Tensor products of complementary series of rank one Lie groups
Genkai Zhang
We consider the tensor product $\pi_{\alpha\otimes \pi_{\beta$ of complementary series representations $\pi_{\alpha$ and $\pi_{\beta$ of classical rank one Lie groups $SO_0(n, 1; \mathbb F)$, $\mathbb F=\br, \bc, \bh$. We prove that there is a discrete component $\pi_{\alpha+\beta$ for small parameters $\alpha, \beta$ (in our parametrization). For the real group $SO_0(n, 1; \mathbb R)$ we give a construction of bilinear intertwining differential operators on tensor products of spherical principal series representations; these operators have been found earlier by Ovsienko and Redou \cite{Ovsienko-Redou by a slightly different method. Using the operators we prove further that there are finitely many complementary series of the form $\pi_{\alpha+\beta + 2j$, $j=0, 1, \cdots, k$, appearing in the tensor product $\pi_{\alpha \otimes \pi_{\beta $ of two complementary series $\pi_{\alpha$ and $\pi_{\beta$, where $k=k(\alpha, \beta, n)$ depends on $\alpha, \beta, n$. For the group $SO_0(2, 1)$ this was proved earlier by Repka where the only discrete components is $\pi_{\alpha +\beta$.
1402.2950.tex



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\baselineskip 1.40pc


\begin{document}
\def\abstractname{Abstract}
\def\chrefname{References}

\title[Tensor products of complementary series
]{Tensor 
products of complementary series
of rank one Lie groups
 }
\author{ Genkai Zhang}
%%%%
%\address{Tata Institute, India}
%\email{}
\address{Mathematical Sciences, Chalmers University of Technology and
Mathematical Sciences, G\"oteborg University, SE-412 96 G\"oteborg, Sweden}
\email{\it genkai@chalmers.se}
\thanks{Research partially supported by 
the Swedish
Science Council (VR)}
%\draft
\begin{abstract}
We consider the tensor product
$\pi_{\alpha}\otimes \pi_{\beta}$ of complementary
series  representations $\pi_{\alpha}$ and 
$\pi_{\beta}$ 
of classical rank one Lie
groups $SO_0(n, 1; \mathbb F)$,
 $\mathbb F=\br, \bc, \bh$.
We prove that there is a discrete 
component $\pi_{\alpha+\beta}$  for small
parameters $\alpha, \beta$ (in our parametrization). For the real group
$SO_0(n, 1; \mathbb R)$
we give a construction
of  bilinear 
 intertwining
differential operators 
on tensor products of spherical
principal series representations;
these operators have been found earlier by Ovsienko and Redou \cite{Ovsienko-Redou}
by a slightly different method.
Using the operators
we prove further that 
there are  finitely many
complementary series of the form $\pi_{\alpha+\beta + 2j}$, $j=0, 1,
\cdots, k$, 
appearing in
the tensor product
$\pi_{\alpha}
\otimes \pi_{\beta}
$ of  two
complementary series $\pi_{\alpha}$
and $\pi_{\beta}$, where
$k=k(\alpha, \beta, n)$ depends on $\alpha, \beta, n$. 
 For the group $SO_0(2, 1)$
this was proved earlier by Repka where the only discrete components
is $\pi_{\alpha +\beta}$.
\end{abstract}

\maketitle

\baselineskip 1.35pc

\section{Introduction}



The purpose of this paper is two fold: First  we shall give elementary
construction of bilinear invariant differential operators
on smoothly induced 
principal series representations of the Lorentz group
$SO_0(n, 1)$, and second we shall prove the boundedness
of the operators on complementary series, which are
 unitarization of principal series.
The study of bilinear invariant differential
operators is of  natural interests
in representation theory of Lie algebras
and in quantization. It can be put, roughly speaking, in the
{\it smooth or algebraic}
regime 
of representations, namely representations on spaces of smooth
functions
on manifolds or algebraic sums of finite dimensional representations.
 A related question in  
representation theory of Lie group is to
find decomposition of tensor products
of {\it unitary}  representations of Lie groups, i.e., decomposition
of  tensor products of pairs of Hilbert spaces under the tensor product actions.
 In certain circumstances
there are  bilinear differential operators defined on dense
subspaces of the tensor products of Hilbert spaces, and it
is thus an immediate question to find their boundedness. 
In this sense the two problems are closely related.

The questions above have been studied for quite some time.
The most
well-known case might be the Rankin-Cohen brackets
on  tensor products
of holomorphic discrete series of $\SL(2, \mathbb R)$, which yield
also a decomposition of the tensor products
in the unitary sense. 
(There
 exist further  formal sums of the brackets 
producing  associative products, or  quantizations; see e.g.
\cite{Connes-Mosc-2}.)
In this case the two regimes of representations, {\it smooth} and {\it
  unitary},
in principle match each other,  and  they can be further  put
under the holomorphic setup.


The representations we consider here
are principal series and complementary series representations of rank one groups.
While 
it is not expected the existence of bounded intertwining 
bilinear differential operators on principal series, one may still
try to find the operators on
complementary series, which exhibit better
smoothness properties than principal
series. Indeed we shall prove 
that this is the case for the real orthogonal group $SO_0(n, 1)$.
We prove also that the natural diagonal restriction $f(x, x)$
of functions $f(x, y)$ defines a bounded linear operator
on  tensor products of complementary series of general
rank one classical Lie groups $SO_0(n, 1; \mathbb F)$,
 $\mathbb F=\br, \bc, \bh$, for appropriate parameters,
and prove thus
the existence  of discrete component
in the tensor product.


We proceed to explain our main results and related  works.
Earlier Ovsienko and Redou
 \cite{Ovsienko-Redou} have
found a family of  differential operators
on tensor product of spherical principal series
representations $\pi_{\alpha}, \alpha\in \mathbb C$,
of the conformal group $O(n, 1)$;
the representations considered there are defined on spaces
of smooth functions on  $\mathbb R^n$
and are viewed as conformal densities.
They found the operators by using an Ansats
expressing the operators as 
polynomials of the Laplacian operators $\mathcal L_x$, $\mathcal L_y$,
and the inner product $\nabla_x\cdot \nabla_y$ evaluated 
on the diagonal $x=y$.
The same operators are obtained in \cite{Clerc-Beckmann}
as residues of a family of integral 
bilinear intertwining operators, which
is based on their earlier works
on trilinear form \cite{Clerc-2011,
Clerc-Orsted, CKOP}. For general rank-one groups the
operators are studied in \cite{BKZ}.
In the present paper we shall give
a direct construction of the operators by using
the Knapp-Stein intertwining operators. While the technical
computations are slightly different from those
in the existing literature, the underlying idea
is still the same, the explicit application of Knapp-Stein 
operators here making the computations conceptually clear.
We prove further
that there are finitely many discrete
components of complementary series of the form
$\pi_{\alpha + \beta+2j}$, $j\ge 0$,
in the tensor product $\pi_{\alpha}\otimes
\pi_{\beta}, \alpha, \beta>0$,
 of complementary
series when the parameters $\alpha$ and $\beta$ 
are relatively small (in our parametrization);
 see Sections \S\S3-4.


In \S5 we treat the  other rank one group $SU(n, 1)$
and $Sp(n, 1)$. The complementary
series representations can be realized as space
of distributions on Heisenberg-type groups,
and certain formal invariant bilinear
differential operators have been constructed in \cite{BKZ, Kob-Pev}.
However it seems the above method of estimating
the  norm of these operators becomes far more complicated.
We shall still prove the existence
of the first component 
 $\pi_{\alpha +\beta} $ in the tensor product for smaller
parameter $\alpha, \beta$ using the method of holomorphic
extension.


In the case of $n=2$ with $\mathfrak{so}(2, 1)=\mathfrak{sl}(2,\mathbb
 R)$  our method yields
a straightforward 
proof for 
the appearance of $\pi_{\alpha +\beta}
$ in 
 $\pi_{\alpha}\otimes \pi_{\beta }$ for small $\alpha, \beta$.
Tensor product
decompositions for representations of $SL(2, \mathbb R)$
had been started
in the work of Pukanszky
\cite{Pukanszky62} and completed by Repka
\cite{Repka78};  see  also \cite{Asmuth-Repka}.
Theses
results combined with the
general theory of Burger-Li-Sarnak \cite{Burger-Li-Sarnak}
have also found applications in automorphic forms \cite{Clozel-ias-park-lec}.
Tensor products of representations of $SL(2, \mathbb C)$ (i.e.,
locally isomorphic to the Lorentz group $SO_0(3, 1)$) have
been studied by Naimark \cite{Naimark-1961}; see also
\cite{Neretin-1995} where some complementary series
representations were constructed using  restriction
of holomorphic representations, the same
idea being used in the present paper in the construction
of discrete components for the groups
$SU(n, 1)$ and $Sp(n, 1)$.



I would like to thank Jean-Louis Clerc 
 for some stimulating discussions. I thank also the anonymous
referee for some expert comments.


\section{Spherical representations of rank one group $G$}
We fix notation and 
 recall  some known results on
induced representations of $G$ 
and the Knapp-Stein intertwining operator.
We shall use the  non-compact realization
of the representations. 
We shall be quite brief, and most of the technical
 formulas can be found e.g. in 
\cite{Johnson-Wallach,CDKR-jga}
where  the general case of rank one groups
is studied.

\subsection{Classical rank one Lie groups}


We recall some standard
facts on rank one Lie groups
and  fix notation. 
Let $G=O(n, 1; \mathbb F)$
for $\mathbb R=\br, \bc, \bh$
be the classical rank one Lie group
in its standard realization \cite{Johnson-Wallach,
Speh-Zhang}.
Let $\fg=\fk +\fp$ be the Cartan decomposition of the Lie algebra $\fg$.
We fix an element $H\in \fp$
and the subspace $\fa:=\mathbb RH_0\in \fp$
such that $\Ad(H)$ has eigenvalues $\pm 2,
\pm 1, 0$.
The root space decomposition of $\fg$ under  $H$ 
is 
$$
\fg=\fn_{-2} + \fn_{-1} +
(\fa+ \fm) + \fn_{1} +\fn_{2} 
$$
with $\pm 2, \pm 1, 0$,
if $\mathbb F=\mathbb C, \bh$,
and with the convention that $\fn_{2} =0$
if $\mathbb F=\mathbb R$.
  Here $\fm\subset \fl$ is the zero root space.
We denote by $\fn=\fn_1 \oplus \fn_2$ the sum of the positive root spaces.
 Then
$\fm+\fa +\fn$
is a maximal parabolic subalgebra
of $\fg$.

Let $\rho$
be the half sum of positive roots. Then
$$
\rho(H)=
\begin{cases}
\frac{n-1}2, &\quad \mathbb F=\br\\
n, &\quad \mathbb F=\bc\\
2n+1, &\quad \mathbb F=\bh
\end{cases}
$$
and we shall identify $\rho=\rho(H)$.





\subsection{Spherical representations and complementary series
for $G=SO_0(n, 1; \mathbb F)$}

Denote $M, A, N$ 
the corresponding 
subgroups of $G$ with Lie algebras
$\fm, \fa, \fn$, and $P=MAN$ the parabolic subgroup.
For $\mu\in \mathbb C$ 
let $\pi_{\mu}^\infty$
 be the 
induced {\it smooth}
representation
of $G$ from the character $
e^{-\mu}:
me^{tH}
n \in P=MAN\mapsto e^{-\mu t}$ consisting of $C^\infty$-functions $f$ on $G$
such that
\begin{equation}
  \label{eq:ind-norm}
f(g 
me^{tH_0}
n
)=
e^{-\mu t}
f(g),  e^{tH_0}mn\in MAN.  
\end{equation} 
In particular  $f$ are determined by their
restriction on $K$ and are identified further
as smooth functions on  $K/M$. We have  $\pi_{\mu}^\infty
=C^\infty(K/M)=C^\infty(S)$ as vector spaces. Restricting 
smooth functions in 
$\pi_{\mu}^\infty$
 to $N^-$ results in an injective
map to a subspace of $C^{\infty}(N^-)
=C^{\infty}(\mathbb R^{n-1})$. We shall fix this realization
of $\pi_{\mu}^\infty$. (Indeed it is only a subspace
of $C^{\infty}(N^-)$ as some matching conditions at infinity are needed.)


The explicit formulas for $\pi_\nu(g)$ can
be found in \cite{Johnson-Wallach} in the compact
picture and in
\cite{Speh-Venk-2} for the non-compact picture. We shall only need
the formula for the real group
$G=SO_0(1, n)$.
$G$ is generated by
the parabolic group
 $ MAN^-$ and 
the Weyl group element
$w$, which as on $\mathbb R{n-1}$
by the defining action
and by inversion, $w(x)=-\frac{1}{x}:=
-\frac{x}{|x|^2}$.
Their actions on $\pi_{\mu}^\infty$
 are  given by
$$
\pi_{\nu}(g)f(x)= e^{-t\nu} f(e^{t}m^{-1}(x-x_0)),
\quad (m, e^{tH}, x_0)
=me^{tH} x_0
\in MAN^-, \quad N^{-}=\mathbb R^{n-1},
$$
and
$$
\pi_{\nu}(w)f(x)= \Vert x\Vert^{-2\nu }
f(-\frac{x}{\Vert x\Vert^2}).
$$
Note also that the Jacobians of 
$g=(m, e^{tH}, x_0)$ and  of the Weyl group element $w$ 
on $N^-=\mathbb R^{n-1}$ 
are given by
\begin{equation}
\label{eq:jac}
J_g(x)=e^{t(n-1)},
\quad
J_w(x)=\frac{1}{|x|^{2(n-1)}}.
\end{equation}


We return now back to $G=SO_0(n, 1; \mathbb F)$.
The representation  $\pi_{\mu}(g), g\in G$,
$\mu\in (\rho +i\mathbb R)$ is already unitary
for the natural unitary norm in $L^2(K/M)$. However
for  $\mu\in  (0, 2\rho)$ 
a different $\fg$-invariant inner product on the space of $K$-finite
vectors can possibly be defined and completed to a unitary representation of
$G$, the {\it complementary series}; see
\cite{Johnson-Wallach}. The precise  parameter is given by
\begin{equation} \label{mu-fixed}
\mu\in  \begin{cases} 
(0, 2\rho), \mathbb F=\br, \bc
\\
(2, 2\rho-2), \mathbb F=\bh.
\end{cases} 
\end{equation} 
 We shall
denote the corresponding representation
still by $\pi_{\mu}$, and shall
 use its non-compact realization for the real case $G=SO_0(n,
 1)$, allowing us to find  (generically) more than one
 discrete components in the tensor product decomposition in \S4.

\subsection{Realization
of complementary series
for $G=SO_0(n, 1)$ on $\mathbb R^{n-1}$}

The unitarization of the complementary series
 is obtained via the Knapp-Stein intertwining operator,
defined preliminarily on $K$-finite vectors (which
can be obtained from $K$-finite vectors on $K/M$ via Cayley transform),
\begin{equation}
  \label{eq:j-n-c}
J_{\mu} f(x)
=\int_{\mathbb R^{n-1}} K_{\mu}(x, y) 
f(y) dy,  
\end{equation}
where 
\begin{equation}
  \label{eq:j-n-c-k}
 K_{\mu}(x, y) = C_{\mu}
\frac1 {|x-y|^{2\mu}}, \quad
C_\mu
=\frac
{\Ga(\rho-\frac{\mu}2)
\Ga(\rho-\frac{\mu}2 +\frac 12)}
{\Ga(\frac{n}2)
\Ga(\rho-\mu)} 
=\frac
{2^{\mu-2\rho +1} \sqrt \pi
\Gamma(2\rho -\mu)
}
{\Ga(\frac{n}2)
\Ga(\rho-\mu)}.
\end{equation}
(The normalization is chosen here so that in the compact picture
$J_\mu 1_S=1_S,$ where $1_S$ is the constant function on $S$ viewed
as a function on $G$ restricted to $\mathbb R^{n-1}.$)

Then $J_\mu$ is a $G$-intertwining operator
$$
\boxed{J_\mu: \pi_{\widetilde \mu}^\infty
\to \pi_{\mu}^\infty, \,
 \quad \widetilde \mu :={2\rho-\mu}}
$$
for $\mu <<0$.  
It has holomorphic continuation
to the whole complex plane, and in particular  holomorphic and non-zero in the two symmetric
strips around $\Re \mu=\rho $,
\begin{equation}
  \label{eq:hol-str}
\{\mu; 0<\Re\mu < \rho\},
\quad
\{\mu; \rho <\Re\mu < 2\rho
\}.
\end{equation}
The formal intertwining property
can be proved by using the following transformation
rule of $K_\mu$,
$$
K_\mu(gz, gw)=  (cz +d)^{-\mu} 
K_\mu(z, w)(cw +d)^{-\mu} 
=  J_g(z)^{\frac{\mu}{n-1}} 
K_\mu(z, w)
 J_g(w)^{\frac{\mu}{n-1}} 
$$
where $J_g$ is the Jacobian of the action of $g\in G$ on
$N^-=\mathbb R^{n-1}$. The holomorphic
continuation can also be done using the 
identity (\ref{eq:L-on-K}) below.
The smooth case  is also consequence of the general
theory 
of intertwining operators \cite{Vogan-Wallach}.

 The inner product
\begin{equation}
  \label{eq:j-n-c-norm}
(f_1, f_2)_\mu=(J_{\widetilde \mu}f_1, f_2)_{L^2(\mathbb R^{n-1})}
\end{equation}
for $f_1, f_2\in  C^\infty_0(\mathbb R^{n-1})$
is  a pre-Hilbert norm, and is invariant under $g\in G$ sufficiently
close
to the identity (depending on 
$f_1, f_2$). The completion  defines the
complementary series, $\mu\in (0, 2\rho)$.
We shall use its description 
in term of Fourier transform $f\mapsto \mathcal Ff$.
The space  $\pi_\mu$  is the completion
of $C_0^\infty(\mathbb R^{n-1})$
under  the (equivalent) norm
\begin{equation}
  \label{eq:Four-norm}
\Vert 
f\Vert_{\mu}^2
=\int_{\mathbb R^{n-1}}
|\mathcal F f(\xi)|^2 |\xi|^{n-1-2\mu} d\xi
=\Vert 
\mathcal F f(\cdot) 
|\cdot |^{\frac{\widetilde\mu}{2}} 
\Vert^2_{
L^2(
\mathbb R^{n-1}
)
},
\end{equation}
for $0<\mu <2\rho$.
See e.g. \cite{Speh-Venk-2}.



\subsection{Complementary series
for $G=SU(n, 1), Sp(n, 1)
$
and their holomorphic extensions}

We shall use a different method for  the cases
 $G=SU(n, 1), Sp(n, 1)$. The method is based on roughly
speaking holomorphic extension,  no explicit
realization  will be  needed.

We consider a Hermitian Lie group
$G_1$ (called overgroup in some Russian literature) containing $G$
as a symmetric subgroup such that
$G/K$ is a real form of the
Hermitian subgroup $G_1/K_1$; see \cite{Dijk-Hille-jfa, Speh-Zhang}.


More precisely let 
$$
G_1=\begin{cases}
SU(n, 1)\times SU(n, 1), \mathbb F=\mathbb C\\
SU(2n, 2), 
 \mathbb F=\bh.
\end{cases}
$$
with $G=SU(n, 1), Sp(n, 1)$ being
realized as the
diagonal  subgroup of $G_1$
and respectively as complex 
transformations via the standard identification
of $\mathbb H=\mathbb C^2$.

The holomorphic discrete series
of $G_1$ can be realized on
the space of holomorphic functions
on $D_1$. To fix notation we let
$$
V_1=\begin{cases} 
\bc^n \oplus \overline{\bc^n} & \mathbb F=\mathbb C\\
\\
 M_{2n, 2}(\mathbb C) & \mathbb F=\bh
\end{cases},
$$
and the space 
$D_1=G_1/K_1$
is realized as a bounded symmetric domain in $V_1$,
$$
D_1=\begin{cases} 
B^n \times \overline{B^n} &  \mathbb F=\mathbb C
\\
\{Z\in M_{2n, 2}(\mathbb C); Z^*Z<I\} & 
 \mathbb F=\bh
\end{cases}.
$$
 Let 
$\mathcal H_\nu(D_1)$ 
be the space
of holomorphic functions on $D_1$ with reproducing kernel
$h(z, w)^{-\nu}$ for $\nu$ sufficiently large,
where $$
h(z, w)=
\begin{cases} 
(1-\langle z_1, w_1\rangle )
(1-\langle w_2, z_2\rangle )
&  \mathbb F=\mathbb C
\\
\det(1- w^\ast z) &  \mathbb F=\mathbb H
\end{cases}
$$

It is now well-known  \cite{FK} that if $\nu$ is in the set
$$
\begin{cases} (0, \infty) & \mathbb F
=\mathbb R, 
 \mathbb C \\
(1, \infty) &  \mathbb F= \mathbb H
\end{cases}
$$
then the kernel $h(z,w)^{-\nu}$
is positive definite   and it defines
a unitary (projective) representation
of $G_1$.
We denote this representation by
$(\mathcal H_\nu(D_1), \tau_\nu, G_1)$. 

We recall the following theorem
\cite{Dijk-Hille-jfa}; see further \cite{Speh-Zhang}
for the present reformulation.
\begin{theo+} 
The complementary series
$(\pi_{\mu}, G)$ appears as 
a discrete summand 
in  
 $(\mathcal H_\nu(D_1),  \tau_\nu, G_1)$ restricted to $G$ if $\nu$ and $\mu$ are
related by
$$2\nu 
=
 \begin{cases} 
\mu, \,\,  \mu \in (0,  n), \quad  & \mathbb F=\bc \\
\mu, \,\, \mu\in (2, {2n-1}), \quad & \mathbb F=\bh.
 \end{cases}
$$
\end{theo+}

Note that the range of $\mu$ is, disregarding the Weyl group symmetry,
precisely the whole range of the complementary series representations.
In other words, any complementary series of $G$ is a discrete component
in the holomorphic representation of $G_1$.


\section{Invariant  bilinear differential operators for general
spherical series representations, $G=SO_0(n, 1)$}



We denote $\pi_{\alpha}^\infty
 \otimes
\pi_{\beta}^\infty$ the induced smooth representation 
of $G\times G$ from the parabolic subgroup 
$P\times P$ and the character $e^{-\alpha}\times  
e^{-\beta}$.
The group $G$ is viewed as the diagonal subgroup
of $G\times G$.

\begin{theo+} For any nonnegative integer $j\ge 0$
there exists
 a    $G$-intertwining  differential operator 
$\mathcal D_{\alpha, \beta, j}
$ of degree $2j$
 meromorphic  in $(\alpha, \beta)\in \mathbb C^2$,
$$
\mathcal D_{\alpha, \beta, j} :
\pi_{\alpha}^\infty
 \otimes
\pi_{\beta}^\infty
 \to  \pi_{\alpha+\beta +2j}^\infty.
$$
The  only possible
poles of $
\mathcal D_{\alpha, \beta, j}
$ appear
when  $\alpha$ or $\beta\in \Lambda_j$, where 
$$
\Lambda_j=\{0, -1, -j+1\} \cup 
\left( \rho -1  +\{0, -1, -j+2\}
\right).
$$
\end{theo+}


The proof will be divided into  a few elementary Lemmas.


Let $S_{\alpha, \beta, j}
(x, y; z, w) $ be the kernel
\begin{equation}
  \label{eq:S-ker}
S_{\alpha, \beta, j}
(x, y; z, w)
=
\left(
    \frac{|(x-z)-(y-w)|^2}
{|x-z|^{2}
|y-w|^{2}
}
\right)^j
  \frac{1}
{|x-z|^{2\alpha}
|y-w|^{2\beta}
},
\end{equation}
and write for simplicity 
$$
S_{\alpha, \beta, j}(x; z, w)=S_{\alpha, \beta, j}(x, x; z, w),
\quad S_{\alpha, \beta, j}(x, y)=S_{\alpha, \beta, j}(x, y; 0, 0)
$$
the diagonal $x=y$ restriction respectively the evaluation
at $z=w=0$.
\begin{lemm+}  The integral operator
$$
T_{j}f(x)=T_{\alpha, \beta, j}f(x):
=
C_\alpha
C_\beta
\int_{\mathbb R^{2(n-1)}}
S_{\alpha, \beta, j}(x; z, w)
 f(z, w) dz dw
$$
defines  an intertwining operator 
$$
\pi_{\widetilde \alpha}^\infty
\otimes \pi_{\widetilde \beta}^\infty
\to \pi_{\alpha+\beta+2j}^\infty
$$
\end{lemm+}
\begin{proof}
Recall the  group $G$ is generated by $P$
and $w$
as a consequence of the
Bruhat decomposition
\cite[Theorem 1.4, Ch.~IX]{He2}.
The formal intertwining property
follows directly from
a  change of variables $(x, y)\mapsto (gx, gy)$
for $g\in P$ and $g=w$
along with the formula 
(\ref{eq:jac})
for the Jacobians.
To prove the meromorphic 
continuation in $\alpha$ and $\beta$
we observe that changing
$(x, y)$ to $(x-z, y-z)$ we need only
to prove that the integral
$$
\int_{\mathbb R^{2(n-1)}}
   \frac{|x-y|^{2j}
}
{|x|^{2j}
|y|^{2j}
}
  \frac{1}
{|x|^{2\alpha}
|y|^{2\beta}
}
 f(x, y) 
dx dy
$$ 
is 
meromorphic in $(\alpha, \beta)$. But this
is just up to normalization  constants
the integral $(J_{\alpha +j} \otimes 
J_{\beta +j})(F)$, $F(x, y)=
|x-y|^{2j} f(x, y) $ and thus has the
continuation.
\end{proof}

In the compact-realization this operator is
$$
T_j f(x)=
\int_{S\times S}
\left(
    \frac{
1-\langle z, w\rangle
}
{
(1-\langle x, z\rangle)
(1-\langle x, w\rangle)
}
\right)^j
  \frac{C_\alpha 
C_\beta }
{
(1-\langle x, z\rangle)^\alpha
(1-\langle x, w\rangle)^\beta
}
f(z, w)dz dw.$$
That the integral is well-defined for $\alpha, \beta <<0$
 can also be  deduced from this formula.


Next  we need some known Bernstein-Sato type identities
for the Laplacian operator $\mathcal L=\partial_1^2 
+\cdots +\partial_n^2$ acting on $|x|^{-2\alpha}$. 
Recall   the Pochammer symbol defined by $(\alpha)_j=\alpha
(\alpha+1)\cdots (\alpha +j-1)$.
\begin{lemm+}
 The following differentiation  formula holds 
\begin{equation}
  \label{eq:L-on-K}
\mathcal L^j |x|^{-2\alpha}
=2^{2j}(\alpha)_j
(\alpha+1-\rho)_j
|x|^{-2(\alpha+j)}, \quad x\ne 0.
\end{equation}
\end{lemm+}


We define a family of differential
operators of constant coefficients on $C^\infty(\mathbb R^{2(n-1)})$
by $$
M_{\alpha, \beta, 0}=I,\quad  M_{\alpha, \beta, 1}=\nabla_x \cdot
\nabla_y$$
 and
$$
\mathcal M_{\alpha, \beta, j+1}
=(\nabla_x \cdot \nabla_y) 
\mathcal M_{\alpha, \beta, j}
-\frac {j( n-1-3j-2 \alpha -2\beta )
 } 
{
 (\alpha +1-\rho)
 (\beta +1-\rho)
}
\mathcal M_{ \alpha+1, \beta+1, j-1
}
\mathcal L_x
\mathcal L_y
$$

It follows from the construction that the only possible poles of 
$M_{\alpha, \beta, j}$, $j\ge 2$,  appear
when   $\alpha $ or     $\beta$ is in 
$$
\{\rho -i; i=1, \cdots, j-1\}
$$


\begin{lemm+} 
The following formula holds for all $(\alpha, \beta)\in \mathbb C^2$
and $m\in \mathbb N$,
\begin{equation}
  \label{eq:M-on-kernel}
  \mathcal M_{\alpha, \beta, m}
S_{\alpha, \beta}(x, y; z, w)
= 2^{2j}(\alpha)_m (\beta)_m
\left(
 \frac{
\langle x-z, y-w\rangle
}
{
|x-z|^{2}
|y-w|^{2}
}
\right)^m
S_{\alpha, \beta}(x, y; z, w)
\end{equation}
\end{lemm+}
\begin{proof} By invariance 
we can assume $z=w=0$.
We prove the identity using induction.
It is trivially true for $m=0$.
Assuming the identity holds for  $0\le m\le j$
for all $\alpha, \beta$
we perform the differentiation 
$\nabla_x \cdot \nabla_y$ on the identity with $m=j$. We have
\begin{equation}
  \label{eq:induct}
\nabla_x \cdot \nabla_y  \mathcal M_{\alpha, \beta, j}
S_{\alpha, \beta}(x, y)
=2^{2N}(\alpha)_j (\beta)_j
(I +II)
\end{equation}
a sum of two terms, with the first term 
\begin{equation*}
\begin{split}
I&=2^{2j}(\alpha)_j (\beta)_j
2^2(\alpha +j)(\beta +j)
\left(
 \frac{
\langle x, y\rangle
}
{
|x|^{2}
|y|^{2}
}
\right)^{j+1}
S_{\alpha, \beta}(x, y)\\
&=
2^{2(j+1)}(\alpha)_{j+1} (\beta)_{j+1}  
\left(
 \frac{
\langle x, y\rangle
}
{
|x|^{2}
|y|^{2}
}
\right)^{j+1}
S_{\alpha, \beta}(x, y)
\end{split}
\end{equation*}
being the RHS of  (\ref{eq:M-on-kernel}) for $m=j+1$,
and 
$$
II=
j (n-1-3j-2\alpha -2\beta) 
\left(
 \frac{
\langle x, y\rangle
}
{
|x|^{2}
|y|^{2}
}
\right)^{j-1}
S_{\alpha+1, \beta+1}(x, y).
$$
We treat the second term using the induction hypothesis for $m=j-1$
with $(\alpha,\beta)
$ being replaced
by  $(\alpha+1,\beta+1)$,
$$
2^{2(j-1)}
(\alpha+1) _{j-1}
(\beta+1) _{j-1}
\left(
 \frac{
\langle x, y\rangle
}
{
|x|^{2}
|y|^{2}
}
\right)^{j-1}
S_{\alpha+1, \beta+1}(x, y)
=\mathcal M_{\alpha+1,\beta+1,
j-1 
}
S_{\alpha+1, \beta+1}(x, y),
$$
which is furthermore
$$
\frac 1{2^2\alpha (\alpha +\rho-1)
\beta (\beta +\rho-1)
}
\mathcal M_{
\alpha+1,\beta+1,
j-1}
\mathcal L_x
\mathcal L_y 
S_{\alpha, \beta}(x, y).
$$
Rewriting 
(\ref{eq:induct}) we find
$$
(\nabla_x \cdot \nabla_y  \mathcal M_{
\alpha, \beta, j}
-\frac {j(n-1-3j-2\alpha -2\beta) 
}
{ (\alpha +\rho-1)
 (\beta +\rho-1)
}
\mathcal M_{
\alpha+1,\beta+1,
j-1}
\mathcal L_x
\mathcal L_y)
S_{\alpha, \beta}(x, y),
$$
which is 
$\mathcal M_{
\alpha, \beta, j+1 
}
S_{\alpha, \beta}(x, y) $ by the definition. This finishes the proof.
\end{proof}



Combining the two Lemmas  we have
\begin{equation*}
\mathcal M_{\alpha+j, \beta+i, k}
\mathcal L_x^j
\mathcal L_x^i
   \frac{1}
{|x|^{2\alpha}
|y|^{2\beta}
}
=c_{i, j, k}(\alpha, \beta)   
\left(
 \frac{\langle x, y\rangle}
{
|x|^{2}
|y|^{2}
}
\right)^k
   \frac{1}
{|x|^{2\alpha+2j}
|y|^{2\beta+2i}
},
\end{equation*}
where 
\begin{equation}
c_{i, j, k}(\alpha, \beta)
=  2^{2k+2j+2i} 
(\alpha)_{j+k}
(\alpha+1-\rho)_j
(\beta)_{i+k}
(\beta+1-\rho)_i.
\end{equation}
Here we have used the fact that
$$
(\gamma)_{j} 
(\gamma+j)_{k}
=(\gamma)_{j+k}.
$$
By translation invariance
we have 
\begin{equation}
  \label{eq:M-L-on-kernel}
\begin{split}
&\quad\,\mathcal M_{
\alpha+j, \beta+i, 
k}
\mathcal L_x^j
\mathcal L_x^i
   \frac{1}
{|x-z|^{2\alpha}
|y-w|^{2\beta}
}
\\
&=c_{i, j, k}(\alpha, \beta)   
\left(
 \frac{\langle x-z, y-w\rangle}
{
|x-z|^{2}
|y-w|^{2}
}
\right)^k
   \frac{1}
{|x-z|^{2\alpha+2j}
|y-w|^{2\beta+2i}
},
\end{split}
\end{equation}




We prove now Theorem 3.1
\begin{proof} 
The operator 
$$
T_{\alpha, \beta, j} 
(J_{\alpha}\otimes  
J_{\beta}):
\pi_{\widetilde \alpha}^\infty
\otimes \pi_{\widetilde \beta}^\infty
\to \pi_{\alpha+\beta+2j}^\infty
$$
 is an intertwining operator by Lemma 3.2. We prove
it is  a differential operator.
The idea is
to  differentiate 
 the identity $f=
(J_\alpha\otimes 
J_\beta)
(J_{\widetilde \alpha}\otimes 
J_{\widetilde \beta}) f$. 
We shall perform formal computations
on the integral first and justify them
in the end. 
Let $f\in \pi_{\alpha}^\infty\otimes
\pi_{\beta}^\infty$ and $
g=
J_{\widetilde \alpha}\otimes
J_{\widetilde \beta} f.
$
We denote
\begin{equation*}
\begin{split}
  \label{eq:2}
&\quad\, \mathcal E_{\alpha, \beta, m}
f(z, w)\\
&=
\sum_{i+j +k=m} 
\varepsi_{i, j, k}(\alpha, \beta)
\mathcal 
M_{\alpha +j,
\beta +i, k}
\mathcal L_x^{j}
\mathcal L_y^{i} f(x, y)
\end{split}
\end{equation*}
and
\begin{equation}
  \label{eq:D-N}
\mathcal D_{\alpha, \beta, m}f(x)=
\mathcal E_{\alpha, \beta, m}f{|}_{x=y},
\end{equation}
for $f\in C^\infty(\mathbb R^{2(n-1)})$,
where
$$
\varepsi_{i, j, k}(\alpha, \beta)
:=\binom{m}{i, j, k}
\frac{(-2)^{k}}
{c_{i, j, k}(\alpha, \beta)
}.
$$
We claim
that 
\begin{equation}
  \label{eq:D-N-TJ}
\mathcal D_{\alpha, \beta, j} 
f=
T_{\alpha, \beta, j} 
(J_{\alpha}\otimes  
J_{\beta})f, \quad f\in \pi_{\widetilde \alpha}^\infty
\otimes \pi_{\widetilde \beta}^\infty
\end{equation}
proving the intertwining property of the differential
operator $ \mathcal D_{\alpha, \beta, m} $.


The binomial expansion of 
$S(x, y; z, w)$
reads as follows
\begin{equation*}
  \begin{split}
S(x, y; z, w)
&=\left(\frac{
|x-z|^2
+|y-w|^2
  -2
\langle x-z, y-w\rangle}
{|x-z|^2 |y-w|^2
}
\right)^m
  \frac{1}
{|x-z|^{2\alpha}
|y-w|^{2\beta}
}
\\
&=\sum_{i+j +k=m}
\binom{m}{i, j, k}
{(-2)^{k}}
\left(\frac
{\langle x-z, y-w\rangle}
{
|x-z|^{2}
|y-w|^{2}
}
\right)^{k}
  \frac{1}
{
|x-z|^{2j+2\alpha}
 |y-w|^{2i+2\beta} 
}
  \end{split}
\end{equation*}
Summing the formula 
(\ref{eq:M-L-on-kernel})
over $(i, j, k)$ we have then
$$
\mathcal E_{\alpha, \beta, m}
   \frac{1}
{|x-z|^{2\alpha}
|y-w|^{2\beta}
}
=S(x, y; z, w)
$$
which further implies that
\begin{equation}
  \label{eq:d-n-a-b}
\mathcal D_{\alpha, \beta, m}
   \frac{1}
{|x-z|^{2\alpha}
|y-w|^{2\beta}
}
=S(x, x; z, w) =S(x; z, w)
\end{equation}




The identity $f=(J_{ \alpha}\otimes
J_{ \beta})
(J_{\widetilde \alpha}\otimes
J_{\widetilde \beta})f
=(J_{ \alpha}\otimes
J_{ \beta})
g$ reads
$$
f(x, y)=(J_{ \alpha}\otimes
J_{ \beta})g
=C_{\alpha}
C_{\beta}
\int_{\mathbb R^{2(n-1)}}
\frac 1{|x-z|^{2\alpha}
|y-w|^{2\beta}} g(z, w) dz dw.
$$
We perform the differentiation 
$\mathcal D_{\alpha, \beta, j}$
on this  identity 
and  find
$$
\mathcal D_{\alpha, \beta, j}f(x)=
C_{\alpha}
C_{\beta}
\int_{\br^{2(n-1)}}
S(x; z, w) g(z, w) dz dw
=T_j g(x) =T_j
J_{\widetilde \alpha}\otimes
J_{\widetilde \beta} f(x),
$$
proving (\ref{eq:D-N-TJ}).

Finally the differentiation
under integral sign can be justified by  taking first
$\alpha, \beta << 0$
and $\alpha\notin \mathbb Z_-, \beta\notin \mathbb Z_-$,
with $\widetilde \alpha >>0, 
\widetilde \beta >>0$,  in which case  Lemma 2.1 implies that all integrals
involved are absolutely convergent. The rest is obtained
by analytic continuation.
\end{proof}



\section{Finitely many discrete components in the tensor product
$\pi_{\alpha}\otimes
\pi_{\beta}$, $G=SO_o(n, 1, \mathbb R)$}


We apply the intertwining operators $\mathcal D_j=\mathcal D_{\alpha,
  \beta, j}$
to
the study of appearance of discrete components
in the tensor product $\pi_{\alpha} \otimes
\pi_{\beta}    $ 
of complementary series. 
 For $\alpha, \beta\in (0, \rho)$ the tensor product 
$\pi_{\alpha} \otimes
\pi_{\beta}    $ in the non-compact picture
is the completion of $C^\infty_0(\mathbb R^{2(n-1)})$
with  norm
$$
\Vert 
f\Vert_{\alpha\otimes \beta}^2
:=\int_{\mathbb R^{2(n-1)}}
|\mathcal F f(\xi, \eta)
|^2 |\xi|^{n-2\alpha}
|\eta|^{n-2\beta} d\xi d\eta;
$$
cf.   (\ref{eq:Four-norm}).


\begin{theo+} Suppose $\alpha>0, \beta> 0$ and $j\in \mathbb N$ satisfy
$0<\alpha<\rho$, 
$0<\beta<\rho$, $\alpha +\beta +2j <\rho$. Then the
intertwining operator $\mathcal D_{\alpha, \beta, j}$ is
a non-zero bounded intertwining operator
$\pi_{\alpha} \otimes
\pi_{\beta}
 \to  \pi_{\alpha+\beta +2j}$, and
thus 
$\pi_{\alpha+\beta +2j}$
 appears in the tensor product
$\pi_{\alpha} \otimes
\pi_{\beta}
$ as an irreducible component.
\end{theo+}

\begin{proof} Noticing that for $\alpha, \beta$ and $j$ as above we have
that the operator  $\mathcal D_j$ is well-defined,
and 
$\pi_{\alpha}$, $\pi_{\beta }$ and $\pi_{\alpha+\beta +2j}$ are
unitary representations.
Recall also the notation $\tilde \alpha
=2\rho -\alpha=n-1-\alpha$
in \S2.3 and the unitary
norm (2.8). Let $f\in
C^\infty_0(\mathbb R^{2(n-1)})
\subset 
\pi_\alpha\otimes
\pi_\beta
$. 
We claim that 
$$ \Vert 
\mathcal D_j
f
\Vert_{\alpha+ \beta+2j}^2
\le C \Vert 
f\Vert_{\alpha\otimes \beta}^2.
$$
Thus $\mathcal D_j $
defines a non-zero intertwining operator from 
$\pi_{\alpha} \otimes
\pi_{\beta} $ into $
\pi_{\alpha+\beta +2j}$, proving our theorem.

Using Fourier inversion we have
$$
f(x, y)=C
\int_{\mathbb R^{2(n-1)}}
e^{i
\langle 
x, \xi
\rangle 
 +i
\langle 
 y, \eta
\rangle 
} 
\mathcal F f(\xi, \eta)
 d\xi d\eta
$$
where $C$ is a normalization constant. We write
the differential operator $\mathcal E_{\alpha, \beta, m}$
in the proof of Theorem 3.1
as $Q(\mathcal L_x,  \mathcal L_y,
\nabla_x\cdot 
\nabla_y
)$
where $Q$ is a homogeneous polynomial of three variables
of degree $j$. 
Thus
$\mathcal D_jf(x)
=
Q(\mathcal L_x,  \mathcal L_y,
\nabla_x\cdot 
\nabla_y
)
f(x, y)|_{x=y}.
$ 
Its action on 
the inversion formula  results in
\begin{equation*}
  \begin{split}
\mathcal D_j
f(x)&=
C
\int_{\mathbb R^{2(n-1)}} 
e^{i
\langle  
x, \xi+ \eta
\rangle }
Q(-|\xi|^2,  -|\eta|^2,  -\langle
\xi, \eta \rangle)
\mathcal F f(\xi, \eta)
d\xi d\eta\\
&=C\int_{\mathbb R^{n-1}}
e^{i(x, \zeta)}
\int_{\mathbb R^{n-1}}
Q(-|\zeta-\eta|^2,  -|\eta|^2,  -\langle
\zeta-\eta, \eta \rangle)
\mathcal F f(\zeta-\eta, \eta)d\eta 
d\zeta.
  \end{split}
\end{equation*}
That is
\begin{equation*}
 \mathcal F(\mathcal D_j
f)(\zeta)=C
\int_{\mathbb R^{n-1}}
Q(-|\zeta-\eta|^2,  -|\eta|^2,  -\langle
\zeta-\eta, \eta \rangle)
\mathcal F f(\zeta-\eta, \eta)d\eta,
\end{equation*}
and furthermore 
\begin{equation*}
| \mathcal F(\mathcal D_j
f)(\zeta)|^2\le 
A(\zeta)
\int_{\mathbb R^{n-1}}
|\mathcal F f(\zeta-\eta, \eta)|^2
|\zeta-\eta|^{2\widetilde\alpha}
|\eta|^{2\widetilde\beta}
d\eta
\end{equation*}
with
$$
A(\zeta):
=C\int_{\mathbb R^{n-1}}
|Q(-|\zeta-\eta|^2,  -|\eta|^2,  -\langle
\zeta-\eta, \eta \rangle)|^2
|\zeta-\eta|^{-2\widetilde\alpha}
|\eta|^{-2\widetilde\beta} 
d\eta.
$$
To estimate the integral $A(\zeta)$ we write $\zeta=|\zeta|u$, $|u|=1$,
and perform a change of variables $\eta=|\zeta|v$. It is
$$
A(\zeta)
=C|\zeta|^{4j-
2\widetilde\alpha
-2\widetilde\beta +(n-1)
}
\int_{\mathbb R^{n-1}}
|Q(-|u-v|^2,  -|v|^2,  -\langle
u-v, v \rangle)|^2
|u-v|^{-2\widetilde\alpha}
|v|^{-2\widetilde\beta} 
dv
$$
and the integral is convergent; indeed it is
locally integrable near  $v=0$, and $v=u$
for $2\widetilde\alpha, 2\widetilde\beta <n-1$ and
is integrable at infinity
for the integrand is dominated by
$$
(1+|v|^2)^{ -(\widetilde\alpha
+\widetilde\beta-2j)}
$$
with $\widetilde\alpha
+\widetilde\beta-2j=n-1+ (n-1-\alpha-\beta-2j) <n-1$.
Thus
$$
| \mathcal F(\mathcal D_j
f)(\zeta)|^2
|\zeta|^{-4j+
2\widetilde\alpha
+2\widetilde\beta -n
}\le C
\int_{\mathbb R^{n-1}}
|\mathcal F f(\zeta-\eta, \eta) |^2
|\zeta-\eta|^{2\widetilde\alpha}
|\eta|^{2\widetilde\beta}
d\eta,
$$
and its integration  over $\zeta$ gives
$$
\int_{\mathbb R^{n-1}}
| \mathcal F(\mathcal D_j
f)(\zeta)|^2
|\zeta|^{-4j+
2\widetilde\alpha
+2\widetilde\beta -n
}d\zeta \le C \int_{\mathbb R^{n-1}}
\int_{\mathbb R^{n-1}}
|\mathcal F f(\zeta-\eta, \eta) |^2
|\zeta-\eta|^{2\widetilde\alpha}
|\eta|^{2\widetilde\beta}
d\eta= C\Vert 
f\Vert_{\alpha\otimes \beta}^2
$$
whereas the LHS is precisely
$ \Vert 
\mathcal D_jf\Vert_{\alpha+ \beta+2j}^2$.
This finishes the proof.
\end{proof}

When $n=2$ then   $j=0$ and the  theorem
states
that 
$\pi_{\alpha+\beta}$ appears in the tensor product
$\pi_{\alpha} \otimes
\pi_{\beta}$ if $\alpha+\beta <1$. This has been
 proved earlier in \cite{Repka78}.


\section{The appearance of 
one component $\pi_{\alpha+\beta}$ in $\pi_{\alpha}\otimes
\pi_{\beta}$ for other rank one groups $G=SU(n, 1),
Sp(n, 1)$}

We treat now the other rank one  groups.

\begin{theo+}  Let $G=SU(n, 1)$ and $Sp(n, 1)$,
$\pi_{\alpha}$ and $\pi_{\beta}$
be the complementary series for $\alpha, \beta$ as in
(\ref{mu-fixed}),
$0<\alpha, \beta <\rho=n$ and respectively
$2<\alpha, \beta <\rho=2n-1$.
Then the complementary series
$(\pi_{\alpha+\beta}, G)$ of
$G$
appears discretely in the tensor product
$\pi_{\alpha}\otimes \pi_{\beta}$ 
if
 $$
\alpha +\beta <
 \begin{cases} 
n  & \mathbb F=\bc \\
2n-1\quad & \mathbb F=\bh.
 \end{cases}
$$
\end{theo+}

\begin{proof} We prove the case for $G=SU(n, 1)$ and the same methods applies also
to $G=Sp(n, 1)$.  We consider
the diagonal imbedding of $G$
in $G_1$. It follows from Theorem 2.1
that for $\alpha, \beta\in (0, \rho)$
the complementary series $\pi_\alpha$
and $\pi_\beta$ appear in $\tau_{\frac \alpha
2 }$
and $\tau_{\frac \beta 2
}$, respectively.
Now $\tau_{\nu}$ of $G_1=SU(n, 1)
\times SU(n, 1)$
is the tensor product
$\lambda_\nu \otimes \overline
{\lambda_{\nu}}$ on 
$\mathcal H_\nu
\otimes \overline
{\mathcal H_\nu}
$
where $\mathcal H_\nu$ is the space
of holomorphic functions on the unit ball
$B^n$ with the reproducing kernel $(1-\langle z, w\rangle)^{-\nu}$.
If $\alpha +\beta<n$
then 
$\pi_\alpha$
 appears in $\tau_{\frac \alpha 2}$, so does
$\pi_\beta$ in
$\tau_{\frac \beta 2}$.
 The tensor product
 $\tau_{\frac \alpha 2
}\otimes
\tau_{\frac \beta 2}$ is now
$$
H:=(\mathcal H_{\frac \alpha 2}
\otimes \overline
{\mathcal H_{\frac \alpha 2}} )
\otimes
(\mathcal H_{\frac \beta 2}
\otimes \overline
{\mathcal H_{\frac \beta 2}} ).
$$
Its restriction to $G$
is
$$H=(\mathcal H_{\frac \alpha 2}
\otimes
\mathcal H_{\frac \beta 2})\otimes
 \overline
{(\mathcal H_{\frac \alpha 2}
\otimes\mathcal H_{\frac \beta 2})
}.
$$
However the tensor product 
$\mathcal H_{\frac \alpha 2}
\otimes
\mathcal H_{\frac \beta 2}$ 
of two holomorphic representation is
decomposed discretely under $G$ and contains
a component 
$\mathcal H_{\frac{\alpha+\beta }2}
$. Thus $H$ contains a discrete component 
$
\mathcal  H_{\frac{\alpha+\beta }2}
\otimes
 \overline
{\mathcal 
 H_{\frac{\alpha+\beta }2}
}.
$
We use again Theorem 2.1 and deduce that
this space has a discrete component $(\pi_{\alpha +\beta}, G)$.
\end{proof}

\def\cprime{$'$} \newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1}
  \newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1}
  \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{10}

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%\bibliographystyle{amsplain}
%\bibliography{../dis-gp-nu-th,../scv-ana-geo,../rep-quan,../gz}

\end{document}


The transfer from compact picture on
$C^\infty(S)$ to the non-compact
picture is done via the Cayley transform $c$. We present
its formula, but shall not need it in the sequel.
The  transform $c$ maps
$\mathbb R^{n-1}$ to $S\setminus \{e_1\}$,
\begin{equation}
\label{s-versus-x}
c: \mathbb R^{n-1} \to S, \quad
x\mapsto s=(s_1, s')=(\frac{|x|^2-1}{|x|^2+1}, \frac{2x}{|x|^2+1}).
\end{equation}
The  representation space
$\pi_\nu^\infty$
 consists now of smooth functions
$f$ on $\mathbb R^{n-1}$
of the form
$$
f(x)=(1+|x|^2)^{-\nu} F(s)
=(\frac{1+s_1}2)^\nu F(s), \quad F\in C^\infty(S).
$$


1402.2950.extracted_bib
{Asmuth-Repka C.~Asmuth and J.~Repka, Tensor products for { ${\rm SL_{2(\mathcal K)$ . {I. {Complementary series and the special representation, Pacific J. Math. 97 (1981), no.~2, 271--282. \MR{641157 (83c:22023)
1402.2950.extracted_bib
{BKZ S. Ben Sa{\" \id, K. Koufany and G. Zhang, Invariant trilinear forms on spherical principal series of real-rank one semisimple Lie groups, Internat. J. Math. 25 (2014), no. 3, 1450017 (35 pages).
1402.2950.extracted_bib
{Burger-Li-Sarnak M.~Burger, J.-S. Li, and P.~Sarnak, Ramanujan duals and automorphic spectrum, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no.~2, 253--257. \MR{1118700 (92h:22023)
1402.2950.extracted_bib
{Clerc-2011 J.-L. Clerc, Singular conformally invariant trilinear forms and covariant differential operators on the sphere, arXiv:1102.1861.
1402.2950.extracted_bib
{Clerc-Beckmann J.-L. Clerc and R.~Beckmann, Singular conformally invariant trilinear forms and generalized rankin cohen operators, arXiv:1104.3461.
1402.2950.extracted_bib
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