a}^{*}$.
By possibly replacing $(\tau,E)$ by its twist $(\tau\circ\theta,E)$ by the
Cartan involution $\theta$ we may assume that
$E^{\prime}\simeq\tau^{\prime}\otimes e^{\mu^{\prime}}\otimes\mathbf{1}$. It
is easy to see that $\mu^{\prime}=-\frac{1}{2}\alpha$. By (4.2) we have

$w_{0}(\tau^{\prime}\otimes
e^{\mu^{\prime}}\otimes\mathbf{1})\simeq(\zeta\otimes\textup{sgn})\otimes
e^{-\mu^{\prime}}\otimes\mathbf{1}.$

Further, using Lemma 3.1 we have the following expression for the translation
kernel:

$\tau^{\prime}(m(\tilde{w}_{0}^{-1}\overline{n}_{x}^{-1})^{-1})\cdot
a(\tilde{w}_{0}^{-1}\overline{n}_{x}^{-1})^{-\mu^{\prime}}=[\zeta\otimes\det](-\tfrac{x}{|x|})\cdot|x|^{-2\mu^{\prime}}=\zeta(x).$

Then Corollary 2.7 gives linear maps

$\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})\to\textup{Hom}_{\widetilde{H}}(\not{\pi}_{\lambda+\frac{1}{2},1-\delta},\textup{Ind}_{\widetilde{P}_{H}}^{\widetilde{H}}(([\zeta\otimes\det]\otimes\textup{sgn}^{\varepsilon})\otimes
e^{\nu-\frac{1}{2}}\otimes\mathbf{1}))$

which are on the level of integral kernels given by

(4.4)
$\mathcal{D}^{\prime}(\mathbb{R}^{n})\to\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{End}_{\mathbb{C}}(\SS)),\quad
K(x)\mapsto\zeta(x)\cdot K(x).$

To obtain intertwining operators into $\not{\tau}_{\nu,\varepsilon}$ note that
$\zeta_{n-1}$ occurs in the restriction
$[\zeta_{n}\otimes\det]|_{\textup{Pin}(n-1)}$ with multiplicity one, and fix a
projection $P:[\zeta_{n}\otimes\det]|_{\textup{Pin}(n-1)}\to\zeta_{n-1}$. Then
we have a linear map

$\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})\to\textup{Hom}_{\widetilde{H}}(\not{\pi}_{\lambda+\frac{1}{2},1-\delta},\not{\tau}_{\nu-\frac{1}{2},\varepsilon})$

which is on the level of integral kernels given by

$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{\pm}\to\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}))_{\lambda+\frac{1}{2},\nu-\frac{1}{2}}^{\mp},\quad
K(x)\mapsto(P\zeta(x))\cdot K(x).$

By Theorems 3.3 and 3.8 all symmetry breaking operators in
$\textup{Hom}_{H}(\pi_{\lambda,\delta}|_{H},\tau_{\nu,\varepsilon})$ are
obtained from the meromorphic family of kernels
$K^{\mathbb{A},\pm}_{\lambda,\nu}\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{\pm}$.
We therefore consider the
$\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1})$-valued kernels

$P\not{K}^{\mathbb{A},\pm}_{\lambda,\nu}(x)=(P\zeta(x))\cdot
K^{\mathbb{A},\mp}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x).$

For $n$ odd the restriction of the $\textup{Pin}(n)$-representation
$\zeta_{n}\otimes\det$ to $\textup{Pin}(n-1)$ is isomorphic to $\zeta_{n-1}$,
so the map $P$ is an isomorphism. Therefore, we can as well study the
$\textup{End}_{\mathbb{C}}(\SS)$-valued distributions

$\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}(x)=\zeta(x)\cdot
K_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}^{\mathbb{A},\mp}(x).$

For $n$ even the restriction of the $\textup{Pin}(n)$-representation
$\zeta_{n}\otimes\det$ to $\textup{Pin}(n-1)$ is isomorphic to the direct sum
of $\zeta_{n-1}$ and $\zeta_{n-1}\otimes\det$. The next result shows that the
poles of
$P\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}\in\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{Hom}_{\mathbb{C}}(\SS_{n},\SS_{n-1}))$
and
$\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}\in\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{End}_{\mathbb{C}}(\SS))$
agree and that the residues of $P\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}$ agree
with the composition of the residues of
$\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}$ with $P$. Note that

$\not{K}_{\lambda,\nu}^{\mathbb{A},\pm}(x)=\sum_{i=1}^{n}\Big{(}x_{i}\cdot
K_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}^{\mathbb{A},\mp}(x)\Big{)}\underline{e_{i}},$

where
$\underline{e_{i}}=\zeta_{n}(e_{i})\in\textup{End}_{\mathbb{C}}(\SS_{n})$.

######  Proposition 4.1.

Assume $n$ is even and let
$u\in\mathcal{D}^{\prime}(\mathbb{R}^{n};\textup{End}_{\mathbb{C}}(\SS_{n}))$
be a distribution of the form

$u(x)=\sum_{i=1}^{n}u_{i}(x)\underline{e_{i}}$

with $u_{i}\in\mathcal{D}^{\prime}(\mathbb{R}^{n})$. Then
$\operatorname{supp}u=\operatorname{supp}Pu$, in particular $Pu=0$ if and only
if $u=0$.

###### Proof.

We have

$(Pu)(x)=\sum_{i=1}^{n}u_{i}(x)\cdot(P\circ\underline{e_{i}}).$

We claim that the operators
$P\circ\underline{e_{1}},\ldots,P\circ\underline{e_{n}}$ are linearly
independent, then the statement follows.
Decompose $\zeta_{n}\otimes\det$ into irreducible
$\textup{Pin}(n-1)$-representations $\SS_{n}=\SS_{n}^{1}\oplus\SS_{n}^{2}$,
where $P:\SS_{n}^{1}\to\SS_{n-1}$ is an isomorphism and $\SS_{n}^{2}=\ker P$.
Now note that for $1\leq i\leq n-1$ we have

$P\circ\underline{e_{i}}=P\circ\zeta_{n}(e_{i})=-P\circ[\zeta_{n}\otimes\det](e_{i})=-\zeta_{n-1}(e_{i})\circ
P$

and therefore $P\circ\underline{e_{i}}$ vanishes on $\SS_{n}^{2}$. On the
other hand, $\underline{e_{n}}=\zeta_{n}(e_{n})$ maps $\SS_{n}^{1}$ to
$\SS_{n}^{2}$, so that $P\circ\underline{e_{n}}$ vanishes on $\SS_{n}^{1}$.
Now assume that

$\sum_{i=1}^{n}\lambda_{i}P\circ\underline{e_{i}}=0.$

Restricting to $\SS_{n}^{1}$ this implies

$\sum_{i=1}^{n-1}\lambda_{i}\zeta_{n-1}(e_{i})=0$

and hence $\lambda_{1}=\ldots=\lambda_{n-1}=0$ since
$\zeta_{n-1}(e_{1}),\ldots,\zeta_{n-1}(e_{n-1})$ are linearly independent.
Then also $\lambda_{n}=0$ and the proof is complete. ∎

To find the right normalization making
$\not{K}^{\mathbb{A},\pm}_{\lambda,\nu}(x)$ holomorphically dependent on
$(\lambda,\nu)\in\mathbb{C}^{2}$ we first study the kernel in
$\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{\pm}$ of the map (4.4).
Note that $\zeta(x)=\sum_{i=1}^{n}x_{i}\underline{e_{i}}$, and the operators
$\underline{e_{1}},\ldots,\underline{e_{n}}\in\textup{End}_{\mathbb{C}}(\SS)$
are linearly independent. Hence, for any distribution
$u\in\mathcal{D}^{\prime}(\mathbb{R}^{n})$ we have

$\zeta\cdot u=0\qquad\Leftrightarrow\qquad x_{i}u=0\quad(1\leq i\leq n)$

and

(4.5) $\operatorname{supp}(\zeta\cdot
u)=\bigcup_{i=1}^{n}\operatorname{supp}(x_{i}u).$

######  Lemma 4.2.

We have

$\displaystyle\\{K\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{+}:\zeta\cdot
K=0\\}$
$\displaystyle=\begin{cases}\mathbb{C}\widetilde{K}^{\mathbb{C},+}_{\lambda,\nu}=\mathbb{C}\delta&\mbox{for
$\lambda-\nu=-\frac{1}{2}$,}\\\ \\{0\\}&\mbox{else,}\end{cases}$
$\displaystyle\\{K\in\mathcal{D}^{\prime}(\mathbb{R}^{n})_{\lambda,\nu}^{-}:\zeta\cdot
K=0\\}$ $\displaystyle=\\{0\\}.$

###### Proof.

It is easy to see that multiples of $K=\delta$ are the only distributions with
$x_{i}K=0$ for all $1\leq i\leq n$. Then the claim follows from Theorems 3.2,
3.3, 3.6 and 3.8. ∎

To state the analogues of Theorems 3.2 and 3.6 for
$\not{K}^{\mathbb{A},\pm}_{\lambda,\nu}(x)$ we write
$\operatorname{\not{D}}_{\mathbb{R}^{n-1}}$ and $\operatorname{\not{D}}_{n}$
for the operators

$\operatorname{\not{D}}_{\mathbb{R}^{n-1}}=\sum_{i=1}^{n-1}\underline{e_{i}}\frac{\partial}{\partial
x_{i}},\qquad\operatorname{\not{D}}_{n}=\underline{e_{n}}\frac{\partial}{\partial
x_{n}}.$

####  4.3.1. The distributions $\not{K}^{\mathbb{A},+}_{\lambda,\nu}(x)$

Let

$\displaystyle\underline{\backslash\backslash}^{+}$
$\displaystyle=\\{(\lambda,\nu):\lambda+\nu=-\tfrac{3}{2}-2k,k\in\mathbb{N}\\},$
$\displaystyle\underline{//}^{+}$
$\displaystyle=\\{(\lambda,\nu):\lambda-\nu=-\tfrac{1}{2}-2\ell,k\in\mathbb{N}\\},$
$\displaystyle\not{L}_{\textup{even}}$
$\displaystyle=\\{(-\rho-\tfrac{1}{2}-i,-\rho_{H}-\tfrac{1}{2}-j):i,j\in\mathbb{N},i-j\in
2\mathbb{N}\\}\subseteq\underline{//}^{+}.$

For $(\lambda,\nu)\in\underline{\backslash\backslash}^{+}$ with
$\lambda+\nu=-\frac{3}{2}-2k$ we further define

$\displaystyle\widetilde{\not{K}}^{\mathbb{B},+}_{\lambda,\nu}(x)={}$
$\displaystyle\frac{1}{\Gamma(\frac{1}{2}(\lambda-\nu+\frac{1}{2}))}\Big{(}\zeta(x^{\prime})(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n}{2}}\delta^{(2k+1)}(x_{n})$
$\displaystyle\hskip
99.58464pt-(2k+1)(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n}{2}}\operatorname{\not{D}}_{n}\delta^{(2k-1)}(x_{n})\Big{)},$
and for $(\lambda,\nu)\in\underline{//}^{+}$ with
$\lambda-\nu=-\frac{1}{2}-2\ell$ we put
$\displaystyle\widetilde{\not{K}}^{\mathbb{C},+}_{\lambda,\nu}(x)={}$
$\displaystyle\sum_{j=0}^{\ell-1}\frac{2^{2\ell-2j-1}(\nu+\frac{1}{2}-\ell)_{\ell-j-1}}{j!(2\ell-2j-1)!}(\Delta_{\mathbb{R}^{n-1}}^{j}\partial_{n}^{2\ell-2j-1}\operatorname{\not{D}}_{\mathbb{R}^{n-1}}\delta)(x)$
$\displaystyle+\sum_{j=0}^{\ell}\frac{2^{2\ell-2j}(\nu+\frac{1}{2}-\ell)_{\ell-j}}{j!(2\ell-2j)!}(\Delta_{\mathbb{R}^{n-1}}^{j}\partial_{n}^{2\ell-2j-1}\operatorname{\not{D}}_{n}\delta)(x).$

Note that both $\widetilde{\not{K}}^{\mathbb{B},+}_{\lambda,\nu}$ and
$\widetilde{\not{K}}^{\mathbb{C},+}_{\lambda,\nu}$ depend holomorphically on
$\lambda\in\mathbb{C}$ (or $\nu\in\mathbb{C}$).

######  Theorem 4.3.

The renormalized distribution

$\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}(x)=\frac{\not{K}^{\mathbb{A},+}_{\lambda,\nu}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu+\frac{3}{2}))\Gamma(\frac{1}{2}(\lambda-\nu+\frac{1}{2}))}$

depends holomorphically on $(\lambda,\nu)\in\mathbb{C}^{2}$ and vanishes only
for $(\lambda,\nu)\in\not{L}_{\textup{even}}$. More precisely,

  1. (1)

For
$(\lambda,\nu)\in\mathbb{C}^{2}-(\underline{\backslash\backslash}^{+}\cup\underline{//}^{+})$
we have
$\operatorname{supp}\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}=\mathbb{R}^{n}$.

  2. (2)

For $(\lambda,\nu)\in\underline{\backslash\backslash}^{+}-\underline{//}^{+}$
with $\lambda+\nu=-\frac{3}{2}-2k$, $k\in\mathbb{N}$, we have

$\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}(x)=\frac{(-1)^{k+1}k!}{(2k+1)!}\widetilde{\not{K}}^{\mathbb{B},+}_{\lambda,\nu}(x).$

In particular,
$\operatorname{supp}\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}=\mathbb{R}^{n-1}$.

  3. (3)

For $(\lambda,\nu)\in\underline{//}^{+}-\not{L}_{\textup{even}}$ with
$\lambda-\nu=-\frac{1}{2}-2\ell$, $\ell\in\mathbb{N}$, we have

$\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}(x)=\frac{(-1)^{\ell}\ell!\pi^{\frac{n-1}{2}}}{2^{2\ell}\Gamma(\nu+\frac{n}{2})}\widetilde{\not{K}}^{\mathbb{C},+}_{\lambda,\nu}(x).$

In particular,
$\operatorname{supp}\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}=\\{0\\}$.

  4. (4)

For $(-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j)\in\not{L}_{\textup{even}}$,
$2\ell=i-j$, restricting the function
$(\lambda,\nu)\mapsto\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}$ to two
different complex hyperplanes in $\mathbb{C}^{2}$ through
$(-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j)$ and renormalizing gives two
holomorphic families of distributions:

$\displaystyle\widetilde{\not{K}}^{\mathbb{C},+}_{\lambda,\nu}(x)$
$\displaystyle=\frac{(-1)^{\ell}2^{2\ell}}{\ell!\pi^{\frac{n-1}{2}}}\Gamma(\nu+\rho_{H}+\tfrac{1}{2})\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}(x),$
$\displaystyle(\lambda-\nu=-\tfrac{1}{2}-(i-j)),$
$\displaystyle{\mathchoice{\widetilde{\hbox{\set@color$\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}}$}}}}^{\mathbb{A},+}_{\lambda,\nu}(x)$
$\displaystyle=\Gamma(\tfrac{1}{2}(\lambda-\nu+\tfrac{1}{2}))\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}(x),$
$\displaystyle(\nu=-\rho_{H}-\tfrac{1}{2}-j).$

Their special values at
$(\lambda,\nu)=(-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j)$ satisfy

$\displaystyle\operatorname{supp}\widetilde{\not{K}}^{\mathbb{C},+}_{-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j}$
$\displaystyle=\\{0\\},$
$\displaystyle\operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color$\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}}$}}}}^{\mathbb{A},+}_{-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j}$
$\displaystyle=\begin{cases}\mathbb{R}^{n}&\mbox{for $n$ even,}\\\
\mathbb{R}^{n-1}&\mbox{for $n$ odd,}\end{cases}$

more precisely, for $n$ odd and $2k=n+i+j-1$

${\mathchoice{\widetilde{\hbox{\set@color$\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}}$}}}}^{\mathbb{A},+}_{-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j}=\frac{(-1)^{k+1}k!}{(2k+1)!}(|x^{\prime}|^{2}+x_{n}^{2})^{j}\Big{(}\zeta(x^{\prime})\delta^{(2k+1)}(x_{n})-(2k+1)\operatorname{\not{D}}_{n}\delta^{(2k-1)}(x_{n})\Big{)}.$

###### Proof.

By Theorem 3.6 the distribution

$\widetilde{\not{K}}^{\mathbb{A},+}_{\lambda,\nu}(x)=\zeta(x)\cdot\widetilde{K}^{\mathbb{A},-}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)=\frac{\zeta(x)\cdot
K^{\mathbb{A},-}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu+\frac{3}{2}))\Gamma(\frac{1}{2}(\lambda-\nu+\frac{1}{2}))}$

depends holomorphically on $(\lambda,\nu)\in\mathbb{C}^{2}$, and by Lemma 4.2
it vanishes if and only if $(\lambda-\frac{1}{2},\nu+\frac{1}{2})\in
L_{\textup{odd}}$, i.e. $(\lambda,\nu)\in\not{L}_{\textup{even}}$. The rest is
similar to the proof of Theorem 4.4 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

$(\lambda,\nu)\in\underline{\backslash\backslash}^{+}\Leftrightarrow(\lambda-\tfrac{1}{2},\nu+\tfrac{1}{2})\in\backslash\backslash^{-},\qquad(\lambda,\nu)\in\underline{//}^{+}\Leftrightarrow(\lambda-\tfrac{1}{2},\nu+\tfrac{1}{2})\in//^{-}.\qed$

####  4.3.2. The distributions $\not{K}^{\mathbb{A},-}_{\lambda,\nu}(x)$

Let

$\displaystyle\underline{\backslash\backslash}^{-}$
$\displaystyle=\\{(\lambda,\nu):\lambda+\nu=-\tfrac{1}{2}-2k,k\in\mathbb{N}\\},$
$\displaystyle\underline{//}^{-}$
$\displaystyle=\\{(\lambda,\nu):\lambda-\nu=-\tfrac{3}{2}-2\ell,k\in\mathbb{N}\\},$
$\displaystyle\not{L}_{\textup{odd}}$
$\displaystyle=\\{(-\rho-\tfrac{1}{2}-i,-\rho_{H}-\tfrac{1}{2}-j):i,j\in\mathbb{N},i-j\in
2\mathbb{N}+1\\}\subseteq\underline{//}^{-}.$

For $(\lambda,\nu)\in\underline{\backslash\backslash}^{-}$ with
$\lambda+\nu=-\frac{1}{2}-2k$ we further define

$\displaystyle\widetilde{\not{K}}^{\mathbb{B},-}_{\lambda,\nu}(x)={}$
$\displaystyle\frac{1}{\Gamma(\frac{1}{2}(\lambda-\nu+\frac{3}{2}))}\Big{(}\zeta(x^{\prime})(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n}{2}}\delta^{(2k)}(x_{n})$
$\displaystyle\hskip
99.58464pt-2k(|x^{\prime}|^{2}+x_{n}^{2})^{-\nu-\frac{n}{2}}\operatorname{\not{D}}_{n}\delta^{(2k-2)}(x_{n})\Big{)},$
and for $(\lambda,\nu)\in\underline{//}^{-}$ with
$\lambda-\nu=-\frac{3}{2}-2\ell$ we put
$\displaystyle\widetilde{\not{K}}^{\mathbb{C},-}_{\lambda,\nu}(x)={}$
$\displaystyle\sum_{j=0}^{\ell}\frac{2^{2\ell-2j}(\nu-\ell-\frac{1}{2})_{\ell-j}}{j!(2\ell-2j)!}(\Delta_{\mathbb{R}^{n-1}}^{j}\partial_{n}^{2\ell-2j}\operatorname{\not{D}}_{\mathbb{R}^{n-1}}\delta)(x)$
$\displaystyle+\sum_{j=0}^{\ell}\frac{2^{2\ell-2j+1}(\nu-\ell-\frac{1}{2})_{\ell-j+1}}{j!(2\ell-2j+1)!}(\Delta_{\mathbb{R}^{n-1}}^{j}\partial_{n}^{2\ell-2j}\operatorname{\not{D}}_{n}\delta)(x).$

Note that both $\widetilde{\not{K}}^{\mathbb{B},-}_{\lambda,\nu}$ and
$\widetilde{\not{K}}^{\mathbb{C},-}_{\lambda,\nu}$ depend holomorphically on
$\lambda\in\mathbb{C}$ (or $\nu\in\mathbb{C}$).

######  Theorem 4.4.

The renormalized distribution

$\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x)=\frac{\not{K}^{\mathbb{A},-}_{\lambda,\nu}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu+\frac{1}{2}))\Gamma(\frac{1}{2}(\lambda-\nu+\frac{3}{2}))}$

depends holomorphically on $(\lambda,\nu)\in\mathbb{C}^{2}$ and vanishes only
for $(\lambda,\nu)\in\not{L}_{\textup{odd}}$. More precisely,

  1. (1)

For
$(\lambda,\nu)\in\mathbb{C}^{2}-(\underline{\backslash\backslash}^{-}\cup\underline{//}^{-})$
we have
$\operatorname{supp}\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}=\mathbb{R}^{n}$.

  2. (2)

For $(\lambda,\nu)\in\underline{\backslash\backslash}^{-}-\underline{//}^{-}$
with $\lambda+\nu=-\frac{1}{2}-2k$, $k\in\mathbb{N}$, we have

$\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x)=\frac{(-1)^{k}k!}{(2k)!}\widetilde{\not{K}}^{\mathbb{B},-}_{\lambda,\nu}(x).$

In particular,
$\operatorname{supp}\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}=\mathbb{R}^{n-1}$.

  3. (3)

For $(\lambda,\nu)\in\underline{//}^{-}-\not{L}_{\textup{odd}}$ with
$\lambda-\nu=-\frac{3}{2}-2\ell$, $\ell\in\mathbb{N}$, we have

$\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x)=\frac{(-1)^{\ell+1}\ell!\pi^{\frac{n-1}{2}}}{2^{2\ell+1}\Gamma(\nu+\frac{n}{2})}\widetilde{\not{K}}^{\mathbb{C},-}_{\lambda,\nu}(x).$

In particular,
$\operatorname{supp}\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}=\\{0\\}$.

  4. (4)

For $(-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j)\in\not{L}_{\textup{odd}}$,
$2\ell=i-j-1$, restricting the function
$(\lambda,\nu)\mapsto\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}$ to two
different complex hyperplanes in $\mathbb{C}^{2}$ through
$(-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j)$ and renormalizing gives two
holomorphic families of distributions:

$\displaystyle\widetilde{\not{K}}^{\mathbb{C},-}_{\lambda,\nu}(x)$
$\displaystyle=\frac{(-1)^{\ell+1}2^{2\ell+1}}{\ell!\pi^{\frac{n-1}{2}}}\Gamma(\nu+\rho_{H}+\tfrac{1}{2})\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x),$
$\displaystyle(\lambda-\nu=-\tfrac{1}{2}-(i-j)),$
$\displaystyle{\mathchoice{\widetilde{\hbox{\set@color$\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}}$}}}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\Gamma(\tfrac{1}{2}(\lambda-\nu+\tfrac{3}{2}))\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x),$
$\displaystyle(\nu=-\rho_{H}-\tfrac{1}{2}-j).$

Their special values at
$(\lambda,\nu)=(-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j)$ satisfy

$\displaystyle\operatorname{supp}\widetilde{\not{K}}^{\mathbb{C},-}_{-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j}$
$\displaystyle=\\{0\\},$
$\displaystyle\operatorname{supp}{\mathchoice{\widetilde{\hbox{\set@color$\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}}$}}}}^{\mathbb{A},-}_{-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j}$
$\displaystyle=\begin{cases}\mathbb{R}^{n}&\mbox{for $n$ even,}\\\
\mathbb{R}^{n-1}&\mbox{for $n$ odd,}\end{cases}$

more precisely, for $n$ odd and $2k=n+i+j$:

${\mathchoice{\widetilde{\hbox{\set@color$\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}}$}}}}^{\mathbb{A},-}_{-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j}=\frac{(-1)^{k}k!}{(2k)!}(|x^{\prime}|^{2}+x_{n}^{2})^{j}\Big{(}\zeta(x^{\prime})\delta^{(2k)}(x_{n})-2k\operatorname{\not{D}}_{n}\delta^{(2k-2)}(x_{n})\Big{)}.$

###### Proof.

By Theorem 3.2 the distribution

$\zeta(x)\cdot\widetilde{K}^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)=\frac{\zeta(x)\cdot
K^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu+\frac{1}{2}))\Gamma(\frac{1}{2}(\lambda-\nu-\frac{1}{2}))}$

depends holomorphically on $(\lambda,\nu)\in\mathbb{C}^{2}$, and by Lemma 4.2
it vanishes if and only if $\lambda-\nu=\frac{1}{2}$ or
$(\lambda-\frac{1}{2},\nu+\frac{1}{2})\in L_{\textup{even}}$. Renormalizing
shows that

$\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x)=\frac{\zeta(x)\cdot\widetilde{K}^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\frac{1}{2}(\lambda-\nu-\frac{1}{2})}=\frac{\zeta(x)\cdot
K^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\Gamma(\frac{1}{2}(\lambda+\nu+\frac{1}{2}))\Gamma(\frac{1}{2}(\lambda-\nu+\frac{3}{2}))}$

depends holomorphically on $(\lambda,\nu)\in\mathbb{C}^{2}$. To prove the
remaining statements, note that

$\displaystyle\zeta(x)\cdot\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\frac{\zeta(x)^{2}\cdot\widetilde{K}^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\frac{1}{2}(\lambda-\nu-\frac{1}{2})}=-\frac{|x|^{2}\cdot\widetilde{K}^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\frac{1}{2}(\lambda-\nu-\frac{1}{2})}\cdot\textup{id}_{\SS}$
$\displaystyle=-\widetilde{K}^{\mathbb{A},+}_{\lambda+\frac{1}{2},\nu-\frac{1}{2}}(x)\cdot\textup{id}_{\SS}$

and hence

(4.6)
$\operatorname{supp}\widetilde{K}^{\mathbb{A},+}_{\lambda+\frac{1}{2},\nu-\frac{1}{2}}\subseteq\operatorname{supp}\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}.$

  1. (1)

Let
$(\lambda,\nu)\in\mathbb{C}^{2}-(\underline{\backslash\backslash}^{-}\cup\underline{//}^{-})$,
then
$(\lambda+\frac{1}{2},\nu-\frac{1}{2})\in\mathbb{C}^{2}-(\backslash\backslash^{+}\cup//^{+})$
and hence
$\operatorname{supp}\widetilde{K}^{\mathbb{A},+}_{\lambda+\frac{1}{2},\nu+\frac{1}{2}}=\mathbb{R}^{n}$.
Now (4.6) implies
$\operatorname{supp}\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}=\mathbb{R}^{n}$.

  2. (2)

Let $(\lambda,\nu)\in\underline{\backslash\backslash}^{-}-\underline{//}^{-}$
with $\lambda+\nu=-\frac{1}{2}-2k$, then
$(\lambda-\frac{1}{2},\nu+\frac{1}{2})\in\backslash\backslash^{+}$ and by
Theorem 3.2 (2) and using
$x_{n}\delta^{(2k)}(x_{n})=-2k\delta^{(2k-1)}(x_{n})$ we have

$\displaystyle\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\frac{\zeta(x)\cdot\widetilde{K}^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\frac{1}{2}(\lambda-\nu-\frac{1}{2})}=\frac{(-1)^{k}k!}{(2k)!}\cdot\frac{\zeta(x)\cdot\widetilde{K}^{\mathbb{B},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}}{\frac{1}{2}(\lambda-\nu-\frac{1}{2})}$
$\displaystyle=\frac{(-1)^{k}k!}{(2k)!}\widetilde{\not{K}}^{\mathbb{B},-}_{\lambda,\nu}(x).$

  3. (3)

Let $(\lambda,\nu)\in\underline{//}^{-}-\not{L}_{\textup{odd}}$ with
$\lambda-\nu=-\frac{3}{2}-2\ell$, then
$(\lambda-\frac{1}{2},\nu+\frac{1}{2})\in//^{+}$ with
$(\lambda-\frac{1}{2})-(\nu+\frac{1}{2})=-\frac{1}{2}-2(\ell+1)$ and by
Theorem 3.2 (3) and using
$[\Delta_{\mathbb{R}^{n-1}}^{j},x_{i}]=2j\Delta^{j-1}\frac{\partial}{\partial
x_{i}}$ we have

$\displaystyle\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\frac{\zeta(x)\cdot\widetilde{K}^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\frac{1}{2}(\lambda-\nu-\frac{1}{2})}=\frac{(-1)^{\ell+1}(\ell+1)!\pi^{\frac{n-1}{2}}}{2^{2\ell+2}\Gamma(\nu+\frac{n}{2})}\cdot\frac{\zeta(x)\cdot\widetilde{K}^{\mathbb{C},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}}{(-\ell-1)}$
$\displaystyle=\frac{(-1)^{\ell+1}\ell!\pi^{\frac{n-1}{2}}}{2^{2\ell+1}\Gamma(\nu+\frac{n}{2})}\widetilde{\not{K}}^{\mathbb{C},-}_{\lambda,\nu}(x).$

  4. (4)

Now let
$(-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j)\in\not{L}_{\textup{odd}}$. The
first statement about $\widetilde{\not{K}}^{\mathbb{C},-}_{\lambda,\nu}(x)$
follows immediately from (3). For the second statement note that
$(-\rho-(i+1),-\rho_{H}-j)\in L_{\textup{even}}$ and we can use Theorem 3.2
(4). Restricting to $\nu=-\rho_{H}-\frac{1}{2}-j$ we have

$\displaystyle{\mathchoice{\widetilde{\hbox{\set@color$\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}}$}}}}^{\mathbb{A},-}_{\lambda,\nu}(x)$
$\displaystyle=\Gamma(\tfrac{1}{2}(\lambda-\nu+\tfrac{3}{2}))\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}$
$\displaystyle=\frac{\Gamma(\tfrac{1}{2}(\lambda-\nu+\tfrac{3}{2}))\zeta(x)\cdot\widetilde{K}^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)}{\frac{1}{2}(\lambda-\nu-\frac{1}{2})}=\zeta(x)\cdot{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},+}_{\lambda-\frac{1}{2},\nu+\frac{1}{2}}(x)$

which clearly depends holomorphically on $\lambda\in\mathbb{C}$. 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. $\mathbb{R}^{n-1}$ 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. $\mathbb{R}^{n-1}$ for $1\leq i\leq n-1$, 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. $\mathbb{R}^{n-1}$ by (4.5). For odd $n$ and $2k=n+i+j$ we further have
by Theorem 3.2 (4)

$\displaystyle{\mathchoice{\widetilde{\hbox{\set@color$\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}}$}}}}^{\mathbb{A},-}_{-\rho-\frac{1}{2}-i,-\rho_{H}-\frac{1}{2}-j}(x)$
$\displaystyle=\zeta(x)\cdot{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{K}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{K}$}}}}^{\mathbb{A},+}_{-\rho-(i+1),-\rho_{H}-j}(x)$
$\displaystyle=\frac{(-1)^{k}k!}{(2k)!}\zeta(x)\cdot(|x^{\prime}|^{2}+x_{n})^{j}\delta^{(2k)}(x_{n})$
$\displaystyle=\frac{(-1)^{k}k!}{(2k)!}(|x^{\prime}|^{2}+x_{n}^{2})^{j}\Big{(}\zeta(x^{\prime})\delta^{(2k)}(x_{n})-2k\operatorname{\not{D}}_{n}\delta^{(2k-2)}\Big{)}.\qed$

######  Remark 4.5.

The identity

$\zeta(x)\cdot\widetilde{\not{K}}^{\mathbb{A},-}_{\lambda,\nu}(x)=-\widetilde{K}^{\mathbb{A},+}_{\lambda+\frac{1}{2},\nu-\frac{1}{2}}(x)\cdot\textup{id}_{\SS}$

that was used in the previous proof can be explained by again applying the
translation principle, now with the dual representation $(\tau^{\prime})^{*}$,
and then projecting onto the trivial constituent in
$\tau^{\prime}\otimes(\tau^{\prime})^{*}=\textup{End}_{\mathbb{C}}(\tau^{\prime})$
which is spanned by $\textup{id}_{\SS}$.

######  Remark 4.6.

The intertwining differential operators $C^{\infty}(\mathbb{R}^{n};\SS)\to
C^{\infty}(\mathbb{R}^{n-1},\SS)$ with integral kernel
$\widetilde{\not{K}}^{\mathbb{C},\pm}_{\lambda,\nu}(x)$ have been obtained
before [8, Theorem 5.7] and it was conjectured that these operators exhaust
the space of all intertwining differential operators. Theorem 5.5 confirms
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 found in Section 4 between spinor-valued principal series span the
space of all symmetry breaking operators. The notation will be as in the
previous section.

###  5.1. Reduction to the pair $(\widetilde{G}_{1},\widetilde{H}_{1})$

To simplify computations we first reduce the study of symmetry breaking
operators for the pair
$(\widetilde{G},\widetilde{H})=(\textup{Pin}(n+1,1),\textup{Pin}(n,1))$ to the
pair
$(\widetilde{G}_{1},\widetilde{H}_{1})=(\textup{Pin}(n+1,1)_{1},\textup{Pin}(n,1)_{1})$,
where

$\textup{Pin}(p,q)_{1}=\textup{Pin}(p,q)_{++}\cup\textup{Pin}(p,q)_{-+}.$

We refer the reader to Appendix A.1 for the definition of
$\textup{Pin}(p,q)_{\pm\pm}$. Note that
$\widetilde{P}_{1}=\widetilde{P}\cap\widetilde{G}_{1}=\widetilde{M}_{1}AN$ is
a parabolic subgroup of $\widetilde{G}_{1}$ with
$\widetilde{M}_{1}=\textup{Pin}(n)$. Similarly,
$\widetilde{P}_{H,1}=\widetilde{P}_{H}\cap\widetilde{H}_{1}=\widetilde{M}_{H,1}AN_{H}$
is a parabolic subgroup of $\widetilde{H}_{1}$ with
$\widetilde{M}_{H,1}=\textup{Pin}(n-1)$. Further,
$\widetilde{G}_{1}/\widetilde{P}_{1}\simeq\widetilde{G}/\widetilde{P}$ and
$\widetilde{H}_{1}/\widetilde{P}_{H,1}\simeq\widetilde{H}/\widetilde{P}_{H}$,
and hence

$\displaystyle\not{\pi}_{\lambda,\delta}|_{\widetilde{G}_{1}}$
$\displaystyle\simeq\not{\pi}_{\lambda}=\textup{Ind}_{\widetilde{P}_{1}}^{\widetilde{G}_{1}}(\zeta_{n}\otimes
e^{\lambda}\otimes\mathbf{1}),$
$\displaystyle\not{\tau}_{\nu,\varepsilon}|_{\widetilde{H}_{1}}$
$\displaystyle\simeq\not{\tau}_{\nu}=\textup{Ind}_{\widetilde{P}_{H,1}}^{\widetilde{H}_{1}}(\zeta_{n-1}\otimes
e^{\nu}\otimes\mathbf{1}).$

Note that the restrictions are independent of $\delta$ and $\varepsilon$.

######  Lemma 5.1.

For $\lambda,\nu\in\mathbb{C}$ and
$\delta,\varepsilon\in\mathbb{Z}/2\mathbb{Z}$ we have

$\dim\textup{Hom}_{\widetilde{H}}(\not{\pi}_{\lambda,\delta},\not{\tau}_{\nu,\varepsilon})+\dim\textup{Hom}_{\widetilde{H}}(\not{\pi}_{\lambda,1-\delta},\not{\tau}_{\nu,\varepsilon})=\dim\textup{Hom}_{\widetilde{H}_{1}}(\not{\pi}_{\lambda},\not{\tau}_{\nu}).$

###### Proof.

The proof of the two identities is analogous to [7, Theorem 2.10] and uses
$\widetilde{G}/\widetilde{G}_{1}\simeq\widetilde{M}/\widetilde{M}_{1}\simeq\mathbb{Z}/2\mathbb{Z}$
and
$\widetilde{H}/\widetilde{H}_{1}\simeq\widetilde{M}_{H}/\widetilde{M}_{H,1}\simeq\mathbb{Z}/2\mathbb{Z}$.
∎