# Delorme’s intertwining conditions for sections of homogeneous vector bundles
on two and three dimensional hyperbolic spaces

Martin Olbrich and Guendalina Palmirotta

###### Abstract

The description of the Paley-Wiener space for compactly supported smooth
functions $C^{\infty}_{c}(G)$ on a semi-simple Lie group $G$, involves certain
intertwining conditions, that are difficult to handle. In the present paper,
we make them completely explicit for $G=\mathbf{SL}(2,\mathbb{R})^{d}$
($d\in\mathbb{N}$) and $G=\mathbf{SL}(2,\mathbb{C})$. Our results are based on
a defining criterion for the Paley-Wiener space, valid for general groups of
real rank one, that we derive from Delorme’s proof of the Paley-Wiener
theorem. In a forthcoming paper, we will show how these results can be used to
study solvability of invariant differential operators between sections of
homogeneous vector bundles over the corresponding symmetric spaces.

###### Contents

  1. 1 Introduction
  2. 2 Intertwining conditions and operators
  3. 3 Sufficient intertwining conditions for rank one
  4. 4 The case $G=\mathbf{SL}(2,\mathbb{R})$ and beyond
  5. 5 The case $G=\mathbf{SL}(2,\mathbb{C})$

##  1 Introduction

Consider a Riemannian symmetric space of non-compact type $X=G/K$, where $G$
is a real connected semi-simple Lie group with finite center of non-compact
type and $K\subset G$ its maximal compact subgroup.
Arthur [Art83] as well as Delorme ([Del05], Thm. 2) established a Paley-Wiener
theorem for ($K$-finite) compactly supported smooth functions on $G$. Their
results involved the so-called Arthur-Campolli and Delorme conditions
respectively. Later van den Ban and Souaifi [vdBS14] proved, without using the
proof nor validity of any associated Paley-Wiener theorems of Arthur or
Delorme, that the two compatibility conditions are equivalent.
In [OlPa22-1], we proved a topological Paley-Wiener(-Schwartz) theorem for
sections of homogeneous vector bundles by adapting Delorme’s intertwining
conditions for our purposes. We considered the intertwining conditions in
three levels, namely (Level 1) refered to Delorme’s condition in the setting
of van den Ban and Souaifi [vdBS14], (Level 2) corresponded to the conditions
for sections over homogeneous vector bundles and (Level 3) stood for spherical
functions.
However, these intertwining conditions are very difficult to check in
practice, even for special $K$-types.
The most important source of such conditions are the Knapp-Stein ([KSt71] &
[KSt80]) and Želobenko [Zelo76] intertwining operators, as well as the
embedding of discrete series into principal series
$H^{\sigma,\lambda}_{\infty}$, for
$(\sigma,\lambda)\in\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}$.

Therefore, in this article, we rewrite them in a more accessible way involving
such intertwining operators and the Harish-Chandra $\mathbf{c}$-functions,
which we introduce in the first Section 2. Moreover, we show that only a part
of them is already sufficient for $G$ of real rank one (Section 3, Thm 3).
This is already implicitly contained in Delorme’s proof of the Paley-Wiener
theorem ([Del05], Thm. 2). For our proof, we essentially use an intermediate
result of Delorme ([Del05], Prop. 1) and his induction procedure ([Del05],
Prop. 2) on the length of minimal $K$-types of a generalized principal series
representation.
To apply it, one has to know more or less the complete composition series of
reducible principle series representations, which is the case for the two
special examples, $\mathbf{SL}(2,\mathbb{R})$ and $\mathbf{SL}(2,\mathbb{C})$,
which are in the focus of the present paper.
In fact, for $G=\mathbf{SL}(2,\mathbb{R})$, in Section 4, by drawing its
principal series representations $H^{\pm,\lambda}_{\infty}$ by ’box-pictures’
(Fig. 1), we can see in which closed $G$-submodule of
$H^{\pm,\lambda}_{\infty}$ there is an intertwining condition in (Level 2)
(Thm. 6). Afterwards, we can deduce the corresponding results also for the
other levels (Thms. 7 & 5) and even go beyond by illustrating the intertwining
conditions for finite products of $\mathbf{SL}(2,\mathbb{R})$ (Thm. 8). As
last example, in Section 5, we consider $G=\mathbf{SL}(2,\mathbb{C})$. The
description of its intertwining conditions (Thms. 10, 11 & 14) are more
difficult then for the previous examples. We remark that
$\mathbf{SL}(2,\mathbb{R})$ and $\mathbf{SL}(2,\mathbb{C})$ are locally
isomorphic to $\mathbf{SO}(1,n),n=2,3$ respectively. In fact, based on Thm. 3,
we have already checked that it is possible to obtain analogous results for
all $n$. We are grateful to P. Delorme for mentioning to us that the results
for odd $n$ (including $\mathbf{SL}(2,\mathbb{C})$) can also be derived from
his earlier Paley-Wiener theorem for groups with only one conjugacy class of
Cartan subgroups [Del82].
This work is part of the second author’s doctoral dissertation [Pa22]. In
fact, our results can be used to study solvability of invariant differential
operators between sections of homogeneous vector bundles on the corresponding
symmetric spaces. The way we expressed the intertwining conditions in (Level
3) is particularly adapted to the solvability questions. In an upcoming paper
[OlPa22-3], these questions will be discussed in detail.

##  2 Intertwining conditions and operators

We adopt the same notations as in [OlPa22-1]. Let $G$ be a real connected
semi-simple Lie group with finite center of non-compact type with Lie algebra
$\mathfrak{g}$. Consider its Iwasawa decomposition $G=KAN$, where $K\subset G$
its maximal compact subgroup with Lie algebra $\mathfrak{k}$,
$A=\exp(\mathfrak{a})$ abelian and $N$ nilpotent. The quotient $X=G/K$ is a
Riemannian symmetric space of non-compact type.
Let $M=Z_{K}(\mathfrak{a})$ be the centralizer of $A$ in $K$. Then $P=MAN$ is
a minimal parabolic subgroup of $G$. Let $(\sigma,E_{\sigma})\in\widehat{M}$
be a finite-dimensional irreducible representation of $M$ and
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$, the complexified dual of the Lie
algebra of $A$. For
$(\sigma,\lambda)\in\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}$, consider
$H^{\sigma,\lambda}_{\infty}$ the space of smooth vectors of the principal
series representations of $G$ induced from the $P$-representation
$\sigma_{\lambda}$ on the vector space $E_{\sigma}$ ([Kna86], p. 168). By
$H^{\sigma}_{\infty}$ we denote its representation space in the compact
picture, which is given by functions on $K$ and is independent of $\lambda$.
Given $\sigma\in\widehat{M}$, let us consider the map

$\pi_{\sigma,\cdot}:G\rightarrow(\mathfrak{a}^{*}_{\mathbb{C}}\rightarrow\text{End}(H^{\sigma}_{\infty})),g\mapsto(\lambda\mapsto\pi_{\sigma,\lambda}(g)).$

In [OlPa22-1], we introduce the $m$-th derived principal series representation
of $G$, which is a very similar definition of derived $G$-representation
([Del05], Déf. 3 (4.4) & [vdBS14], Sect. 4.5).

######  Definition 1 ($m$-th derived representation, [OlPa22-1], Def. 2).

For $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$, let
$\emph{\text{Hol}}_{\lambda}$ be the set of germs at $\lambda$ of
$\mathbb{C}$-valued holomorphic functions $\mu\mapsto f_{\mu}$ and
$m_{\lambda}\subset\emph{\text{Hol}}_{\lambda}$ the maximal ideal of germs
vanishing at $\lambda$.
Denote by $H^{\sigma}_{[\lambda]}$ the set of germs at $\lambda$ of
$H^{\sigma}_{\infty}$-valued holomorphic functions $\mu\mapsto\phi_{\mu}\in
H^{\sigma}_{\infty}$ with $G$-action

$(g\phi)_{\mu}=\pi_{\sigma,\mu}(g)\phi_{\mu},\;\;\;g\in G.$

For $m\in\mathbb{N}_{0}$, it induces a representation
$\pi^{(m)}_{\sigma,\lambda}$ on the space

$H^{\sigma,\lambda}_{\infty,(m)}:=H^{\sigma}_{[\lambda]}/m_{\lambda}^{m+1}H^{\sigma}_{[\lambda]},$
(2.1)

which is equipped with the natural Fréchet topology. We call this
representation the $m$-th derived principal series representation of $G$.

Let us briefly recall the intertwining conditions in the three levels, for
more details we refer to [OlPa22-1].

Intertwining conditions and Paley-Wiener theorem in (Level 1). We denote by
$\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}})$ the space of holomorphic functions
on $\mathfrak{a}^{*}_{\mathbb{C}}$ and by
$\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}(H^{\sigma}_{\infty}))$
the space of maps
$\mathfrak{a}^{*}_{\mathbb{C}}\ni\lambda\mapsto\phi(\lambda)\in\text{End}(H^{\sigma}_{\infty})$
such that for $\varphi\in H^{\sigma}_{\infty},$ the function
$\lambda\mapsto\phi(\lambda)\varphi\in H^{\sigma}_{\infty}$ is holomorphic.

######  Definition 2 (Delorme’s intertwining condition in (Level 1),
[OlPa22-1], Def. 3).

Let $\Xi$ be the set of all 3-tuples $(\sigma,\lambda,m)$ with
$\sigma\in\widehat{M}$, $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ and
$m\in\mathbb{N}_{0}$. Consider the $m$-th derived $G$-representation
$H^{\sigma,\lambda}_{\infty,(m)}$ defined in (2.1). For every finite sequence
$\xi=(\xi_{1},\xi_{2},\dots,\xi_{s})\in\Xi^{s},s\in\mathbb{N}$, we define the
$G$-representation

$H_{\xi}:=\bigoplus_{i=1}^{s}H^{\sigma_{i},\lambda_{i}}_{\infty,(m_{i})}.$

We consider proper closed $G$-subrepresentations $W\subseteq H_{\xi}$.
Such a pair $(\xi,W)$ with $\xi\in\Xi^{s}$ and $W\subset H_{\xi}$ as above, is
called an intertwining datum. Every function
$\phi\in\prod_{\sigma\in\widehat{M}}\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{End}}(H^{\sigma}_{\infty}))$
induces an element

$\phi_{\xi}\in\bigoplus_{i=1}^{s}\emph{\text{End}}(H^{\sigma_{i},\lambda_{i}}_{\infty,(m_{i})})\subset\emph{\text{End}}(H_{\xi}).$

  * (D.a)

We say that $\phi$ satisfies Delorme’s intertwining condition, if
$\phi_{\xi}(W)\subseteq W$ for every intertwining datum $(\xi,W)$.

Moreover, for $r>0$, one can introduce a corresponding Paley-Wiener space

$PW_{r}(G)\subset\prod_{\sigma\in\widehat{M}}\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}(H^{\sigma}_{\infty})),$

which is characterised by the usual growth condition ([Del05], Déf. 3 (4.3)
and [OlPa22-1], Def. 4 $(1.ii)_{r}$) and intertwining condition $(D.a)$. Let
$\overline{B}_{r}(o)$ be the preimage of the closed ball of radius $r>0$
centered at $o\in X$ under the projection $p:G\rightarrow X$. Write by
$C^{\infty}_{r}(G)$ the space of smooth complex functions on $G$ with support
in $\overline{B}_{r}(o)$. Then, by taking the union over all $r>0$, and
considering the Fourier transform of $f\in\leavevmode\nobreak\
C^{\infty}_{c}(G)$:

$\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}\in(\sigma,\lambda)\mapsto\mathcal{F}_{\sigma,\lambda}(f):=\pi_{\sigma,\lambda}(f)=\int_{G}f(g)\pi_{\sigma,\lambda}(g)\;dg\in\prod_{\sigma\in\widehat{M}}\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}(H^{\sigma}_{\infty})),$

we have the following isomorphism.

######  Theorem 1 (Delorme’s Paley-Wiener Theorem in (Level 1), [Del05], Thm.
2).

For any $r>0$, the Fourier transform

$C_{r}^{\infty}(G)\ni f\mapsto\mathcal{F}_{\sigma,\lambda}(f)\in
PW_{r}(G),\;\;\;(\sigma,\lambda)\in\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}$

is a topological isomorphism between the two Fréchet spaces
$C_{r}^{\infty}(G)$ and $PW_{r}(G)$. ∎

Intertwining conditions and Paley-Wiener theorems in (Level 2) and (Level 3).
Next, consider two finite-dimensional, not necessary irreducible,
$K$-representations $(\tau,E_{\tau})$ and $(\gamma,E_{\gamma})$. We consider
the space of smooth compactly supported sections of homogeneous vector bundles
$\mathbb{E}_{\tau}$ over $X$ by

$C_{c}^{\infty}(X,\mathbb{E}_{\tau})=\bigcup_{r>0}C_{r}^{\infty}(X,\mathbb{E}_{\tau})\cong\bigcup_{r>0}\\{f:G\stackrel{{\scriptstyle
C^{\infty}}}{{\longrightarrow}}E_{\tau}\;|\;f(gk)=\tau^{-1}(k)(f(g)),\forall
g\in,k\in K\;\&\;supp(f)\subset\overline{B}_{r}(0)\\}.$

The group $G$ acts on $C_{c}^{\infty}(X,\mathbb{E}_{\tau})$ by left
translations, $(g\cdot f)(g^{\prime})=f(g^{-1}g^{\prime}),\forall
g,g^{\prime}\in G$. It is not difficult to see that we have the
$G$-isomorphisms $C^{\infty}(X,\mathbb{E}_{\tau})\cong
C^{\infty}(G,E_{\tau})^{K}\cong[C^{\infty}(G)\otimes E_{\tau}]^{K}.$ Moreover,
we also consider the space of $(\gamma,\tau)$-spherical functions on $G$

$\displaystyle
C_{c}^{\infty}(G,\gamma,\tau)=\bigcup_{r>0}C_{r}^{\infty}(G,\gamma,\tau)$
$\displaystyle:=$
$\displaystyle\bigcup_{r>0}\\{f:G\rightarrow\text{Hom}(E_{\gamma},E_{\tau})\;|\;f(k_{1}gk_{2})=\tau(k_{2})^{-1}f(g)\gamma(k_{1})^{-1},\forall
k_{1},k_{2}\in K\;\&\;supp(f)\subset\overline{B}_{r}(0)\\}.$

Note that by taking topological linear duals, we obtain the spaces of
distributional sections $C^{-\infty}(X,\mathbb{E}_{\tau})$ and
$C^{-\infty}(G,\gamma,\tau)$. We are particularly interested in distributional
sections with compact support $C_{c}^{-\infty}(X,\mathbb{E}_{\tau})$ and
$C_{c}^{-\infty}(G,\gamma,\tau)$.

The Fourier transform for (distributional) sections of homogeneous vector
bundles in (Level 2) and (Level 3) is given in the following definition.

######  Definition 3 (Fourier transforms, [OlPa22-1], Def. 5 & 6).

Let $g=\kappa(g)a(g)n(g)\in KAN=G$ be the Iwasawa decomposition. For fixed
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ and $k\in K$, we define the function
$e^{\tau}_{\lambda,k}$ by

$\displaystyle e^{\tau}_{\lambda,k}:G$ $\displaystyle\rightarrow$
$\displaystyle\emph{\text{End}}(E_{\tau})\cong E_{\tilde{\tau}}\otimes
E_{\tau}$ $\displaystyle g$ $\displaystyle\mapsto$ $\displaystyle
e^{\tau}_{\lambda,k}(g):=\tau(\kappa(g^{-1}k))^{-1}a(g^{-1}k)^{-(\lambda+\rho)},$
(2.2)

where $\rho$ is the half sum of the positive roots of
$(\mathfrak{g},\mathfrak{a})$.

  * (a)

(Level 2) For $f\in C^{\infty}_{c}(X,\mathbb{E}_{\tau})$, the Fourier
transformation is given by

$\mathcal{F}_{\tau}f(\lambda,k)=\int_{G}e^{\tau}_{\lambda,k}(g)f(g)\;dg=\int_{G/K}e^{\tau}_{\lambda,k}(g)f(g)\;dg,$
(2.3)

where the last equality makes sense, since the integrand is right
$K$-invariant. Similar, for distributional sections $T\in
C^{-\infty}_{c}(X,\mathbb{E}_{\tau})$:
$\mathcal{F}_{\tau}T(\lambda,k):=\langle
T,e^{\tau}_{\lambda,k}\rangle=T(e^{\tau}_{\lambda,k})\in E_{\tau},$ with
$(\lambda,k)\in\mathfrak{a}^{*}_{\mathbb{C}}\times K/M.$

  * (b)

(Level 3) The Fourier transformation for $f\in C^{\infty}_{c}(G,\gamma,\tau)$
is given by

$\prescript{}{\gamma}{\mathcal{F}}_{\tau}f(\lambda):=\int_{G}e^{\tau}_{\lambda,1}(g)f(g)\;dg,\;\;\;\;\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}.$
(2.4)

Similar, for $T\in C^{-\infty}_{c}(G,\gamma,\tau)$, we have
$\prescript{}{\gamma}{\mathcal{F}}_{\tau}T(\lambda):=\langle
T,e^{\tau}_{\lambda,1}\rangle.$

Note that $\mathcal{F}_{\tau}(f)$,
$\mathcal{F}_{\tau}(T)\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},H^{\tau|_{M}}_{\infty})$,
where $H^{\tau|_{M}}_{\infty}:=\\{f:K\stackrel{{\scriptstyle
C^{\infty}}}{{\rightarrow}}E_{\tau}\;|\;f(km)=\tau(m)^{-1}f(k)\\}$ and that
$\prescript{}{\gamma}{\mathcal{F}}_{\tau}(f)$,
$\prescript{}{\gamma}{\mathcal{F}}_{\tau}(T)\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{Hom}_{M}(E_{\gamma},E_{\tau}))$.
We also remark that
$\text{Hom}_{K}(E_{\gamma},H^{\tau|_{M}}_{\infty})\cong\text{Hom}_{M}(E_{\gamma},E_{\tau}),$
by Frobenius reciprocity. We will also consider
$H^{\tau|_{M},\lambda}_{\infty,(m)}$ defined as in Def. 1 with $\sigma$
replaced by $\tau|_{M}$.

######  Definition 4.

We define

$\emph{\text{Hom}}_{M}(E_{\tau},E_{\sigma})^{\lambda}_{(m)}:=\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{Hom}}_{M}(E_{\tau},E_{\sigma}))/m_{\lambda}^{m+1}\;\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{Hom}}_{M}(E_{\tau},E_{\sigma}))$

as in (2.1). For $\tau\in\widehat{K}$ and an intertwining datum $(\xi,W)$, we
set

$\displaystyle D^{\tau}_{W}$ $\displaystyle:=$
$\displaystyle\\{t\in\bigoplus_{i=1}^{s}\emph{\text{Hom}}_{M}(E_{\tau},E_{\sigma_{i}})^{\lambda_{i}}_{(m_{i})}\;|\;T=\widetilde{Frob}^{-1}(t)\in\emph{\text{Hom}}_{K}(E_{\tau},W)\subset\emph{\text{Hom}}_{K}(E_{\tau},H_{\xi})\\}$
(2.5) $\displaystyle\subset$
$\displaystyle\bigoplus_{i=1}^{s}\emph{\text{Hom}}_{M}(E_{\tau},E_{\sigma_{i}})^{\lambda_{i}}_{(m_{i})}.$

We have shown ([OlPa22-1], Def. 7+8, Prop. 7 & Thm. 2) that Delorme’s
intertwining condition $(D.a)$ correspond to the following intertwining
conditions in (Level 2) and (Level 3).

######  Definition 5 (Intertwining conditions in (Level 2) and (Level 3),
[OlPa22-1], Thm. 2 ).

Let $\Xi$ be as in Def. 2 and let $\overline{\Xi}$ the set of all tuples
$(\lambda,m)$ with $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ and
$m\in\mathbb{N}_{0}$. We define a map

$\Xi\longrightarrow\overline{\Xi},\;\;\;\xi=(\sigma,\lambda,m)\mapsto\overline{\xi}=(\lambda,m).$

For $s\in\mathbb{N}$ and $\xi\in\Xi^{s}$, we have the corresponding element
$\overline{\xi}\in\overline{\Xi}^{s}$.

  * (D.2)

(Level 2) We say that
$\psi\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},H^{\tau|_{M}}_{\infty})$
satisfies the intertwining condition, if for each intertwining datum $(\xi,W)$
and each non-zero $t=(t_{1},t_{2},\dots,t_{s})\in D^{\tau}_{W}$, the induced
element
$\psi_{\overline{\xi}}\in\bigoplus_{i=1}^{s}H^{\tau|_{M},\lambda_{i}}_{\infty,(m_{i})}=:H^{\tau|_{M}}_{\overline{\xi}}$
satisfies

$t\circ\psi_{\overline{\xi}}=(t_{1}\circ\psi_{1},\dots,t_{2}\circ\psi_{s})\in
W.$

  * (D.3)

(Level 3) We say that
$\varphi\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{Hom}}_{M}(E_{\gamma},E_{\tau}))$
satisfies the intertwining condition, if for each intertwining datum $(\xi,W)$
and each non-zero $t=(t_{1},t_{2},\dots,t_{s})\in D^{\tau}_{W}$, the induced
element
$\varphi_{\overline{\xi}}\in\bigoplus_{i=1}^{s}\emph{\text{Hom}}_{M}(E_{\gamma},E_{\tau})^{\lambda_{i}}_{(m_{i})}=:H^{\gamma,\tau}_{\overline{\xi}}$
satisfies

$t\circ\varphi_{\overline{\xi}}=(t_{1}\circ\varphi_{1},\dots,t_{2}\circ\varphi_{s})\in
D^{\gamma}_{W}.$

######  Example 1 ([OlPa22-1], Example 1).

Consider now $s=2$ and $m_{1}=m_{2}=0$. Let

$L:H^{\sigma_{1},\lambda_{1}}_{\infty}\longrightarrow
H^{\sigma_{2},\lambda_{2}}_{\infty}$

be an intertwining operator between the two principal series representations.
The corresponding intertwining data is given by
$\xi:=((\sigma_{1},\lambda_{1},0),(\sigma_{2},\lambda_{2},0))\in\Xi^{2}$ and
$W=graph(L)\subset H^{\sigma_{1},\lambda_{1}}_{\infty}\oplus
H^{\sigma_{2},\lambda_{2}}_{\infty}$. Moreover, define
$l^{\tau}:\text{Hom}_{M}(E_{\tau},E_{\sigma_{1}})\longrightarrow\text{Hom}_{M}(E_{\tau},E_{\sigma_{2}})$
by

$l^{\tau}(t)(v)=L(t\tau(\cdot)^{-1}v)(e)=L(\phi_{v}(t))(e),$

where the element $\phi_{v}(t)\in H_{\infty}^{\sigma}$ is given by the
function $\phi_{v}(t)(k):=t\tau(k^{-1})v,$ for $k\in K,v\in E_{\tau}$ and
$t\in\text{Hom}_{M}(E_{\tau},E_{\sigma_{1}})$. Then

$\displaystyle D^{\tau}_{W}=\\{(t_{1},t_{2})\;|\;t_{2}=l^{\tau}(t_{1})\\}$
$\displaystyle=$
$\displaystyle\\{(t,l^{\tau}(t))\;|\;t\in\text{Hom}_{M}(E_{\tau},E_{\sigma_{1}})\\}$
$\displaystyle\subset$
$\displaystyle\text{Hom}_{M}(E_{\tau},E_{\sigma_{1}})\oplus\text{Hom}_{M}(E_{\tau},E_{\sigma_{2}}).$

In this situation, we get the following intertwining conditions.

  * (D2.L)

(Level 2) For each $t\in\text{Hom}_{M}(E_{\tau},E_{\sigma_{1}})$, we have for
$\psi(\lambda_{i},\cdot)\in H^{\tau|_{M},\cdot}_{\infty},i=1,2$

$L(t\circ\psi(\lambda_{1},\cdot))=l^{\tau}(t)\circ\psi(\lambda_{2},\cdot).$
(2.6)

  * (D3.L)

(Level 3) For each $t\in\text{Hom}_{M}(E_{\tau},E_{\sigma_{1}})$, we have for
$\varphi(\lambda_{i})\in\text{Hom}_{M}(E_{\gamma},E_{\tau}),i=1,2$

$l^{\gamma}(t\circ\varphi(\lambda_{1}))=l^{\tau}(t)\circ\varphi(\lambda_{2}).$
(2.7)

Similar as in (Level 1), for $r>0$, we introduce a Paley-Wiener space in:

  * $-$

(Level 2) $PW_{r,\tau}(\mathfrak{a}^{*}_{\mathbb{C}}\times
K/M)\subset\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},H^{\tau|_{M}}_{\infty})$,

  * $-$

(Level 3)
$\prescript{}{\gamma}{PW}_{\tau}(\mathfrak{a}^{*}_{\mathbb{C}})\subset\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{Hom}_{M}(E_{\gamma},E_{\tau}))$,

which is characterised by the usual growth condition ([OlPa22-1], Def.
$(2.ii)_{r}$ resp. $(3.ii)_{r}$) and the corresponding intertwining condition
$(D.2)$ and $(D.3)$, respectively. The Paley-Wiener-Schwartz space for
distributional sections is denoted by
$PWS_{\tau}(\mathfrak{a}^{*}_{\mathbb{C}}\times K/M)$ respectively
$\prescript{}{\gamma}{PWS}_{\tau}(\mathfrak{a}^{*}_{\mathbb{C}})$. It is
characterised in the same way as above except that we replace the growth
condition by a weaker one ([OlPa22-1], Def. $(2.iis)_{r}$ resp.
$(3.iis)_{r}$). As in ([OlPa22-1], Sect. 6) we equip the Paley-
Wiener(-Schwartz) space with the inductive limit topology and the space of
distributional sections $C_{c}^{-\infty}(X,\mathbb{E}_{\tau})$ and
$C^{-\infty}_{c}(G,\gamma,\tau)$ with the strong dual topology. In [OlPa22-1],
we derived the following.

######  Theorem 2 (Topological Paley-Wiener(-Schwartz) theorem for sections in
(Level 2) and (Level 3), [OlPa22-1] Thms. 3 & 4).

The Fourier transform

$C_{c}^{\pm\infty}(X,\mathbb{E}_{\tau})\ni\psi\mapsto\mathcal{F}_{\tau}(\psi)(\lambda,k)\in
PW(S)_{\tau}(\mathfrak{a}^{*}_{\mathbb{C}}\times
K/M),\;\;\;(\lambda,k)\in\mathfrak{a}^{*}_{\mathbb{C}}\times K$

is a topological isomorphism between $C_{c}^{\pm\infty}(X,\mathbb{E}_{\tau})$
and $PW(S)_{\tau}(\mathfrak{a}^{*}_{\mathbb{C}}\times K/M)$.
Moreover, by considering an additional $K$-representation
$(\gamma,E_{\gamma})$ with associated homogeneous vector bundle
$\mathbb{E}_{\gamma}$, then the Fourier transform

$C^{\pm\infty}_{c}(G,\gamma,\tau)\ni\varphi\mapsto\prescript{}{\gamma}{\mathcal{F}}_{\tau}(\varphi)(\lambda)\in\prescript{}{\gamma}{PW(S)}_{\tau}(\mathfrak{a}^{*}_{\mathbb{C}}),\;\;\;\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$

is a topological isomorphism between $C^{\pm\infty}_{c}(G,\gamma,\tau)$ and
$\prescript{}{\gamma}{PW(S)}_{\tau}(\mathfrak{a}^{*}_{\mathbb{C}})$.
$\square$

In Example 1, we worked out the intertwining conditions coming from
intertwining operators between principal series representations of $G$. We
want to make them more explicit for the most important case, the Knapp-Stein
intertwining operators, which we now recall.

Knapp-Stein intertwining operator. Let $W_{A}:=N_{K}(\mathfrak{a})/M$ be the
Weyl group. Note that $W_{A}$ acts on $\mathfrak{a}^{*}_{\mathbb{C}}$ as well
as on $\widehat{M}$. Let $w\in W_{A}$ be represented by $m_{w}\in
M^{\prime}:=N_{K}(\mathfrak{a})$ and $\sigma\in\widehat{M}$. We realise
$\sigma$ on the vector space $E_{\sigma}.$ We define a new representation
$w\sigma\in\widehat{M}$ of $M$ acting on $E_{\sigma}$ by

$w\sigma(m):=\sigma(m_{w}^{-1}mm_{w}),\;\;m\in M.$

Its equivalence class only depends on $w\in W_{A}$ and not on the choice of
$m_{w}$.

######  Definition 6 (Knapp-Stein intertwining operator, [KSt71], [KSt80] &
[Kna02], Chap. V).

Let $\Delta^{+}_{\mathfrak{a}}$ be the positive root system of
$(\mathfrak{g},\mathfrak{a})$ corresponding to $N$. Let
$\Delta^{-}_{\mathfrak{a}}:=-\Delta^{+}_{\mathfrak{a}}$ and $\overline{N}$ be
the unipotent subgroup coming from the associated Iwasawa decomposition
corresponding to $\Delta^{-}_{\mathfrak{a}}$. Write $N_{w}:=N\cap
w\overline{N}w^{-1}$. For
$(\sigma,\lambda)\in\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}$ with
$(\emph{\text{Re}}(\lambda),\alpha)>0$, for all
$\alpha\in\Delta^{+}_{\mathfrak{a}}\cap w^{-1}\Delta^{-}_{\mathfrak{a}}$ and
for a fixed representative $m_{w}\in M^{\prime}$, we define the intertwining
operator

$J_{w,\sigma,\lambda}:H^{\sigma,\lambda}_{\infty}\longrightarrow
H^{w\sigma,w\lambda}_{\infty}$

by the convergent integral

$J_{w,\sigma,\lambda}(\varphi(g)):=\int_{{N}_{w}}\varphi(g{n}m_{w})\;d{n},\;\;g\in
G,\varphi\in H^{\sigma,\lambda}_{\infty},$

which depends holomorphically on $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$.
This operator has a meromorphic continuation to
$\mathfrak{a}^{*}_{\mathbb{C}}$.

For later reference, let us state the intertwining conditions coming from the
Knapp-Stein operators.

######  Example 2 (Knapp-Stein intertwining condition in (Level 1).

For $w\in W_{A}$ and fixed
$(\sigma,\lambda)\in\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}$, we
consider the Knapp-Stein intertwining operator $J_{w,\sigma,\lambda}$ as in
Def. 6.
Let
$\phi\in\prod_{\sigma\in\widehat{M}}\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}(H^{\sigma}_{\infty}))$.
Then, the condition

$J_{w,\sigma,\lambda}\circ\phi(\sigma,\lambda)=\phi(w\sigma,w\lambda)\circ
J_{w,\sigma,\lambda},$ (2.8)

is a special intertwining condition of $(D.a)$ in Def. 2. Note that the
corresponding intertwining datum $(\xi,W)$ is given by
$\xi=((\sigma,\lambda,0),(w\sigma,w\lambda,0))$ and
$W=graph(J_{w,\sigma,\lambda})\subset H^{\sigma,\lambda}_{\infty}\oplus
H^{w\sigma,w\lambda}_{\infty},$ (compare Example 1).

######  Definition 7 (Harish-Chandra $\mathbf{c}$-function, (e.g. [Olb95],
Def. 3.8)).

Let $\tau\in\widehat{K},w\in W_{A}$, $\overline{N}_{w}:=\overline{N}\cap
w^{-1}Nw$ and $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ with
$(\emph{\text{Re}}(\lambda),\alpha)>0$, for all
$\alpha\in\Delta^{+}_{\mathfrak{a}}\cap w^{-1}\Delta_{\mathfrak{a}}^{-}$. The
$\mathbf{c}$-function is defined by

$\mathbf{c}_{w,\tau}(\lambda):=\int_{\overline{N}_{w}}a(\overline{n})^{-(\lambda+\rho)}\tau(\kappa(\overline{n}))\;d\overline{n}\in\emph{\text{End}}_{M}(E_{\tau}),$

which can be extended to a meromorphic function on
$\mathfrak{a}^{*}_{\mathbb{C}}$.
Furthermore, we have
$\mathbf{c}_{w,\tau}(\sigma,\lambda):=pr_{\sigma}\circ\mathbf{c}_{w,\tau}(\lambda)\circ
pr_{\sigma}\in\emph{\text{End}}_{M}(E_{\tau}(\sigma)),$ where
$pr_{\sigma}:E_{\tau}\longrightarrow E_{\tau}(\sigma)$ is the projection on
the $\sigma$-isotypic compenent.
Consider now $w\in W_{A}$ as a Weyl element with maximal length, then, we set

$\mathbf{c}_{\tau}(\lambda):=\mathbf{c}_{w,\tau}(\lambda)\;\;\;\;\;\text{ and
}\;\;\;\;\;\mathbf{c}_{\tau}(\sigma,\lambda):=\mathbf{c}_{w,\tau}(\sigma,\lambda).$

One can express the Knapp-Stein intertwining operators, by the Harish-Chandra
$\mathbf{c}$-functions ([Olb95], Lem. 3.12 & Satz 3.13):

$\displaystyle
J_{w,\sigma,\lambda}(\phi_{v}(t))=\phi_{v}(t\circ\tau(m_{w}^{-1})\mathbf{c}_{w^{-1},\tau}(-w\lambda)),$
(2.9)

where $\phi_{v}(t)$ is defined as in Example 1 for $v\in E_{\tau}$ and
$t\in\text{Hom}_{M}(E_{\tau},E_{\sigma}).$ Set
$J_{w,\tau,\lambda}:=\tau(m_{w})J_{w,\tau|_{M},\lambda}$, which is independent
of the choice of $w\in W_{A}$. Hence, this leads to the following statement.

######  Proposition 1 (Knapp-Stein intertwining condition in (Level 2) and
(Level 3)).

  * (a)

(Level 2) Let
$\psi\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},H_{\infty}^{\tau|_{M}})$
satisfying $(D.2)$. Then, we have

$J_{w,\tau,\lambda}\psi(\lambda,\cdot)=\mathbf{c}_{w^{-1},\tau}(-w\lambda)\psi(w\lambda,\cdot),\;\;\;\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},w\in
W_{A}.$ (2.10)

  * (b)

(Level 3) Let
$\varphi\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{Hom}}_{M}(E_{\gamma},E_{\tau}))$
satisfying $(D.3)$. Then, we have

$\varphi(\lambda)\gamma(m_{w}^{-1})\mathbf{c}_{w^{-1},\gamma}(-w\lambda)=\tau(m_{w}^{-1})\mathbf{c}_{w^{-1},\tau}(-w\lambda)\circ\varphi(w\lambda),\;\;\;\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},w\in
W_{A}.$ (2.11)

###### Proof.

We consider the operator
$l^{\tau}:\text{Hom}_{M}(E_{\tau},E_{\sigma})\longrightarrow\text{Hom}_{M}(E_{\tau},E_{\sigma})$
as in Example 1 associated by $L=J_{w,\sigma,\lambda}$. Them (2.9) says that

$l^{\tau}(t)=t\circ\tau(m_{w}^{-1})\circ c_{w^{-1},\tau}(-w\lambda).$

Now the proposition follows from (2.6) and (2.7), as in Example 1. ∎

######  Example 3.

  * (a)

Let $\tau$ be a trivial one-dimensional represenation. Then,
$C_{c}^{\infty}(X,\mathbb{E}_{\tau})=C^{\infty}_{c}(X)$ and
$H^{\tau|_{M}}_{\infty}=C^{\infty}(K/M)$. Helgason showed in ([Hel89], Thm.
5.1.) that $\beta\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},C^{\infty}(K/M))$
belongs to the Paley-Wiener space if, and only, if it satisfies the usual
growth condition and the intertwining condition:

$\int_{K/M}e_{w\lambda,k}^{\tau}(g)\beta(w\lambda,k)dk=\int_{K/M}e_{\lambda,k}^{\tau}(g)\beta(\lambda,k)dk,\;\;\;\;\;w\in
W_{A}.$ (2.12)

It is not difficult to show that Helgason’s intertwining condition is
equivalent to (2.10). In fact, for $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$,
consider the Poisson transform (e.g. [Hel89] or [Olb95], Def. 3.2)
$P_{\lambda}:C^{\infty}(K/M)\longrightarrow C^{\infty}(X)$ given by
$\mathcal{P}_{\lambda}(f)(g):=\int_{K}e^{\tau}_{\lambda,k}(g)f(k)\;dk,$ for
$g\in G$. Then, Helgason’s condition (2.12) can be expressed in terms of
Poisson transform

$\mathcal{P}_{\lambda}\circ\beta_{\lambda}=\mathcal{P}_{w\lambda}\circ\beta_{w\lambda},\;\;\;\;\;w\in
W_{A},\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},$

where $\beta_{\lambda}:=\beta(\lambda,\cdot)$. The result now follows from the
functional equation of the Poisson transform ([Olb95], Satz 3.15).

  * (b)

Let $\tau$ and $\gamma$ be two trivial one-dimensional representations.
Consider a function $\beta\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}})$ which
satisfies the usual growth condition. Helgason and Gangolli ([Gan71] &
[Hel20], Thm. 7.1) proved that
$\beta\in\prescript{}{\gamma}{PW}_{\tau}(\mathfrak{a}^{*}_{\mathbb{C}})$, if
and only, if

$\beta(\lambda)=\beta(w\lambda),\text{ for
}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},w\in W_{A}.$

This condition is equivalent to (2.11) in the case $\gamma,\tau$ are trivial.

##  3 Sufficient intertwining conditions for rank one

We want to reduce the amount of intertwining data in $(D.a)$ of Def. 2 to a
minimum. In this section, we assume that $G$ has real rank one. In this
situation, the set of positive restricted roots $\Delta^{+}_{\mathfrak{a}}$
consists of at most two elements, namely $\alpha$ and possibly $2\alpha$. The
Weyl group is reduced to $\\{-1,1\\}$ acting on
$\mathfrak{a}^{*}_{\mathbb{C}}$ by multiplication.

We have the following special intertwining condition. An irreducible unitary
representation $(\pi,E_{\pi})$ of $G$ is called a representation of the
discrete series if there is a $G$-invariant embedding $E_{\pi}\hookrightarrow
L^{2}(G)$. Here, $L^{2}(G)$ denote the space of all square integrable
functions with respect to invariant measure $dg$ on $G$. Write
$\widehat{G}_{d}$ the set of equivalence classes of discrete series
representations of $G$. Let $H_{\pi}$ be any Hilbert space, where the
representation $\pi\in\widehat{G}_{d}$ is realized. Let
$H^{\infty}_{\pi}\subset H_{\pi}$ be the corresponding space of smooth
vectors. For every representation of discrete series $\pi\in\widehat{G}_{d}$,
we choose an embedding

$i_{\pi}:H^{\infty}_{\pi}\hookrightarrow
H^{\sigma_{\pi},\lambda_{\pi}}_{\infty}$

into some principal series representation (Casselman’s subrepresentation
theorem, [Wal88], Thm. 3.8.3. & Casselman’s and Wallach’s globalization Thm.
[Wal92], Chap. 11) and set

$W_{\pi}:=i_{\pi}(H^{\infty}_{\pi})\subset
H^{\sigma_{\pi},\lambda_{\pi}}_{\infty}.$

It is a closed $G$-invariant subspace. Hence, the condition

$\phi(\sigma_{\pi},\lambda_{\pi})(W_{\pi})\subset
W_{\pi},\;\;\;\pi\in\widehat{G}_{d}$ (3.1)

is also of the form $(D.a)$, with $s=1$ and $m=0$, and it permits us to define
that

$\phi(\pi):=\phi(\sigma_{\pi},\lambda_{\pi})|_{W_{\pi}}\in\text{End}(W_{\pi}).$
(3.2)

Thus, for $r>0$, we define the ’special’ Paley-Wiener space $PW^{+}_{r}(G)$ by
replacing Delorme’s intertwining condition $(D.a)$ by the conditions (2.8) and
(3.1) only.
Let $w\in W_{A}$ be the non-trivial element. For
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ with
$(\text{Re}(\lambda),\alpha)>0$, let $m\in\mathbb{N}_{0}$ be the maximal order
of the zeros of $J_{w,\sigma,\mu}(f_{\mu})$ at $\mu=\lambda$, where
$\mu\mapsto f_{\mu}\in H^{\sigma,\mu}_{\infty}$ runs over all germs of
holomorphic functions at $\lambda$ with $f_{\lambda}\neq 0$.
We consider the induced operator

$J_{w,\sigma,\lambda}^{(m-1)}:H^{\sigma,\lambda}_{\infty,(m-1)}\longrightarrow
H^{w\sigma,-\lambda}_{\infty,(m-1)}$ (3.3)

and its kernel $\text{Ker}(J_{w,\sigma,\lambda}^{(m-1)})\subset
H^{\sigma,\lambda}_{\infty,(m-1)}.$ By convention, we set
$H^{\sigma,\lambda}_{\infty,(-1)}=\\{0\\}$, for $m=0$. Notice that due to
condition (2.8), we have for
$\phi^{(m-1)}(\sigma,\lambda)\in\text{End}(H^{\sigma,\lambda}_{\infty,(m-1)})$

$\phi^{(m-1)}(\sigma,\lambda)(\text{Ker}(J_{w,\sigma,\lambda}^{(m-1)}))\subset\text{Ker}(J_{w,\sigma,\lambda}^{(m-1)})\subset
H^{\sigma,\lambda}_{\infty,(m-1)},\;\;\;\;(\sigma,\lambda)\in\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}.$

Let $\tau\in\widehat{K}$ with highest weight $\mu_{\tau}\in
i\mathfrak{t}^{*}$, where $\mathfrak{t}\subset\mathfrak{k}$ is the Lie algebra
of a maximal torus $T\subset K$. We define

$2\rho_{c}:=\sum_{\alpha\in\Delta^{+}(\mathfrak{k},\mathfrak{t})}\alpha\in
i\mathfrak{t}^{*}$

the sum of all positive roots of $\mathfrak{t}_{\mathbb{C}}$ in
$\mathfrak{k}_{\mathbb{C}}$. For a $K$-representation $V$, we denote by
$V(\tau)$ its corresponding isotypic component. For $\sigma\in\widehat{M}$ and
$\pi\in\widehat{G}_{d}$, we define $|\sigma|,|\pi|\in[0,\infty]$ by

$|\sigma|:=\min_{\\{\tau\;|\;H^{\sigma}_{\infty}(\tau)\neq\\{0\\}\\}}||\mu_{\tau}+2\rho_{c}||\text{
and
}|\pi|:=\min_{\\{\tau\;|\;H^{\infty}_{\pi}(\tau)\neq\\{0\\}\\}}||\mu_{\tau}+2\rho_{c}||$

i.e. $|\sigma|,|\lambda|$ are the lengths of ’the’ minimal $K$-type $\tau$ of
$H^{\sigma}_{\infty}$ and $H^{\infty}_{\pi}$ respectively, ([Del05], Sect.
1.3). Denote by $B(\sigma),B(\pi)\subset\widehat{K}$ the finite set of all
minimal $K$-types of $H^{\sigma}_{\infty},H^{\infty}_{\pi}.$

######  Example 4.

Let $G=SL(2,\mathbb{R})$ and $K=SO(2)$ its maximal compact subgroup. With the
notations introduced in Sect. 4 below, we have that
$\widehat{K}\cong\mathbb{Z}$ and $\rho_{c}=0.$

  * (i)

$M=\\{\pm 1\\}$, thus

    * -

if $\sigma$ is trivial, then $B(\sigma)=\\{0\\}\subset\mathbb{Z}$, (trivial
$K$-type) and $|\sigma|=0,$

    * -

if $\sigma$ is non-trivial, then $B(\sigma)=\\{+1,-1\\}\subset\mathbb{Z}$ and
$|\sigma|=1$.

  * (ii)

Let $\pi=D_{k},k\in\mathbb{Z}\backslash\\{0\\}$ be the discrete series
representation of $G$ parametrized as in Thm. 4 below, then

$B(D_{k})=\begin{cases}\\{k+1\\},&k>0,\\\ \\{k-1\\},&k<0\end{cases}$

and $|\pi|=k+1$.

The following result tells us that the intertwining conditions (2.8) and (3.1)
with an additional ’vanishing’ condition are sufficient for semi-simple Lie
group $G$ of real rank one.

######  Theorem 3.

Let $\emph{\text{rk}}_{\mathbb{R}}(G)=1$. For $r>0$, let $\mathcal{A}$ be a
linear closed and $K\times K$ invariant subspace of $PW^{+}_{r}(G)$ satisfying
$\mathcal{F}_{\sigma,\lambda}(C^{\infty}_{r}(G))\subset\mathcal{A}$ and the
following condition:

  * (D.b)

Let $\sigma\in\widehat{M}$ and $\phi\in\mathcal{A}$ such that

    * (i)

$\phi(\sigma^{\prime},\lambda)=0$, for all $\sigma^{\prime}\in\widehat{M}$
with $|\sigma^{\prime}|>|\sigma|$ and
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$,

    * (ii)

$\phi(\pi)=0$, for all $\pi\in\widehat{G}_{d}$ with $|\pi|>|\sigma|$.

Then, for all $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ with
$(\emph{\text{Re}}(\lambda),\alpha)>0$, $\phi$ induces the zero-operator on
$\emph{\text{Ker}}(J_{w,\sigma,\lambda}^{(m-1)})$:

$\phi^{(m-1)}(\sigma,\lambda)\Big{|}_{\emph{\text{Ker}}(J_{w,\sigma,\lambda}^{(m-1)})}=0.$

Here, $(m-1)$ depends on $(\sigma,\lambda)$ as defined above (3.3) and
$\phi(\pi)$ is defined in (3.2).

Then,

$\mathcal{A}=PW_{r}(G)\cong\mathcal{F}_{\sigma,\lambda}(C^{\infty}_{r}(G)).$

###### Proof of Thm. 3.

By Delorme’s Paley-Wiener Thm. 1, we already know that
$PW_{r}(G)\cong\mathcal{F}_{\sigma,\lambda}(C^{\infty}_{r}(G))$ is a closed
and $K\times K$ invariant subspace of $PW^{+}_{r}(G)$. Therefore, it suffices
to show that
$\mathcal{F}_{\sigma,\lambda}(C^{\infty}_{r}(G))\subset\mathcal{A}$ is dense.
Thus for every $K\times K$-finite element $\phi\in\mathcal{A}$, we need to
find a function $f\in C^{\infty}_{r}(G)_{K\times K}$ such that

$\pi_{\sigma,\lambda}(f)=\phi(\sigma,\lambda),\;\;\;\;\forall(\sigma,\lambda)\in\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}.$

Let $\phi\in\mathcal{A}_{K\times K}$. It is given by a collection
$(\phi_{\sigma}),\sigma\in\widehat{M}$. By $K\times K$-finiteness, only
finitely many $\phi_{\sigma}$ are non-zero. Similar, by $K\times
K$-finiteness, $\phi(\pi)=0$, for all but finitely many
$\pi\in\widehat{G}_{d}$. Indeed, for any given $K$-type $\tau$, there are only
finitely many $\pi\in\widehat{G}_{d}$, with $H^{\infty}_{\pi}(\tau)\neq 0$
(e.g. [Wal88], Cor. 7.7.3).
We define $l(\phi)\in[0,\infty)$ by

$l(\phi):=\max\\{|\sigma|,|\pi|\;|\;\sigma\in\widehat{M},\phi_{\sigma}\neq
0;\pi\in\widehat{G}_{d},\phi(\pi)\neq 0\\}.$

We can now imitate the inductive proof of Prop. 2 in Delorme’s paper [Del05].
Assume, as induction hypothesis, that for all $\psi\in\mathcal{A}$ with
$l(\psi)<l(\phi)$, there are $f\in C^{\infty}_{r}(G)$ with
$\mathcal{F}(f)=\psi.$ We enumerate

$\\{\sigma\in\widehat{M}\;|\;|\sigma|=l(\phi)\\}=\\{\sigma_{1},\dots,\sigma_{n}\\}\cup\\{w\sigma_{1},\dots,w\sigma_{n}\\}$

and

$\\{\pi\in\widehat{G}_{d}\;|\;|\pi|=l(\phi)\\}=\\{\pi_{1},\dots,\pi_{s}\\}.$

Condition $(ii)$ together with (2.8) says, in particular, that
$\phi_{\sigma_{i}}$ belongs to a space that Delorme denotes by
$\mathcal{K}_{\sigma_{i}}$, ([Del05], Def. 1). Strictly speaking Delorme has a
condition for $(\text{Re}(\lambda),\alpha)\geq 0.$ But if
$\text{rk}_{\mathbb{R}}(G)=1$, only $(\text{Re}(\lambda),\alpha)>0$ matters.
Note, that $\phi(\pi_{j})$ belongs automatically to $\mathcal{K}_{\pi_{j}}$.
We can apply Prop. 1 together with Eq. (1.38) of Delorme’s paper [Del05], to
deduce the existence of $f_{1},f_{2},\dots,f_{n}\in C^{\infty}_{r}(G)$ and
$g_{1},g_{2},\dots,g_{s}\in C^{\infty}_{r}(G)$ with

$\displaystyle\pi_{\sigma_{i},\lambda}(f_{i})$ $\displaystyle=$
$\displaystyle\phi(\sigma_{i},\lambda),\;\;\;i\in\\{1,2,\dots,n\\},$
$\displaystyle\pi_{j}(g_{j})$ $\displaystyle=$
$\displaystyle\phi(\pi_{j}),\;\;\;\;\;\;j\in\\{1,2,\dots,s\\},$

for $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$. Moreover, the discussion after
Eq. (3.9) in ([Del05], p.1018), makes clear that we can choose the $f_{i}$ and
$g_{j}$ such that

  * (i)

$l(\mathcal{F}(f_{i}))=l(\mathcal{F}(g_{j}))=l(\phi),\forall
i\in\\{1,2,\dots,n\\},j\in\\{1,2,\dots,s\\}$ and

  * (ii)

$\pi_{\sigma_{k},\lambda}(f_{i})=0,\forall k\neq i,$

  * (iii)

$\pi_{\sigma_{i},\lambda}(g_{j})=0,\forall i,j,$

  * (iv)

$\pi_{j}(f_{i})=0,\forall i,j,$

  * (v)

$\pi_{k}(g_{j})=0,\forall k\neq j.$

Now, we set

$\psi:=\phi-\sum_{i=1}^{n}\mathcal{F}(f_{i})-\sum_{j=1}^{s}\mathcal{F}(g_{j}).$

Then, by (i)-(v) we have $l(\psi)<l(\phi)$. Thus, by induction hypothesis
$\psi=\mathcal{F}(f_{0})$. We conclude that $\phi=\mathcal{F}(f)$ with
$f=f_{0}+f_{1}+\cdots+f_{n}+g_{1}+g_{2}+\cdots+g_{s}.$ ∎

######  Remark 1.

The result above can be extended to higher real rank. The extension involves
representations induced from all cuspidal parabolic subgroups $P$ as well as
the Knapp-Stein intertwining operator for them.

##  4 The case $G=\mathbf{SL}(2,\mathbb{R})$ and beyond

We consider $G=\mathbf{SL}(2,\mathbb{R})=\Big{\\{}g:=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathbf{GL}(2,\mathbb{R})\;\Big{|}\;\text{det}(g)=1\Big{\\}}$
the special linear group of $\mathbb{R}^{2}$. It has dimension three. We fix
the Iwasawa decomposition $G=KAN$, where

$K=\mathbf{SO}(2)=\Bigg{\\{}k_{\theta}:=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}\;\Big{|}\;\theta\in\mathbb{R}\Bigg{\\}},\;\;\;A=\Bigg{\\{}a_{t}:=\begin{pmatrix}e^{t}&0\\\
0&e^{-t}\end{pmatrix}\;\Big{|}\;t\in\mathbb{R}\Bigg{\\}},$
$N=\Bigg{\\{}n_{x}:=\begin{pmatrix}1&x\\\
0&1\end{pmatrix}\;\Big{|}\;x\in\mathbb{R}\Bigg{\\}}.$

$G$ is a connected and simple Lie group with maximal compact subgroup $K$ of
dimension one. Clearly, $K$ is isomorphic to the unit circle $S^{1}$. Hence

$\widehat{K}=\\{\delta_{n}\;|\;n\in\mathbb{Z}\\}\cong\mathbb{Z},\;\;\delta_{n}(k_{\theta}):=e^{in\theta}\in\mathbf{GL}(1,\mathbb{C})\cong\mathbb{C}\backslash\\{0\\}$

is the set of all irreducible representations of $K$. The representation space
$E_{\delta_{n}}$ is one-dimensional and equal to $\mathbb{C}$. We sometimes
denote the $K$-representation $\delta_{n}$ simply by $n$. If
$H=\begin{pmatrix}t&0\\\ 0&-t\end{pmatrix}\in\mathfrak{a}$, then the positive
root $\alpha$ is given by $\alpha(H)=2t$ and $\rho(H)=t$, for all
$t\in\mathbb{R}$. We identify $\mathfrak{a}^{*}_{\mathbb{C}}$ isometricaly to
$\mathbb{C}$ with respect to the norm of the invariant twice trace form on
$\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$, i.e., $z\alpha\mapsto z$ and
$\rho\mapsto\frac{1}{2}$.

Since $M=\\{\pm\text{Id}\\}$, we have
$\widehat{M}\cong\mathbb{Z}/2\mathbb{Z}$. Let $+$ be the trivial and $-$ the
non-trivial element of $\widehat{M}$. For $\sigma=\pm\in\widehat{M}$ and
$\lambda\in\mathbb{C}\cong\mathfrak{a}^{*}_{\mathbb{C}}$, we write
$(\pi_{\pm,\lambda},H^{\pm,\lambda}_{\infty})$ for the principal series
representations of $G$. Its restriction to $K$ is the set of Fourier series on
$S^{1}$ with only non-zero even or odd Fourier coefficients

$H^{\pm}_{\infty}=\\{f\in C^{\infty}(K,\mathbb{C})\;|\;f\text{ even or odd
}\\}\stackrel{{\scriptstyle K}}{{\cong}}\bigoplus_{n\text{ even or
odd}}\delta_{n}.$

In order to write down the composition series of $H^{\pm,\lambda}_{\infty}$,
we will from now on denote, for convenience, an exact, non-splitting, module
sequence

$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0,$

shortly by a ’boxes-picture’

$B=$$A$$C$.

A proof of the following classical result can be found for example in
([Wal88], 5.6) or in ([La75], Ch. VI). Note that the referenced proof is also
valid for $G$-representations of smooth vectors instead of
$(\mathfrak{g},K)$-modules, if we apply Casselman’s and Wallach’s
globalization theorem ([Wal92], Prop. 11).

######  Theorem 4 (Structure of principal series representations of
$\mathbf{SL}(2,\mathbb{R})$).

The principal series representations $H^{\pm,\lambda}_{\infty}$ of
$\mathbf{SL}(2,\mathbb{R})$ is exactly reducible, if

$\lambda\in I_{\pm}:=\begin{cases}\frac{1}{2}+\mathbb{Z},&\sigma=+,\\\
\mathbb{Z},&\sigma=-.\end{cases}$

For $\lambda=\frac{k}{2}\in I_{\pm},k\in\mathbb{N}$, we have

$H^{\pm,-\frac{k}{2}}_{\infty}=$$F_{k}$$D_{-k}\oplus
D_{k}$$H^{\pm,\frac{k}{2}}_{\infty}=$$D_{-k}\oplus D_{k}$$F_{k}$

where $F_{k}:=\bigoplus^{k-1}_{l=-(k-1)}\delta_{2l}$ are the finite-
dimensional irreducible $\mathbf{SL}(2,\mathbb{R})$-representation of
dimension $k$ and $D_{k}$ resp. $D_{-k}$ are smooth vectors of a
representation of the discrete series, which are characterized by the $K$-type
decomposition $D_{\pm k}=\overline{\bigoplus}_{j\geq 0}\delta_{\pm(k+1+2j)}.$
Furthermore, for $\lambda=0$, we have

$H^{-,0}_{\infty}=$$D_{-}\oplus D_{+}$

where $D_{\pm}=\overline{\bigoplus}_{j\geq 0}\delta_{\pm(1+2j)}$ are the
limits of the discrete series.  $\square$

######  Remark 2.

Let $W_{\lambda}$ be a proper closed invariant $G$-submodule of
$H^{\pm,\lambda}_{\infty}$, for $\lambda\in I_{\pm}$, as in Thm. 4. Then, one
can observe that

  * -

for $\lambda>0$, $W_{\lambda}\in\\{D_{-k},D_{k},D_{-k}\oplus D_{k}\\}$,
$k=2\lambda$,

  * -

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$k=2\lambda$,

  * -

while for $\lambda=0$ and $\sigma=-$, $W_{\lambda}\in\\{D_{+},D_{-}\\}.$

To describe the intertwining conditions for $G=\mathbf{SL}(2,\mathbb{R})$ in
the three levels, we first need some preparation. The Harish-Chandra
$\mathbf{c}$-function for $G$ is denoted by $\mathbf{c}_{n}(\lambda)$ for
$n\in\mathbb{Z}$. Due Cohn ([Co74], App. 1), it is given explicitly in terms
of gamma function $\Gamma(\cdot)$, by the formula (for a suitable
normalization of the Haar measure $d\overline{n}$)

$\mathbf{c}_{n}(\lambda)=\mathbf{c}_{-n}(\lambda)=\frac{1}{\sqrt{\pi}}\frac{\Gamma(\lambda)\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda+\frac{1+n}{2})\Gamma(\lambda+\frac{1-n}{2})},\;\;\;\lambda\in\mathfrak{a}_{\mathbb{C}}^{*}.$
(4.1)

Let $n\equiv m$ (mod 2), not necessary distinct, then using the gamma function
recurrence formula

$\Gamma(\lambda+a)=(\lambda+(a-1))\Gamma(\lambda+(a-1)),\;\;\;a\in\mathbb{Z},\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$
(4.2)

repeatedly, the quotient of the $\mathbf{c}$-functions is given by

$\frac{\mathbf{c}_{n}(\lambda)}{\mathbf{c}_{m}(\lambda)}=\frac{\Gamma(\lambda+\frac{1+m}{2})\Gamma(\lambda+\frac{1-m}{2})}{\Gamma(\lambda+\frac{1+n}{2})\Gamma(\lambda+\frac{1-n}{2})}=\begin{cases}1,&\text{
for }|n|=|m|,\\\
\frac{(\lambda-\frac{|n|-1}{2})(\lambda-\frac{|n|-3}{2})\cdots(\lambda-\frac{|m|+1}{2})}{(\lambda+\frac{|n|-1}{2})(\lambda+\frac{|n|-3}{2})\cdots(\lambda+\frac{|m|+1}{2})},&\text{
for }|n|>|m|,\\\
\frac{(\lambda+\frac{|m|-1}{2})(\lambda+\frac{|m|-3}{2})\cdots(\lambda+\frac{|n|+1}{2})}{(\lambda-\frac{|m|-1}{2})(\lambda-\frac{|m|-3}{2})\cdots(\lambda-\frac{|n|+1}{2})},&\text{
for }|n|<|m|.\end{cases}$ (4.3)

Note that the quotient has zeros
$\lambda\in\\{\frac{|n|-1}{2},\frac{|n|-3}{2},\dots,\frac{|m|+1}{2}\\}$ and
poles in
$\\{-\frac{|n|-1}{2},-\frac{|n|-3}{2},\dots,-\frac{|m|+1}{2}\\}$, for
$|n|>|m|$, and inversely for $|n|<|m|$. We know by (2.11) that the matrix
coefficient of the Knapp-Stein intertwining operator
$J_{w,\pm,\lambda}:H^{\pm,\lambda}_{\infty}\longrightarrow
H^{\pm,-\lambda}_{\infty}$ with respect to the Fourier decomposition of
$H^{\pm,\lambda}_{\infty}\cong\bigoplus_{n\text{ even or odd}}\delta_{n}$ is
given by $c_{n}(\lambda)$ (up to sign).

######  Theorem 5 (Intertwining conditions in (Level 1)).

For $r>0$, let $A$ be the space of all
$\phi\in\prod_{\sigma\in\widehat{M}}\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{End}}(H^{\pm}_{\infty}))$
such that $\phi$ statisfies the corresponding growth condition as well as the
two intertwining conditions (2.8) and

  * (D.b’)

$\phi$ leaves every proper closed $G$-submodule $W_{\lambda}$ of
$H^{\pm}_{\infty}$, listed in Remark 2, invariant.

Then, $\mathcal{A}$ satisfies the conditions of Thm. 3, this means that
$\mathcal{A}=PW_{r}(G).$

###### Proof.

Note first, that the space $\mathcal{A}$ is $K\times K$ invariant and linear
closed, due the intertwining conditions (2.8) and $(D.b^{\prime})$.
We have that $(D.b)$ of Thm. 3 gives a condition for each
$\sigma\in\widehat{M}=\\{\pm\\}$. Let us first consider
$\sigma=\\{+\\}\in\widehat{M}.$ By Example 4, we have $|+|=0$ and
$|\pi|=k+1>0$. Now let $\phi\in\mathcal{A}$ satisfying the assumption
$(D.b)\leavevmode\nobreak\ (ii)$, i.e., in particular

$\phi^{(0)}\Big{(}+,\frac{k}{2}\Big{)}\Big{|}_{D_{-k}\oplus
D_{k}}=0,\;\;\;k\in 2\mathbb{Z}+1.$ (4.4)

Let us check that:

  * (a)

for $\text{Re}(\lambda)>0$, the intertwining operator $J_{-,+,\lambda}$ has
zeros of order at most one.

  * (b)

the kernel of $J_{w,+,\lambda}$ is equal to $0$ or $D_{-k}\oplus D_{k}$ for
$\text{Re}(\lambda)>0$.

Consider the $K$-type $n\in 2\mathbb{Z}$ and the Harish-Chandra
$\mathbf{c}$-function $\mathbf{c}_{n}$ as in (4.1). If $n=0$, then
$\mathbf{c}_{0}(\lambda)=\frac{1}{\sqrt{\pi}}\frac{\Gamma(\lambda)}{\Gamma(\lambda+\frac{1}{2})}$
and we see that $\mathbf{c}_{0}(\lambda)$, for $\text{Re}(\lambda)>0$, has no
zeros and no poles. Thus, we can consider the quotient

$\frac{\mathbf{c}_{n}(\lambda)}{\mathbf{c}_{0}(\lambda)}=\frac{\Gamma(\lambda+\frac{1}{2})^{2}}{\Gamma(\lambda+\frac{1+n}{2})\Gamma(\lambda+\frac{1-n}{2})}=\frac{(\lambda-\frac{|n|-1}{2})\cdots(\lambda-\frac{1}{2})}{(\lambda+\frac{|n|-1}{2})\cdots(\lambda+\frac{1}{2})}.$

It has zeros $\lambda\in\\{\frac{1}{2},\cdots,\frac{|n|-1}{2}\\}$ of first
order. Due to (2.9), we know that the intertwining operator $J_{w,+,\lambda}$
is in relation with the $\mathbf{c}$-function. If on all $K$-types, we have
zeros of first order, then $J_{w,+,\lambda}$ should also have zeros of first
order. Hence $J_{w,+,\lambda}$ has zeros of at most order one, this proves the
first assertation (a) of the claim.

Concerning (b), we need to check for which $K$-type $n$, the quotient
$\frac{\mathbf{c}_{n}(\lambda)}{\mathbf{c}_{0}(\lambda)}$ has a zero, for
fixed $\text{Re}(\lambda)>0$. It is clear that, if $\lambda\notin I_{+}$, then
$\frac{\mathbf{c}_{n}(\lambda)}{\mathbf{c}_{0}(\lambda)}$ has no zeros, i.e.
that $\text{Ker}(J_{w,+,\lambda})=0$. For fixed $\lambda=\frac{k}{2},k\in
2\mathbb{Z}+1$, the $\mathbf{c}$-quotient
$\frac{\mathbf{c}_{n}(\lambda)}{\mathbf{c}_{0}(\lambda)}$ has zeros if, and
only if, $n$ is a $K$-type of $D_{-k}$ and $D_{k}$, i.e.
$\text{Ker}(J_{w,+,\lambda})=D_{-k}\oplus D_{k}$. Thus, this implies (b).

By (b) and (4.4), we have that the operator $\phi^{(0)}(+,\lambda)$
annihiliates $\text{Ker}(J_{w,+,\lambda})$ for $\sigma=\\{+\\}\in\widehat{M}$
and $\text{Re}(\lambda)>0$. By (a), we have that the order $m$ is equal to the
one, thus this condition is sufficient.
By arguing in a similar way as above for $\sigma=\\{-\\}\in\widehat{M}$ with
$|-|=1$, and $\pi=D_{k}\in\widehat{G}_{d},k\in 2\mathbb{Z}\backslash\\{0\\}$,
with $|\pi|=|k|+1$, we can conclude that $\mathcal{A}$ satisfies the condition
$(D.b)$ of Thm. 3. ∎

Now let us move to (Level 2).

######  Theorem 6 (Intertwining conditions in (Level 2)).

Let $m\in\mathbb{Z}$ be a $K$-type, Then,
$\psi\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},H^{m|_{M}}_{\infty})$
satisfies the intertwining condition (D.2) of Def. 5 if, and only if,

  * (2.a)

$J_{w,m,\lambda}\psi(\lambda,\cdot)=\mathbf{c}_{m}(\lambda)\psi(-\lambda,\cdot),$
for all $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},$

  * (2.b)

$\psi(\lambda,\cdot)\in W_{\lambda}$, where $W_{\lambda}$ is the invariant
(colored in blue) $G$-submodule of $H^{\pm,\lambda}_{\infty}$ represented by
the boxes-pictures in Fig. 1. Here the choice of $\pm$ depends on the parity
of $m$.

Notice that if $W_{\lambda}$ is the whole colored blue box, then there are no
intertwining condition.

  * •

for $m=0$:

...${-\frac{3}{2}}$${-\frac{1}{2}}$${\frac{1}{2}}$${\frac{3}{2}}$${\frac{5}{2}}$...$\lambda$

  * •

for $m\in 2\mathbb{Z}$:

  * $\cdot$

$m>0$:

...${-\frac{m+3}{2}}$${-\frac{m+1}{2}}$${-\frac{m-1}{2}}$...${-\frac{1}{2}}$${\frac{1}{2}}$...${\frac{m-1}{2}}$${\frac{m+1}{2}}$${\frac{m+3}{2}}$...$\lambda$

  * $\cdot$

$m<0$:

...${-\frac{m+3}{2}}$${-\frac{m+1}{2}}$${-\frac{m-1}{2}}$...${-\frac{1}{2}}$${\frac{1}{2}}$...${\frac{m-1}{2}}$${\frac{m+1}{2}}$${\frac{m+3}{2}}$...$\lambda$

  * •

for $m\in 2\mathbb{Z}+1$:

  * $\cdot$

$m>0$:

...${-\frac{m+3}{2}}$${-\frac{m+1}{2}}$${-\frac{m-1}{2}}$...$-1$$0$$1$...${\frac{m-1}{2}}$${\frac{m+1}{2}}$${\frac{m+3}{2}}$...$\lambda$

  * $\cdot$

$m<0$:

...${-\frac{m+3}{2}}$${-\frac{m+1}{2}}$${-\frac{m-1}{2}}$...$-1$$0$$1$...${\frac{m-1}{2}}$${\frac{m+1}{2}}$${\frac{m+3}{2}}$...$\lambda$

Figure 1: Boxes-pictures for $G=\mathbf{SL}(2,\mathbb{R})$.

###### Proof.

We need to show that the conditions $(2.a)$ and $(2.b)$ correspond to the
condition $(D.2)$ in Def. 5. In fact, by Prop. 1(a), we have that $(2.a)$ is a
special case of $(D.2)$.

Concerning $(2.b)$, condition $(D.2)$ says that for each $W$ we have an
intertwining condition corresponding to $(D.b^{\prime})$ in Thm. 5. Now we
need to extract in which of these $W_{\lambda}$, there is an intertwining
condition. If the $K$-type $m$ is in a closed $G$-submodule $W_{\lambda}$ of
$H^{\pm,\lambda}_{\infty}$, then $D^{m}_{W}$ is one-dimensional. Hence, by
$(D.2)$, $\psi$ has values in this $G$-submodule $W_{\lambda}$. By
$(D.b^{\prime})$ in Thm. 5, we thus take the smallest closed proper invariant
$G$-submodule of them. Otherwise, if the $K$-type is not in a closed
$G$-submodule $W_{\lambda}$ of $H^{\pm,\lambda}_{\infty}$, then
$D^{m}_{W_{\lambda}}=\\{0\\}$ and thus there are no intertwining conditions.
Consequently, we obtain the boxes-pictures in Fig. 1. ∎

The final step will be to move in (Level 3). Note that
$\text{Hom}_{M}(E_{n},E_{m})=\\{0\\}$ if $n\equiv m$ (mod 2).

######  Definition 8.

Let $n\equiv m$ (mod 2) $K$-types. We define the polynomial $q_{n,m}$ in
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ with values in
$\emph{\text{Hom}}_{M}(E_{n},E_{m})\cong\mathbb{C}$ by

$q_{n,m}(\lambda):=\begin{cases}1,&\text{ if }n=m,\\\
(\lambda+\frac{|m|+1}{2})(\lambda+\frac{|m|+3}{2})\cdots(\lambda+\frac{|n|-1}{2}),&\text{
if }|n|>|m|\text{ and same signs},\\\
(\lambda-\frac{|n|+1}{2})(\lambda-\frac{|n|+3}{2})\cdots(\lambda-\frac{|m|-1}{2}),&\text{
if }|n|<|m|\text{ and same signs},\\\
(\lambda+\frac{|n|-1}{2})(\lambda+\frac{|n|-3}{2})\cdots(\lambda-\frac{|m|-1}{2}),&\text{
else, with different signs.}\end{cases}$ (4.5)

######  Theorem 7 (Intertwining conditions in (Level 3)).

Let $n\equiv m$ (mod 2) be two $K$-types.
Then,
$\varphi\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{Hom}}_{M}(E_{n},E_{m}))$
satisfies the intertwining condition (D.3) of Def. 5 if, and only if, there
exists an even holomorphic function $h\in\emph{\text{Hol}}(\lambda^{2})$ such
that

$\varphi(\lambda)=h(\lambda)\cdot
q_{n,m}(\lambda),\;\;\;\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},$ (4.6)

where $q_{n,m}$ is the polynomial (4.5).

###### Proof.

By Thm. 6, it is sufficient to prove that the conditions $(2.a)$ and $(2.b)$
correspond to (4.6). In particular, we want to show that
$(\lambda,k_{\theta})\mapsto\varphi(\lambda)e^{in\theta}$ satisfies $(2.b)$
if, and only if, $\varphi$ has zeros at the zeros of the polynomial $q_{n,m}$.
From Thm. 6 $(2.b)$, we know that the invariant $G$-submodule $W_{\lambda}$
are represented by the boxes-pictures in Fig. 1. Thus, we need to check, where
the $K$-type $n$ is not in the colored blue invariant $G$-submodule
$W_{\lambda}$. We leave it to the reader to check that this happens exactly at
the zeros of $q_{n,m}$. Thus, we can deduce that $\varphi$ is of the form
(4.6) with $h$ an arbitrary holomorphic function.

Concerning the correspondence between the conditions $(2.a)$ and (4.6), by
Prop. 1(b), we observe that $(2.a)$ corresponds to $(D.3)$:

$(-1)^{(m-n)/2}\frac{\mathbf{c}_{n}(\lambda)}{\mathbf{c}_{m}(\lambda)}\varphi(\lambda)=\varphi(-\lambda),\;\;\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},n,m\in\mathbb{Z}.$
(4.7)

By using Def. 8, we observe that
$\frac{q_{n,m}(-\lambda)}{q_{n,m}(\lambda)}=(-1)^{(m-n)/2}\frac{\mathbf{c}_{n}(\lambda)}{\mathbf{c}_{m}(\lambda)},$
for $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},n,m\in\mathbb{Z}.$ Hence, we
obtain

$\frac{\varphi(\lambda)}{q_{n,m}(\lambda)}=\frac{\varphi(-\lambda)}{q_{n,m}(-\lambda)}.$

This means, (4.7) is satisfied if, and only if, $h(\lambda)=h(-\lambda)$, for
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$. ∎

We have completely determined the Paley-Wiener-(Schwartz) spaces for
$G=\mathbf{SL}(2,\mathbb{R})$ in (Level 2) and (Level 3).

### The case $G=\mathbf{SL}(2,\mathbb{R})\times\mathbf{SL}(2,\mathbb{R})$

Let

$G:=G^{\prime}\times
G^{\prime}=\mathbf{SL}(2,\mathbb{R})\times\mathbf{SL}(2,\mathbb{R})=\\{g=(g_{1},g_{2})\;|\;g_{1},g_{2}\in
G^{\prime}\\}$

be the Cartesian product of two copies of semi-simple non-compact connected
real Lie group $G^{\prime}=\mathbf{SL}(2,\mathbb{R})$. Since
$K^{\prime}=\mathbf{SO}(2)$ is the maximal compact subgroup of $G^{\prime}$,
$K:=K^{\prime}\times K^{\prime}$ is maximal compact in $G$.

The irreducible representations of $K$ are given by pairs
$(n_{1},n_{2}),n_{i}\in\widehat{K}^{\prime}\cong\mathbb{Z}$ ([Wal73], Sect.
2.36). More precisely, we denote by a tuple of integers
$n:=(n_{1},n_{2})\in\mathbb{Z}\times\mathbb{Z}\cong\mathbb{Z}^{2}\cong\widehat{K}$
the $K$ representions on the vector space $E_{n}:=E_{n_{1}}\otimes
E_{n_{2}}\in\mathbb{C}$ with action
$[n_{1},n_{2}](k_{1},k_{2})=n_{1}(k_{1})\otimes n_{2}(k_{2})$. The associated
homogeneous line bundle over $X:=X^{\prime}\times X^{\prime}$ is denoted by
$\mathbb{E}_{n}$. For $l,n\in\widehat{K}$, we observe that
$\text{Hom}_{M}(E_{l},E_{n})=\\{0\\}$, if $l_{1}\not\equiv n_{1}$ (mod 2) or
$l_{2}\not\equiv n_{2}$ (mod 2). Note that
$\mathfrak{a}^{*}_{\mathbb{C}}\cong\mathbb{C}\times\mathbb{C}$. By using Def.
8, we define for $l,n\in\mathbb{Z}^{2}$, $l_{1}\equiv n_{1}$ (mod 2),
$l_{2}\equiv n_{2}$ (mod 2), the polynomial $q_{l,n}$ given by

$q_{l,n}(\lambda_{1},\lambda_{2}):=q_{l_{1},n_{1}}(\lambda_{1})\cdot
q_{l_{2},n_{2}}(\lambda_{2}),\;\;\;\;(\lambda_{1},\lambda_{2})\in\mathbb{C}\times\mathbb{C}\cong\mathfrak{a}^{*}_{\mathbb{C}},$
(4.8)

where $q_{l_{i},n_{i}}(\lambda_{i})$ is the ’intertwining’ polynomial (4.5)
for $\lambda_{i}\in\mathbb{C},i=1,2$.

######  Theorem 8 (Intertwining condition in (Level 3)).

Let $l,n\in\mathbb{Z}^{2}$ be two tuples of integers. Then,
$\varphi\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{Hom}}_{M}(E_{l},E_{n}))$
satisfies the intertwining condition (D.3) of Def. 5 if, and only if, there
exists an even holomorphic function
$h\in\emph{\text{Hol}}(\lambda_{1}^{2},\lambda_{2}^{2})$:
$h(\lambda_{1},\lambda_{2})=h(-\lambda_{1},\lambda_{2})=h(\lambda_{1},-\lambda_{2})$
such that

$\varphi(\lambda_{1},\lambda_{2}):=h(\lambda_{1},\lambda_{2})\cdot
q_{l,n}(\lambda_{1},\lambda_{2}),$ (4.9)

where $q_{l,n}$ is the polynomial (4.8) in
$(\lambda_{1},\lambda_{2})\in\mathfrak{a}^{*}_{\mathbb{C}}$.

To prove Thm. 8, we need first a density argument, which permits us to
approximate the even holomorphic function $h$ in (4.9) by even polynomials.

######  Lemma 1.

Consider the subset $P$ of polynomial functions in
$A:=\\{h\in\emph{\text{Hol}}(\mathbb{C}^{2})\;|\;h(\lambda_{1},\lambda_{2})=h(-\lambda_{1},\lambda_{2})=h(\lambda_{1},-\lambda_{2}),\forall(\lambda_{1},\lambda_{2})\in\mathbb{C}^{2}\\}.$
Then, $P\subset A$ is dense with respect to uniform convergence on compact
subset of $\mathbb{C}^{2}$.

###### Proof.

Let $h(\lambda_{1},\lambda_{2})\in A$ be a holomorphic function. Consider the
Taylor series at the point $0=(0,0)$ in two variables
$(\lambda_{1},\lambda_{2})\in\mathfrak{a}^{*}_{\mathbb{C}}$:

$\sum_{\alpha}a_{\alpha}\lambda^{\alpha}=\sum_{\alpha_{1},\alpha_{2}}a_{\alpha_{1},\alpha_{2}}\lambda_{1}^{\alpha_{1}}\lambda_{2}^{\alpha_{2}},$

where $a_{\alpha}$ are constants and the sum runs over multi-indices
$\alpha=(\alpha_{1},\alpha_{2}),\alpha_{j}\in\mathbb{N}_{0}$. Let
$|\alpha|=\sum_{j=1}^{2}\alpha_{j}$ be the length. Note that
$a_{\alpha_{1},\alpha_{2}}=0$ if $\alpha_{1}$ or $\alpha_{2}$ is odd. Thus the
corresponding Taylor polynomials $\sum_{|\alpha|\leq
k}a_{\alpha}\lambda^{\alpha}$ belong to $P$. The Taylor polynomials converge
uniformly for $k$ going to infinity on each ball $\overline{B}_{r}(0)$. ∎

Next, by using the Iwasawa decomposition of $g=(g_{1},g_{2})\in G$, the
’exponential’ function $e^{n}_{\lambda,k}$ can be rewritten as follows.

######  Proposition 2.

For fixed $\lambda=(\lambda_{1},\lambda_{2})\in\mathfrak{a}^{*}_{\mathbb{C}}$
and $k=(k_{1},k_{2})\in K$, the function $e^{n}_{\lambda,k}\in C^{\infty}(G)$
defined as in Def. 3, is a product of the corresponding functions on
$G^{\prime}$:

$e^{n}_{\lambda,k}(g_{1},g_{2})=e^{n_{1}}_{\lambda_{1},k_{1}}(g_{1})\cdot
e^{n_{2}}_{\lambda_{2},k_{2}}(g_{2}),\;\;\;\;(g_{1},g_{2})\in G.$

###### Proof.

Consider the Iwasawa decomposition of

$g=(g_{1},g_{2})=(n^{\prime}_{1}a_{1}k^{\prime}_{1},n^{\prime}_{2}a_{2}k^{\prime}_{2})=n^{\prime}ak^{\prime}\in
G$

so that $n^{\prime}=(n^{\prime}_{1},n^{\prime}_{2})\in N,a=(a_{1},a_{2})\in A$
and $k^{\prime}=(k^{\prime}_{1},k^{\prime}_{2})\in K.$ One can easily deduce
that for $\lambda=(\lambda_{1},\lambda_{2})\in\mathfrak{a}^{*}_{\mathbb{C}}$,
we get

$e^{(\lambda+\rho)\log(a)}=e^{(\lambda_{1}+\rho)\log(a_{1})+(\lambda_{2}+\rho)\log(a_{2})}=a_{1}^{\lambda_{1}+\rho}\cdot
a_{2}^{\lambda_{2}+\rho}.$

Hence, for $g\in G$, we then have

$\displaystyle
e^{n}_{\lambda,k}(g)=e^{n}_{\lambda,k}(g_{1},g_{2})=a_{1}^{\lambda_{1}+\rho}a_{2}^{\lambda_{2}+\rho}e^{n}_{\lambda,1}(k_{1},k_{2})$
$\displaystyle=$ $\displaystyle
a_{1}^{\lambda_{1}+\rho}a_{2}^{\lambda_{2}+\rho}n_{1}(k_{1})n_{2}(k_{2})$
$\displaystyle=$ $\displaystyle
a_{1}^{\lambda_{1}+\rho}a_{2}^{\lambda_{2}+\rho}e^{n_{1}}_{\lambda_{1},1}(k_{1})e^{n_{2}}_{\lambda_{2},1}(k_{2})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:exptau})}}{{=}}$ $\displaystyle
e^{n_{1}}_{\lambda_{1},k_{1}}(g_{1})\cdot
e^{n_{2}}_{\lambda_{2},k_{2}}(g_{2}).\qed$

Let
$\prescript{}{l}{PWS}_{n,H}(\mathfrak{a}^{*}_{\mathbb{C}})=\\{\varphi\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{Hom}_{M}(E_{l},E_{n}))\;|\;\varphi$
satisfies $(D.3)\\}$ be the ’pre’-Paley-Wiener-Schwartz space. Note that
$\prescript{}{l}{PW}_{n}(\mathfrak{a}^{*}_{\mathbb{C}})\subset\prescript{}{l}{PWS}_{n}(\mathfrak{a}^{*}_{\mathbb{C}})\subset\prescript{}{l}{PWS}_{n,H}(\mathfrak{a}^{*}_{\mathbb{C}})$.

###### Proof of Thm. 8.

It suffices to show that

  * (a)

every function
$\varphi\in\prescript{}{l}{PW}_{n}(\mathfrak{a}^{*}_{\mathbb{C}})$ is of the
form (4.9) and

  * (b)

(inversely) if $\varphi$ is of the form (4.9) then it is in
$\prescript{}{l}{PWS}_{n,H}(\mathfrak{a}^{*}_{\mathbb{C}})$.

Let $\varphi\in\prescript{}{l}{PW}_{n}(\mathfrak{a}^{*}_{\mathbb{C}})$. By the
Paley-Wiener Thm. 2, there exists $f\in C^{\infty}_{c}(G,l,n)$ with
$\prescript{}{l}{\mathcal{F}}_{n}(f)=\varphi$. By Fubini’s theorem and Prop.
2, we have

$\displaystyle\prescript{}{l}{\mathcal{F}}_{n}(f)(\lambda)=\int_{G^{\prime}}e^{n_{1}}_{\lambda_{1},1}(g_{1})\Big{(}\int_{G^{\prime}}e^{n_{2}}_{\lambda_{2},1}(g_{2})f(g_{1},g_{2})\;dg_{2}\Big{)}\;dg_{1}$
$\displaystyle=$
$\displaystyle\int_{G^{\prime}}e^{n_{1}}_{\lambda_{1},1}(g_{1})\tilde{f}_{\lambda_{2}}(g_{1})\;dg_{1}$
(4.10) $\displaystyle=$
$\displaystyle\prescript{}{l_{1}}{\mathcal{F}}_{n_{1}}\tilde{f}_{\lambda_{2}}(\lambda_{1}),$

where we set
$\tilde{f}_{\lambda_{2}}(g_{1}):=\int_{G^{\prime}}e^{n_{2}}_{\lambda_{2},1}(g_{2})f(g_{1},g_{2})\;d{g_{2}}.$
Note that $\tilde{f}_{\lambda_{2}}\in
C^{\infty}_{c}(X^{\prime},\mathbb{E}_{n_{1}}).$ Similarly, if we fix
$\lambda_{1}\in\mathfrak{a}^{*}_{\mathbb{C}}$, we have
$\tilde{f}_{\lambda_{1}}(g_{2}):=\int_{G^{\prime}}e^{n_{1}}_{\lambda_{1},k_{1}}(g_{1})f(g_{1},g_{2})\;d{g_{1}}\in
C^{\infty}_{c}(X^{\prime},\mathbb{E}_{n_{2}})$ and
$\prescript{}{l}{\mathcal{F}}_{n}(f)(\lambda)=\prescript{}{l_{2}}{\mathcal{F}}_{n_{2}}\tilde{f}_{\lambda_{1}}(\lambda_{2})$.
Thus, by Thm. 7, the Fourier transform has the form

$\prescript{}{l}{\mathcal{F}}_{n}(f)(\lambda)=h_{\lambda_{2}}(\lambda_{1})\cdot
q_{l_{1},n_{1}}=h_{\lambda_{1}}(\lambda_{2})\cdot
q_{l_{2},n_{2}}(\lambda_{2}),$ (4.11)

where $h_{\lambda_{i}}$ is an even holomorphic function in
$\lambda_{i}\in\mathfrak{a}^{*}_{\mathbb{C}}$ and $q_{l_{i},n_{i}}$ is the
’intertwining’ polynomial in $\lambda_{i}\in\mathfrak{a}^{*}_{\mathbb{C}}$,
defined in (4.5). By (4.8), we deduce that there exists
$h\in\text{Hol}(\lambda_{1}^{2},\lambda_{2}^{2})$ such that

$\varphi(\lambda)=\prescript{}{l}{\mathcal{F}}_{n}f(\lambda)=h(\lambda_{1},\lambda_{2})\cdot
q_{l,n}(\lambda_{1},\lambda_{2}),\;\;\;\lambda=(\lambda_{1},\lambda_{2})\in\mathfrak{a}^{*}_{\mathbb{C}},$

as desired.

Concerning (b), let $\varphi$ of the form (4.9), by (4.8), we have that

$\varphi(\lambda_{1},\lambda_{2})=h(\lambda_{1},\lambda_{2})q_{l_{1},n_{1}}(\lambda_{1})q_{l_{2},n_{2}}(\lambda_{2}),$

for $(\lambda_{1},\lambda_{2})\in\mathfrak{a}^{*}_{\mathbb{C}}$. By Lem. 1, we
can approximate $h$ by the sum of products of two monomials
$\lambda_{1}^{\alpha_{1}}$ and $\lambda_{2}^{\alpha_{2}}$. By Thm. 7 and the
Paley-Wiener-Schwartz Thm. 2:

$\lambda_{i}^{\alpha_{i}}q_{l_{i},n_{i}}(\lambda_{i})=\prescript{}{l_{i}}{\mathcal{F}}_{n_{i}}(f_{i})(\lambda_{i})\in\prescript{}{l_{i}}{PWS}_{n_{i}}(\mathfrak{a}^{*}_{\mathbb{C}})$

is the image of the Fourier transform of a distribution $f_{i}\in
C^{-\infty}_{c}(G^{\prime},l_{i},n_{i}),i=1,2$. Consider now the tensor
product of these two distributions: $f_{1}\otimes f_{2}\in
C^{-\infty}_{c}(G,l,n).$ By taking the Fourier transform and using the
computations involving Fubini’s theorem, we obtain that

$\prescript{}{l}{\mathcal{F}}_{n}(f_{1}\otimes
f_{2})(\lambda)=\prescript{}{l_{1}}{\mathcal{F}}_{n_{1}}(f_{1})(\lambda_{1})\cdot\prescript{}{l_{2}}{\mathcal{F}}_{n_{2}}(f_{2})(\lambda_{2})=\lambda^{\alpha}q_{l,n}(\lambda).$

By using Paley-Wiener-Schwartz Thm. 2, we have
$\prescript{}{l}{\mathcal{F}}_{n}(f_{1}\otimes
f_{2})=\lambda^{\alpha}q_{l,n}\in\prescript{}{l}{PWS}_{n}(\mathfrak{a}^{*}_{\mathbb{C}})\subset\prescript{}{l}{PWS}_{n,H}(\mathfrak{a}^{*}_{\mathbb{C}})$.
Since
$\prescript{}{l}{PWS}_{n,H}(\mathfrak{a}^{*}_{\mathbb{C}})\subset\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}})$
is closed with respect to uniform convergence on compact subsets, we conclude
that $\varphi=h\cdot
q_{l,n}\in\prescript{}{l}{PWS}_{n,H}(\mathfrak{a}^{*}_{\mathbb{C}}).$ ∎

By Thm. 8, we have explicitly determined the Paley-Wiener(-Schwartz) spaces
for $G=\mathbf{SL}(2,\mathbb{R})\times\mathbf{SL}(2,\mathbb{R})$ in (Level 3).
Moreover, all the previous results can be generalized to
$G=\mathbf{SL}(2,\mathbb{R})^{d},d\geq 2$.

##  5 The case $G=\mathbf{SL}(2,\mathbb{C})$

Let
$G=\mathbf{SL}(2,\mathbb{C})=\\{g\in\mathbf{GL}(2,\mathbb{C})\;|\;\det(g)=1\\}$
be the special linear group of $\mathbb{C}^{2}$ with maximal compact subgroup

$K=\mathbf{SU}(2)=\Big{\\{}\begin{pmatrix}\alpha&\beta\\\
-\overline{\beta}&\overline{\alpha}\end{pmatrix}\in\mathbf{GL}(2,\mathbb{C})\;\Big{|}\;|\alpha|^{2}+|\beta|^{2}=1\Big{\\}}.$

Note that $K$ is homeomorphic to the 3-sphere, therefore simply connected.
Furthermore, the one-dimensional abelian subgroup $A$ of $G$
($A\subset\mathbf{SL}(2,\mathbb{R})\subset G$) is the same as in Sect. 4 and
the nilpotent subgroup $N$ of $G$ is given by
$N=\Big{\\{}\begin{pmatrix}1&t\\\
0&1\end{pmatrix}\;\Big{|}\;t\in\mathbb{C}\Big{\\}}.$ We identify
$\mathfrak{a}^{*}_{\mathbb{C}}\cong\mathbb{C}$ by sending $\lambda$ to
$\lambda(H)$, where $H=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\in\mathfrak{a}$. Note that $\rho(H)=2.$ For the irreducible
complex representations of $K$, we have (e.g. [Wal88], Sect. 5.7)

$\widehat{K}=\\{\delta_{n}\;|\;n\in\mathbb{N}_{0}\\}\cong\mathbb{N}_{0}\text{
with }d_{\delta_{n}}:=\dim(\delta_{n})=n+1.$

The tensor product of two irreducible $K$-representations decomposes into
irreducibles according to the classical Clebsch-Gordan rule (e.g. [Wal88],
5.7.1 (1)):

$\delta_{n}\otimes\delta_{m}=\bigoplus_{0\leq
j\leq\min(n,m)}\delta_{n+m-2j},\;\;\;\,n,m\in\mathbb{N}_{0}.$ (5.1)

In addition, $M=Z_{K}(A)=\Big{\\{}m_{\theta}:=\begin{pmatrix}e^{i\theta}&0\\\
0&e^{-i\theta}\end{pmatrix}\;\Big{|}\;\theta\in\mathbb{R}\Big{\\}},$ which is
abelian and a maximal torus in $K$. We parametrize
$\widehat{M}:=\\{\sigma_{l}\;|\;l\in\mathbb{Z}\\}\cong\mathbb{Z}$ by the
integers with $\sigma_{l}(m_{\theta})=e^{il\theta}\in\mathbb{Z}$. Moreover,
let $\chi:M\rightarrow\mathbb{C}$ denotes the character of a finite-
dimensional irreducible representation $(\delta_{n},E_{n})$ of $K$. Then, the
Weyl character formula for $m_{\theta}\in M$ (e.g. [Kna02], Chap. V.6)

$\chi(m_{\theta})=\text{Tr}(\delta_{n}(m_{\theta}))=\frac{e^{i(n+1)\theta}-e^{-i(n+1)\theta}}{e^{i\theta}-e^{-i\theta}}=\frac{\sin((n+1)\theta)}{\sin(\theta)}=e^{-in\theta}+e^{-i(n-2)\theta}+\dots+e^{in\theta}$

tells us that the weights of $(\delta_{n},E_{n})$ have the form
$-n,-(n-2),\dots,n-2,n$, each with multiplicity one, with $n\in\mathbb{N}_{0}$
a highest weight. The following important result clarifies the reducibility of
the principal series representations of $\mathbf{SL}(2,\mathbb{C})$. We refer
for example to Wallach’s book ([Wal88], Sect. 5.7) for a proof. Note that
Wallach’s proof is also valid for $G$-representation on smooth vectors (see
remark before Thm. 4).

######  Theorem 9 (Structure of principal series representations of
$\mathbf{SL}(2,\mathbb{C})$).

The principal series representations $H^{\sigma,\lambda}_{\infty}$ of
$\mathbf{SL}(2,\mathbb{C})$ is exactly reducible, if $\lambda$ is real and

$|\lambda|>|\sigma|\;\;\;\&\;\;\;|\lambda|-|\sigma|\text{ even integer}.$

In this case, for $\lambda>0$, there is an unique irreducible
subrepresentation $R^{\sigma,\lambda}$ of each $H^{\sigma,\lambda}_{\infty}$.
Then, we have

$H^{-\sigma,-\lambda}_{\infty}=$$F_{m,n}$$R^{\sigma,\lambda}$$H^{\sigma,\lambda}_{\infty}=$$R^{\sigma,\lambda}$$F_{m,n}$

where $m=\frac{\sigma+\lambda}{2}-1,n=\frac{\lambda-\sigma}{2}-1$ and
$F_{m,n}$ is an irreducible finite-dimensional $G$-representation that is
isomorphic to $\delta_{m}\otimes\delta_{n}$ as a $K$-representation. Moreover,
there is an intertwining opertator

$L_{\sigma,\lambda}:H^{-\lambda,-\sigma}_{\infty}\longrightarrow
H^{\sigma,\lambda}_{\infty}$

so that
$\emph{\text{Ker}}(J_{w,\sigma,\lambda})=\emph{\text{Im}}(L_{\sigma,\lambda})=R^{\sigma,\lambda}$.
In particular $R^{\sigma,\lambda}$ is isomorphic to
$H^{-\lambda,-\sigma}_{\infty}$. Here,
$J_{w,\sigma,\lambda}:H^{\sigma,\lambda}\longrightarrow H^{-\sigma,-\lambda}$
denotes the Knapp-Stein intertwining operator defined in Def. 6 with $w=-1$
and $m_{w}=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}.$

Furthermore, from the intertwining operator $L_{\sigma,\lambda}$, we can
deduce the existence of further intertwining operators:

$H^{\lambda,\sigma}_{\infty}$$H^{-\sigma,-\lambda}_{\infty}$$H^{\sigma,\lambda}_{\infty}$$H^{-\lambda,-\sigma}_{\infty}$$\tilde{L}_{\sigma,\lambda}$$L_{\sigma,\lambda}$$L^{\prime}_{\sigma,\lambda}$$\tilde{L}^{\prime}_{\sigma,\lambda}$$J_{-,\sigma,\lambda}$$J_{-,\sigma,\lambda}$

Note that the operators
$L_{\sigma,\lambda},\tilde{L}_{\sigma,\lambda},L^{\prime}_{\sigma,\lambda}$
and $\tilde{L}^{\prime}_{\sigma,\lambda}$ are precisely the Želobenko
operators (also called BGG-operators) for $G=\mathbf{SL}(2,\mathbb{C})$
([Zelo76], [BGG71], [BGG75] & [BGG76]). In Fig. 2, we illustrate the principal
series representations (with regular integral infinitesimal character) in a
grid, where the horizontal axis represents the values of
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ and the vertical one the values of
$\sigma\in\widehat{M}$. Note that inside the region
$\\{\pm|\sigma|>\lambda\\}$, we have the irreducible principal series
representations $H_{\infty}^{-\lambda,-\sigma}$ respectively
$H_{\infty}^{\lambda,\sigma}$ colored in gray and outside the reducible ones,
colored in black.

$\sigma$$\lambda$$\sigma=\lambda$$\sigma=-\lambda$$H^{-2,0}_{\infty}$$H^{-4,0}_{\infty}$$H^{-3,-1}_{\infty}$$H^{-3,1}_{\infty}$$H^{-4,-2}_{\infty}$$H^{-4,2}_{\infty}$$H^{2,0}_{\infty}$$H^{4,0}_{\infty}$$H^{3,1}_{\infty}$$H^{-3,1}_{\infty}$$H^{4,2}_{\infty}$$H^{4,-2}_{\infty}$$H^{0,-2}_{\infty}$$H^{0,2}_{\infty}$$H^{0,-4}_{\infty}$$H^{0,4}_{\infty}$$H^{1,-3}_{\infty}$$H^{-1,3}_{\infty}$$H^{-1,-3}_{\infty}$$H^{1,3}_{\infty}$$H^{2,4}_{\infty}$$H^{-2,-4}_{\infty}$$H^{-2,4}_{\infty}$$H^{2,-4}_{\infty}$
Figure 2: Principal series representations, where the colored one indacte the
intertwining relations that occures between each others with the same colors.

######  Example 5.

Fix $(\sigma,\lambda)\in\widehat{M}\times\mathfrak{a}^{*}_{\mathbb{C}}$ such
that $\lambda-|\sigma|\in 2\mathbb{N}$, we have the special condition

$L_{\sigma,\lambda}\circ\phi(-\lambda,-\sigma)=\phi(\sigma,\lambda)\circ
L_{\sigma,\lambda}.$ (5.2)

######  Theorem 10 (Intertwining condition in (Level 1)).

For $r>0$, let $\mathcal{A}$ be the space of all
$\phi\in\prod_{\sigma\in\widehat{M}}\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{End}}(H^{\sigma}_{\infty}))$
such that $\phi$ statisfies the corresponding growth condition as well as the
two intertwining conditions (2.8) and (5.2). Then, $\mathcal{A}$ satisfies the
conditions of Thm. 3, this means that $\mathcal{A}=PW_{r}(G).$

Before we proceed with the proof of Thm. 10, let us first state the explicit
expression of the Harish-Chandra $\mathbf{c}$-function ([Co74], App. 2) for
$G=\mathbf{SL}(2,\mathbb{C})$, which is given by the following formula, for
$|\sigma|\leq n,\sigma\equiv n$ (mod 2):

$\mathbf{c}_{n,\sigma}(\lambda):=\delta_{n}(\sigma_{\lambda})=\frac{\Gamma(\frac{1}{2}(\lambda+\sigma))\Gamma(\frac{1}{2}(\lambda-\sigma))}{\Gamma(\frac{1}{2}(\lambda+n+2))\Gamma(\frac{1}{2}(\lambda-n))},\;\;\;\lambda\in\mathfrak{a}_{\mathbb{C}}^{*}.$

Consider an additional, not necessary distinct, $K$-type $m$ and fix
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$. Then, by using repeatly the
relation (4.2), we obtain for $n\equiv m$ (mod 2), the following quotient:

$\frac{\mathbf{c}_{n,\sigma}(\lambda)}{\mathbf{c}_{m,\sigma}(\lambda)}=\frac{\Gamma(\frac{\lambda}{2}+\frac{m}{2}+1)\Gamma(\frac{\lambda}{2}-\frac{m}{2})}{\Gamma(\frac{\lambda}{2}+\frac{n}{2}+1)\Gamma(\frac{\lambda}{2}-\frac{n}{2})}=\begin{cases}1,&\;\text{if
}n=m\\\
\frac{(\lambda+m)(\lambda+m-2)(\lambda+m-4)\cdots(\lambda+n+2)}{(\lambda-m)(\lambda-(m+2))(\lambda-(m+4))\cdots(\lambda-(n+2))},&\;\text{if
}n<m\\\
\frac{(\lambda-n)(\lambda-(n+2))(\lambda-(n+4))\cdots(\lambda-(m+2))}{(\lambda+n)(\lambda+n-2)(\lambda+n-4)\cdots(\lambda+m+2)},&\;\text{if
}n>m.\end{cases}$ (5.3)

Hence, we can directly see that the quotient has zeros in
$\\{-m,-m+2,\dots,-n-2\\}$ and poles in $\\{m,m+2,\dots,n+2\\}$ for $n<m$ and
inversely for $n>m$.

###### Proof of Thm. 10.

Note that $\mathcal{A}$ is a $K\times K$ invariant closed linear subspace, due
the intertwining conditions. We proceed similar as in the proof of Thm. 5.
Note that $\widehat{G}_{d}=\emptyset$. Consider $\phi\in\mathcal{A}$ such that
for each $\sigma\in\widehat{M}$, the assumption $(D.b)(i)$ of Thm. 3:

$\phi_{\sigma^{\prime}}=0\text{, for all }\sigma^{\prime}\in\widehat{M}\text{
with }|\sigma^{\prime}|>|\sigma|$ (5.4)

is satisfied. Analogous to the proof of Thm. 5, we need to check that:

  * (a)

for $\text{Re}(\lambda)>0$, the intertwining opertator $J_{w,\sigma,\lambda}$
has a zero of order at most one.

  * (b)

the kernel of $J_{w,\sigma,\lambda}$ is equal to 0 or $R^{\sigma,\lambda}$,
for $\text{Re}(\lambda)>0$.

The minimal $K$-type of the principal series representation
$H^{\sigma,\lambda}_{\infty}$ is $n=|\sigma|$, hence its Harish-Chandra
$\mathbf{c}$-function (5) $\mathbf{c}_{\sigma,\sigma}$ is regular for
$\text{Re}(\lambda)>0$. Let $n\in\hat{K}$, then we see that for
$n\geq|\sigma|$ and for $\text{Re}(\lambda)>0$:

$\frac{\mathbf{c}_{n,\sigma}(\lambda)}{\mathbf{c}_{\sigma,\sigma}(\lambda)}=\frac{(\lambda-n)(\lambda-(n+2))(\lambda-(n+4))\cdots(\lambda-(\sigma+2))}{(\lambda+n)(\lambda+n-2)(\lambda+n-4)\cdots(\lambda+\sigma+2)}$

is also regular and has no poles, but zeros
$\lambda\in\\{n,n+2,\dots,\sigma+2\\}$ of first order. Hence due (2.9),
$J_{w,\sigma,\lambda}$ has zeros of order one, this proves the first
assertation (a) of the claim.

By Thm. 9, we have
$\text{Ker}(J_{w,\sigma,\lambda})=\text{Im}(L_{\sigma,\lambda})=R^{\sigma,\lambda}$,
thus, this implies (b).

Now, by putting everyting together and using the intertwining condition (5.2)
as well as (5.4), we have, for each $\text{Re}(\lambda)>0$ with
$|\lambda|=|-\lambda|>|\sigma|$, that

$\phi_{\sigma}(\lambda)\circ
L_{\sigma,\lambda}=L_{\sigma,\lambda}\circ\underbrace{\phi_{-\lambda}(-\sigma)}_{=0}=0.$

Thus, by (b) and the assumption, we deduce that the operator
$\phi^{(0)}_{\sigma}(\lambda)$ annihiliates
$\text{Ker}(J_{w,\sigma,\lambda})=\text{Im}(L_{\sigma,\lambda})=R^{\sigma,\lambda}$,
for $|\lambda|>|\sigma|$ and $|\lambda|-|\sigma|$ even. Moreover, by (a), this
condition is sufficient since the order of $m$ is one. This completes the
proof. ∎

Now let us move to (Level 2) and state the corresponding intertwining
conditions there. In fact, this will determine explicitly the Paley-
Wiener(-Schwartz) spaces for $G=\mathbf{SL}(2,\mathbb{C})$ in (Level 2). In
order to distinguish between representation spaces of $K$ and $M$, we denote
$E_{\delta_{n}}$ by $E_{n}$. while the one-dimensional space $E_{\sigma_{l}}$
is denoted by $\mathbb{C}_{l}$.

######  Theorem 11 (Intertwining condition in (Level 2)).

Let $n\in\mathbb{N}_{0}$ be a $K$-type and $k,l\in\widehat{M}\cong\mathbb{Z}$
so that $n\geq|k|,|l|,l>|k|$ and $k\equiv l\equiv n$ (mod 2). Consider the
operator

$l^{n}_{k,l}:\emph{\text{Hom}}_{M}(E_{n},\mathbb{C}_{-l})\longrightarrow\emph{\text{Hom}}_{M}(E_{n},\mathbb{C}_{k})$

defined as in Example 1 corresponding to $L=L_{k,l}$ and $\tau=\delta_{n}$.
Then,
$\psi\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},H^{n|_{M}}_{\infty})$
satisfies the intertwining condition (D.2) of Def. 5 if, and only if,

  * (2.a)

$J_{w,n,\lambda}t\psi(\lambda)=\begin{pmatrix}c_{n,n}(\lambda)&0&\dots&0\\\
0&c_{n,n-2}(\lambda)&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&c_{n,-n}(\lambda)\end{pmatrix}\psi_{-k}(-\lambda)$, for all
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},$

  * (2.b)

$L_{k,l}(t\circ\psi(-k))=l^{n}_{k,l}(t)\circ\psi(l)$, for all
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ and
$t\in\emph{\text{Hom}}_{M}(E_{n},E_{-l})$.

###### Proof of Thm. 11.

By Prop. 1, we have that $(2.a)$ corresponds to (2.6), hence it corresponds to
the intertwining condition $(D.2)$ in Def. 5 Similar for $(2.b)$, which is, by
Example 1 (2.6), a special intertwining condition of $(D.2)$. Hence, we have
equivalence between the conditions $(2.a)$ & $(2.b)$ and $(D.2)$. ∎

In order to move to (Level 3), let us first consider the case where the two
$K$-type $(n,E_{n})$ and $(m,E_{m})$ are equal and then progress to the case
of distinct $K$-types.

### Initial case: The $K$-types $m$ and $n$ are equal

######  Definition 9.

Let $(m,E_{m})$ be a fixed $K$-type and $k=-m,-(m-2),\dots,m-2,m$.
We define $\prescript{}{m}{\mathcal{A}}_{m}$ the space of all elements in
$\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{End}}_{M}(E_{m}))$,
which are given by holomorphic functions
$\varphi_{k}:\mathfrak{a}_{\mathbb{C}}^{*}\rightarrow\mathbb{C},$ ordered to a
$(m+1)\times(m+1)$ matrix with respect to $M$-weight vectors as a basis of
$E_{m}$:

$\varphi:=\begin{pmatrix}\varphi_{m}(\lambda)&0&\cdots&0\\\
0&\varphi_{m-2}(\lambda)&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\varphi_{-m}(\lambda)\\\
\end{pmatrix}\in\emph{\text{Hol}}(\mathfrak{a}^{*}_{\mathbb{C}},\emph{\text{End}}_{M}(E_{m}))$

such that

$\displaystyle\varphi_{k}(\lambda)$ $\displaystyle=$
$\displaystyle\varphi_{-k}(-\lambda),\;\text{ for
}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ $\displaystyle\varphi_{k}(l)$
$\displaystyle=$ $\displaystyle\varphi_{l}(k),\;\text{ for }k\equiv l\equiv
m\text{ (mod }2)\;\&\;|k|,|l|\leq m.$ (5.5)

Note that
$\prescript{}{m}{\mathcal{A}}_{m}\subset\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}_{M}(E_{m}))$
is even an algebra. Let
$\prescript{}{m}{PWS}_{m,H}(\mathfrak{a}^{*}_{\mathbb{C}})$ be the pre-Paley-
Wiener-Schwartz space such that
$\varphi\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}_{M}(E_{m}))$
and satisifies the intertwining condition $(D.3)$ of Def. 5.

######  Theorem 12.

With the previous notations, we have that

$\prescript{}{m}{\mathcal{A}}_{m}\cong\prescript{}{m}{PWS}_{m,H}(\mathfrak{a}^{*}_{\mathbb{C}}).$

###### Proof.

We need to check that the intertwining conditions (9) of
$\prescript{}{m}{\mathcal{A}}_{m}$ correspond to the intertwining condition
$(D.3)$ of $\prescript{}{m}{PWS}_{m,H}(\mathfrak{a}^{*}_{\mathbb{C}})$. More
precisely, by Example 1, it suffices to show that the intertwining condition
(2.7) for $L=J_{w,\sigma,\lambda}$ and $L=L^{\prime}_{\sigma,\lambda}$
corresponds to (9).

We know from Prop. 1 with $m=\gamma=\tau\in\mathbb{N}_{0}$, that for each
intertwining datum $((\sigma,\lambda,0),(-\sigma,-\lambda,0);W)$ and
$t\in\text{Hom}_{M}(E_{m},E_{\sigma})$, we have

$\delta_{m}(m_{w})^{-1}\varphi(\lambda)\delta_{m}(m_{w})=\varphi(-\lambda),\;\;\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},\varphi\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}_{M}(E_{m})),$

Note that the complex hull of $\mathfrak{a}$ is the sum of $\mathfrak{a}$ and
$\mathfrak{m}$, where $\mathfrak{m}$ is in the maximal torus $M$. This means
that the Weyl group $W_{A}$ acts on $\mathfrak{a}$ as well as on
$\mathfrak{m}$ by $-1$. Thus, also on $i\mathfrak{m}^{*}$ by $-1.$ Let
$-k,-(k-2),\dots,k-2,k$ be the weights of the representation
$m\in\mathbb{N}_{0}$. Then, by the $\delta_{m}(m_{w})$-component is sent to
$\delta_{m}(-m_{w})$-component. The matrix
$\text{diag}(\varphi_{m}(\lambda),\dots,\varphi_{-m}(\lambda))$ is reversed by
conjugation by $\delta_{m}(m_{w})$, i.e., we get
$\text{diag}(\varphi_{-m}(-\lambda),\dots,\varphi_{m}(-\lambda))$. Hence,
$\varphi_{-k}(-\lambda)=\varphi_{k}(\lambda)$, for all $|k|\leq m$ and
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$.

Let $|l|,|k|\leq m$ and $k\equiv l\equiv m$ (mod 2). Let $l^{m}_{k,l}$ as in
Thm. 11. We know that for the intertwining datum $((k,l,0),(l,k,0);W)$ and
$t\in\text{Hom}_{M}(E_{m},\mathbb{C}_{-l})$, we have (2.7) for
$m=\gamma=\tau\in\mathbb{N}_{0}$ and $l>|k|$:

$l^{m}_{k,l}(t\circ\varphi(-k))=l^{m}_{k,l}(t)\circ\varphi(l).$

Normalized $t\in\text{Hom}_{M}(E_{m},\mathbb{C}_{-l})$ and
$t^{\prime}\in\text{Hom}_{M}(E_{m},\mathbb{C}_{k})$ such that
$\varphi_{-l}=t\circ\varphi$ and $\varphi_{k}=t^{\prime}\circ\varphi.$ We have
that $l^{m}_{k,l}(t)=c\cdot t^{\prime}$, where $c\in\mathbb{C}$. Note that,
$c\neq 0$. In fact, by Thm. 9, the intertwining operator $L_{k,l}$ on the
$K$-type $m$ is not zero, hence $l^{m}_{k,l}$ too. Consequently, we have

$\displaystyle
l^{m}_{k,l}(t\circ\varphi(-k))=l^{m}_{k,l}(t)\circ\varphi(l)\iff
c\cdot\varphi_{-l}(-k)=c\cdot\varphi_{k}(l)\iff\varphi_{-l}(-k)=\varphi_{k}(l).$
(5.6)

For $0\leq l<|k|$, we consider $L_{l,|k|}$ instead. Combined with
$\varphi_{-l}(-k)=\varphi_{l}(k)$, we obtain (5.6) for every pair $k,l$ as
above. This completes the proof. ∎

We also consider the corresponding situation in polynomial functions:

$\prescript{}{m}{\text{Pol}}_{m}:=\\{\varphi\in\text{Pol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}_{M}(E_{m}))\;|\;\varphi_{k}\text{
satisfies }(\ref{eq:propSL2C})\;\forall|k|\leq m\\}.$

Note that $\prescript{}{m}{\text{Pol}}_{m}$ is equal to the vector space
$\prescript{}{m}{PWS}_{m,0}(\mathfrak{a}^{*}_{\mathbb{C}})$. We will sometimes
write element of $\prescript{}{m}{\text{Pol}}_{m}$ and
$\prescript{}{m}{\mathcal{A}}_{m}$ as functions of two-variables:
$\varphi(\lambda,k)=\varphi_{k}(\lambda)$. We consider the subalgebra of
$\prescript{}{m}{\text{Pol}}_{m}$ generated by $\lambda^{2}+k^{2}$. It is the
subalgebra generated by the Fourier image of the Casimir operator and
isomorphic to $\text{Pol}(\mathbb{C})$. Thus , we can view
$\prescript{}{m}{\text{Pol}}_{m}$ as a $\text{Pol}(\mathbb{C})$-module.
Similarly, $\prescript{}{m}{\mathcal{A}}_{m}$ has the structure of
$\text{Hol}(\mathbb{C})$-module. Here, $h\in\text{Hol}(\mathbb{C})$ acts on
$\prescript{}{m}{\mathcal{A}}_{m}$ by

$(h\cdot\varphi)_{k}(\lambda)=h(\lambda^{2}+k^{2})\varphi_{k}(\lambda),\;\;h\in\text{Hol}(\mathbb{C}),\varphi\in\prescript{}{m}{\mathcal{A}}_{m}.$

######  Theorem 13.

The algebra $\prescript{}{m}{\mathcal{A}}_{m}$ is a free
$\emph{\text{Hol}}(\mathbb{C})$-module with the $m+1$ generators
$(k\lambda)^{l}\in\prescript{}{m}{\emph{\text{Pol}}}_{m}\subset\prescript{}{m}{\mathcal{A}}_{m},l=0,\dots,m$.
Furthermore, we have

$\prescript{}{m}{\mathcal{A}}_{m}\cong\emph{\text{Hol}}(\mathbb{C})\otimes_{\emph{\text{Pol}}(\mathbb{C})}\prescript{}{m}{\emph{\text{Pol}}}_{m}.$
(5.7)

Analogously, $\prescript{}{m}{\emph{\text{Pol}}}_{m}$ is a free
$\emph{\text{Pol}}(\mathbb{C})$-module with same generators as
$\prescript{}{m}{\mathcal{A}}_{m}$.

Note that Thm. 13 also tells that the two elements $\lambda^{2}+k^{2}$ and
$kl$ generate $\prescript{}{m}{\text{Pol}}_{m}$ as a algebra. Observe also
that $\prescript{}{m}{\text{Pol}}_{m}$ is isomorphic to
$\mathcal{D}_{G}(\mathbb{E}_{m},\mathbb{E}_{m})$, the set of all invariant
differential operators $D:C^{\infty}(X,\mathbb{E}_{m})\longrightarrow
C^{\infty}(X,\mathbb{E}_{m})$ ([OlPa22-1], Sect. 7).

###### Proof.

Consider $\varphi\in\prescript{}{m}{\mathcal{A}}_{m}$. It is sufficient to
show the existence and the uniqueness of holomorphic functions
$h_{0},\dots,h_{m}\in\text{Hol}(\mathbb{C})$ so that

$\varphi_{k}(\lambda):=\sum_{l=0}^{m}h_{l}(\lambda^{2}+k^{2})\cdot(k\lambda)^{l},\;\;\;\text{
for }k=-m,\dots,m$ (5.8)

and

$\varphi\in\prescript{}{m}{\text{Pol}}_{m}\text{ implies
}h_{l}\in\text{Pol}(\mathbb{C}),\;\;\;\text{ for }l=0,\dots,m.$ (5.9)

Then, $\prescript{}{m}{\mathcal{A}}_{m}$ is a free
$\text{Hol}(\mathbb{C})$-module with generators $(k\lambda)^{l},l=0,\dots,m$.
Similarly for $\prescript{}{m}{\text{Pol}}_{m}$. Note that
$\text{Hol}\otimes_{\text{Pol}}\text{Pol}\cong\text{Hol}$. Since, we have
$m+1$ free generators,

$\prescript{}{m}{\mathcal{A}}_{m}\cong\text{Hol}(\mathbb{C})^{m+1}\text{ and
}\prescript{}{m}{\text{Pol}}_{m}\cong\text{Pol}(\mathbb{C})^{m+1}$

we thus have that (5.8) & (5.9) imply (5.7).

We proceed by a step two induction on $m\in\mathbb{N}_{0}$, for the existence
of $h_{l}\in\text{Hol}(\mathbb{C}),$ $l=0,\dots,m$.
For $m=0$:

$\prescript{}{0}{\mathcal{A}}_{0}=\\{\varphi_{0}\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}})\;|\;\varphi_{0}(\lambda)=\varphi_{0}(-\lambda),\forall\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}\\}$

we see immediately that there is exactly one holomorphic function $h_{0}$ such
that $\varphi_{0}(\lambda)=h_{0}(\lambda^{2})$. $\varphi_{0}$ is a polynomial
function, if and only, if $h_{0}$ is one.
For $m=1$, we have that
$\prescript{}{1}{\mathcal{A}}_{1}=\\{\varphi_{1},\varphi_{-1}\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}})\;|\;\varphi_{-1}(\lambda)=\varphi_{1}(-\lambda),\forall\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}\\}$
and $\varphi_{1}(\lambda)$ can be decomposed into an even and odd part as
follows:

$\varphi_{1}(\lambda)=\varphi^{\text{even}}_{1}(\lambda)+\varphi^{\text{odd}}_{1}(\lambda)=h_{0}(\lambda^{2}+1)+\lambda
h_{1}(\lambda^{2}+1).$

Then,
$\varphi_{-1}(\lambda)=\varphi_{1}(-\lambda)=h_{0}(\lambda^{2}+1)-\lambda
h_{1}(\lambda^{2}+1).$ Hence, this leads us to desired relation
$\varphi_{k}(\lambda)=h_{0}(\lambda^{2}+k^{2})+(k\lambda)h_{1}(\lambda^{2}+k^{2})$,
for all $k=\pm 1$. $h_{0}$ and $h_{1}$ are polynomials, if and only, if
$\varphi_{\pm 1}$ are polynomials as well.
Assume now the existence of $h_{l}$ satisfying (5.8) & (5.9) for $m$ replaced
by $m-2$. Let $\varphi\in\prescript{}{m}{\mathcal{A}}_{m}$. We have
$\overline{\varphi}:=\text{diag}(\varphi_{m-2}(\lambda),\varphi_{m-4}(\lambda),\dots,\varphi_{-(m-4)}(\lambda),\varphi_{-(m-2)}(\lambda))\in\prescript{}{m-2}{\mathcal{A}}_{m-2}$
so that, by induction hypothesis, there exsits
$h_{l}\in\text{Hol}(\mathbb{C})$ with

$\varphi_{k}(\lambda)=\sum_{l=0}^{m-2}h_{l}(\lambda^{2}+k^{2})(k\lambda)^{l}\;\;\;\text{
for }|k|\leq m-2,$

and $h_{l}\in\text{Pol}(\mathbb{C})$ if
$\varphi\in\prescript{}{m}{\text{Pol}}_{m}$. Consider
$\tilde{\varphi}:=\text{diag}(\tilde{\varphi}_{m}(\lambda),\varphi_{m-2}(\lambda),\dots,\varphi_{-(m-2)}(\lambda),\tilde{\varphi}_{-m}(\lambda))\in\prescript{}{m}{\mathcal{A}}_{m}$
with

$\tilde{\varphi}_{\pm
m}(\lambda):=\sum_{l=0}^{m-2}h_{l}(\lambda^{2}+m^{2})(\pm m\lambda)^{l}.$

By taking the difference of $\varphi$ and $\tilde{\varphi}$, we get that
$\varphi-\tilde{\varphi}=\text{diag}(\varphi^{+}_{m}(\lambda),0,\dots,0,\varphi^{+}_{-m}(\lambda))\in\prescript{}{m}{\mathcal{A}}_{m},$
where we have set $\varphi^{+}_{\pm m}(\lambda):=\varphi_{\pm
m}(\lambda)-\tilde{\varphi}_{\pm m}(\lambda)$. Notice that $\varphi^{+}_{\pm
m}(l)=0$ for $|l|\leq m-2,l\equiv m$ (mod 2). We introduce the polynomial
function:

$p_{m}(\lambda,k)=\prod_{\begin{subarray}{c}|l|\leq m-2\\\ l\equiv m\text{
(mod 2)}\end{subarray}}(k-l)(\lambda-l)\in\prescript{}{m}{\text{Pol}}_{m}.$

Note that $p_{m}(\lambda,k)\equiv 0$ for $|k|\leq m-2,k\equiv m$ (mod 2).
Moreover, if $k=m$, then
$p_{m}(\lambda,m)=c_{m}\prod_{\begin{subarray}{c}|l|\leq m-2\\\ l\equiv
m\text{ (mod 2)}\end{subarray}}(\lambda-l),$ where $c_{m}$ is a non-zero
constant depending on the integer $m$. We conclude that there exist
$h^{+}_{0},h^{+}_{1}\in\text{Pol}(\mathbb{C})$ if
$\varphi\in\prescript{}{m}{\text{Pol}}_{m}$ such that

$\varphi^{+}_{m}(\lambda)=[h^{+}_{0}(\lambda^{2}+m^{2})+h^{+}_{1}(\lambda^{2}+m^{2})(m\lambda)]p_{m}(\lambda,m).$

This implies that
$(\varphi-\tilde{\varphi})(\lambda,k)=[h^{+}_{0}(\lambda^{2}+k^{2})+h^{+}_{1}(\lambda^{2}+k^{2})(k\lambda)]p_{m}(\lambda,k)$.
In addition,
$(k-\lambda)(\lambda-l)(k+l)(\lambda+l)=(k^{2}-l^{2})(\lambda^{2}-l^{2})=(k\lambda)^{2}-l^{2}(\lambda^{2}+k^{2})+l^{4}$
and thus $p_{m}(\lambda,k)$ is of the form:

$p_{m}(\lambda,k)=(k\lambda)^{m-1}+\sum_{\begin{subarray}{c}l=0\\\ l\equiv
m-1\text{ (mod
2)}\end{subarray}}^{m-3}p^{m}_{l}(\lambda^{2}+k^{2})(k\lambda)^{l},$ (5.10)

where $p^{m}_{l}$ are certain polynomials. We obtain

$\displaystyle(\varphi-\tilde{\varphi})(\lambda,k)$ $\displaystyle{=}$
$\displaystyle
h^{+}_{0}(\lambda^{2}+k^{2})(k\lambda)^{m-1}+h^{+}_{0}(\lambda^{2}+k^{2})\sum_{\begin{subarray}{c}l=0\\\
l\equiv m-1\text{ (mod
2)}\end{subarray}}^{m-3}p^{m}_{l}(\lambda^{2}+k^{2})(k\lambda)^{l}$
$\displaystyle+h^{+}_{1}(\lambda^{2}+k^{2})(k\lambda)^{m}+h^{+}_{1}(\lambda^{2}+k^{2})\sum_{\begin{subarray}{c}l=0\\\
l\equiv m-1\text{ (mod
2)}\end{subarray}}^{m-3}p^{m}_{l}(\lambda^{2}+k^{2})(k\lambda)^{l+1}.$

This implies that $\varphi-\tilde{\varphi}$ is of the desired form, hence
$\varphi$.

Concerning the uniqueness, we need to show that

$\sum_{l=0}^{m}h_{l}(\lambda^{2}+k^{2})(k\lambda)^{l}=0\;\;\;\forall|k|\leq
m,\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}\text{ implies }h_{l}=0,l=0,\dots,m.$
(5.11)

We proceed again by a two step induction on $m\in\mathbb{N}_{0}$. For the
initial cases $m=0,1$, the assertation is clear, see above. Assume that (5.11)
holds true for $m-2$ and let us prove that it holds for $m$. Let
$\varphi\in\prescript{}{m}{\mathcal{A}}_{m}$. By using the polynomial function
(5.10), we have for $|k|\leq m-2$:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{m}h_{l}(\lambda^{2}+k^{2})(k\lambda)^{l}-[h_{m-1}(\lambda^{2}+k^{2})+(k\lambda)h_{m}(\lambda^{2}+k^{2})p_{m}(\lambda,k)]$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{m-2}h_{l}(\lambda^{2}+k^{2})(k\lambda)^{l}-h_{m-1}(\lambda^{2}+k^{2})\sum_{\begin{subarray}{c}l=0\\\
l\equiv m-1\text{ (mod
2)}\end{subarray}}^{m-3}p^{m}_{l}(\lambda^{2}+k^{2})(k\lambda)^{l}$
$\displaystyle-h_{m}(\lambda^{2}+k^{2})\sum_{\begin{subarray}{c}l=0\\\ l\equiv
m-1\text{ (mod
2)}\end{subarray}}^{m-3}p^{m}_{l}(\lambda^{2}+k^{2})(k\lambda)^{l+1}$

in $\prescript{}{m-2}{\mathcal{A}}_{m-2}$. By induction hypothesis, this
implies that for $l\leq m-2$:

$h_{l}(\mu)=\begin{cases}h_{m-1}(\mu)p^{m}_{l}(\mu),&l\equiv m-1\text{ (mod
2)}\\\ h_{m}(\mu)p^{m}_{l-1}(\mu),&l\equiv m\text{ (mod 2)}.\end{cases}$

Then, for $k=m$, this implies

$0=\sum_{l=0}^{m}h_{l}(\lambda^{2}+m^{2})(k\lambda)^{l}=[h_{m-1}(\lambda^{2}+m^{2})+h_{m}(\lambda^{2}+m^{2})(m\lambda)]p_{m}(\lambda,m).$

Since $p_{m}(\lambda,m)$ is not identical zero on
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$, we obtain that $h_{m-1}$ and
$h_{m}$ are zero. Hence $h_{l}=0$ for $l\leq m$. This completes the proof. ∎

### General case: The $K$-types $n$ and $m$ are distinct

Consider now two distinct $K$-types $(n,E_{n})$ and $(m,E_{m})$. Since
$\text{Hom}_{M}(E_{n},E_{m})=0$ for $n\not\equiv m$ (mod 2), we assume
throughout the section that $n\equiv m$ (mod 2). We define

$\prescript{}{n}{\mathcal{A}}_{m}:=\prescript{}{n}{PWS}_{m,H}(\mathfrak{a}^{*}_{\mathbb{C}})\subset\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{Hom}_{M}(E_{n},E_{m}))$

and
$\prescript{}{n}{\text{Pol}}_{m}:=\prescript{}{n}{PWS}_{m,0}(\mathfrak{a}^{*}_{\mathbb{C}}).$
Here, $\prescript{}{n}{PWS}_{m,H}(\mathfrak{a}^{*}_{\mathbb{C}})$ denotes the
pre-Paley-Wiener-Schwartz space such that
$\varphi\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{Hom}_{M}(E_{n},E_{m}))$
and satisfies the intertwining condition $(D.3)$ of Def. 5, same for
$\prescript{}{n}{PWS}_{m,0}(\mathfrak{a}^{*}_{\mathbb{C}})$ but for polynomial
functions
$\text{Pol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{Hom}_{M}(E_{n},E_{m}))$. Note
that after a choice of basis vectors in $E_{n},E_{m}$ consisting of $M$-weight
vectors, we can write the elements in the following way:

$\varphi:=\begin{cases}\begin{pmatrix}0&\cdots&0\\\ \vdots&\vdots&\vdots\\\
0&\cdots&0\\\ \varphi_{n}(\lambda)&\cdots&0\\\ \vdots&\ddots&\vdots\\\
0&\cdots&\varphi_{-n}(\lambda)\\\ 0&\cdots&0\\\ \vdots&\vdots&\vdots\\\
0&\cdots&0\end{pmatrix}_{(m+1)\times(n+1)}&\text{ if }n<m,\\\
\begin{pmatrix}0&\cdots&0&\varphi_{m}(\lambda)&\cdots&0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&\cdots&0&0&\cdots&\varphi_{-m}(\lambda)&0&\cdots&0\\\
\end{pmatrix}_{(m+1)\times(n+1)}&\text{ if }n>m.\end{cases}$

We start with the case $|n-m|=2$.

######  Proposition 3.

Let $m$ be a $K$-type. There exists an unique, up to normalization, operator
of first order $q^{+}_{m}$ in $\prescript{}{m}{\emph{\text{Pol}}}_{m+2}$
(resp. $q^{-}_{m}$ in $\prescript{}{m+2}{\emph{\text{Pol}}}_{m}$ by taking the
adjoint), which corresponds to

$\displaystyle(\lambda+m+2)\cdot$ $\displaystyle\begin{pmatrix}0&\cdots&0\\\
1&\cdots&0\\\ \vdots&\ddots&\vdots\\\ 0&\ldots&1\\\
0&\cdots&0\end{pmatrix}_{(m+2)\times m}$ $\displaystyle\Bigg{(}\text{resp.
}(\lambda-(m+2))\cdot$ $\displaystyle\begin{pmatrix}0&d(m,m)&\cdots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\ldots&d(m,-m)&0\\\
\end{pmatrix}_{m\times(m+2)}\Bigg{)}$

under some appropriate basis choice and where $d(m,k)=(m+2)^{2}-k^{2}$, for
all $|k|\leq m$.

###### Proof.

Consider the symmetric algebra $S(\mathfrak{p})$ of
$\mathfrak{p}\cong\mathbb{R}^{3}$ over $\mathbb{C}$. From ([Olb95], Folgerung
2.5), we deduce that the associated graded module of
$\prescript{}{m}{\text{Pol}}_{m+2}$ is isomorphic to
$[S(\mathfrak{p})\otimes\text{Hom}_{M}(E_{m},E_{m+2})]^{K}.$ By using Clebsch-
Gordan rule (5.1), we have

$\displaystyle[S^{\leq 1}(\mathfrak{p})\otimes\text{Hom}(E_{m},E_{m+2})]^{K}$
$\displaystyle\cong$ $\displaystyle[(S^{0}(\mathfrak{p})\oplus
S^{1}(\mathfrak{p}))\otimes(E_{2}\oplus E_{4}\oplus\dots\oplus E_{2m+2})]^{K}$
$\displaystyle\cong$ $\displaystyle[S^{1}(\mathfrak{p})\otimes E_{2}]^{K},$

where $[S^{1}(\mathfrak{p})\otimes E_{2}]^{K}$ is one-dimensional. Note that
$S^{1}(\mathfrak{p})\cong\mathfrak{p}_{\mathbb{C}}\cong E_{2}$. This means
that $\prescript{}{m}{\text{Pol}}_{m+2}$ contains exactly (up to
normalization) one element $q^{+}_{m}$ of filter degree 1. According to an
appriopate chose of basis, we can write $q^{+}_{m,n}(\lambda)$ in a matrix
form as above. Since $q^{+}_{m,k}(\lambda)$ is of first order (but the
individual components could be constants), this means that all elements, under
normalization, are of the form

$q^{+}_{m,k}(\lambda)=\begin{cases}\tilde{c}(m,k),&|k|\leq m\text{ or }\\\
(\lambda+c(m,k)),&|k|\leq m,\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}\text{ (at
least for one $k$)}.\end{cases}$

Here $\tilde{c}(m,k)$ are constants depending on $k$ and $m$. Consider
$\varphi_{k}\in\prescript{}{m}{\mathcal{A}}_{m+2}$. By Prop. 1 (b) and Harish-
Chandra $\mathbf{c}$-functions (5.3), we have that

$\displaystyle\varphi_{-k}(-\lambda)=(-1)^{(m+2-m)/2}\frac{\mathbf{c}_{m+2,k}(\lambda)}{\mathbf{c}_{m,k}(\lambda)}\varphi_{k}(\lambda)$
$\displaystyle=$
$\displaystyle(-1)\frac{\Gamma(\frac{\lambda}{2}+\frac{m}{2}+1)\Gamma(\frac{\lambda}{2}-\frac{m}{2})}{\Gamma(\frac{\lambda}{2}+\frac{m}{2}+2)\Gamma(\frac{\lambda}{2}-\frac{m}{2}-1)}\varphi_{k}(\lambda)$
$\displaystyle=$
$\displaystyle(-1)\frac{(\lambda-(m+2))}{(\lambda+(m+2))}\varphi_{k}(\lambda)$
$\displaystyle=$
$\displaystyle\frac{(-\lambda+(m+2))}{(\lambda+(m+2))}\varphi_{k}(\lambda).$

Since $\varphi_{-k}$ has no singularities, we conclude that $\varphi_{k}$ has
a zero at $-(m+2)$. Applying this to $\varphi=q^{+}_{m}$, we see that

$q^{+}_{m,k}(\lambda)=\begin{cases}0,\\\ \lambda+(m+2).\end{cases}$

Analogously, there exists exactly one (up to normalization)
$q^{-}_{m}\in\prescript{}{m+2}{\mathcal{A}}_{m}$ of degree 1. We consider
$q^{+}_{m}q^{-}_{m}\in\prescript{}{m+2}{\mathcal{A}}_{m+2}$. Then
$q^{+}_{m}q^{-}_{m}(\lambda,\pm(m+2))=0$. This implies that
$q^{+}_{m}q^{-}_{m}(\pm(m+2),k)=0$ for $|k|\leq m,k\equiv m$ (mod 2) of degree
2. Hence, $q^{+}_{m}q^{-}_{m}(\lambda,k)=d(m,k)(\lambda^{2}-(m+2)^{2})$ for
some constants $d(m,k)$. Since $q^{-}_{m}=c\cdot(q^{+}_{m})^{*}$, where $*$
stands for the adjoint $p^{*}(\lambda):=p(-\overline{\lambda})^{*}$, we have
$d(m,k)\neq 0$ for all $|k|\leq m$. In addition, by (9)

$d(m,k)=\begin{cases}d(m,0)\cdot\frac{(m+2)^{2}-k^{2}}{(m+2)^{2}}\neq
0,&m\text{ even}\\\ d(m,1)\cdot\frac{(m+2)^{2}-k^{2}}{(m+2)^{2}-1}\neq
0,&m\text{ odd}.\end{cases}$

We conclude that $q^{+}_{m}(\lambda,k)=\lambda+c(m,k)=\lambda+(m+2)^{2}$ and
$q^{-}_{m}(\lambda,k)=((m+2)^{2}-k^{2})(\lambda-(m+2))$ (up to normalization).
∎

Now we consider general $n,m\in\mathbb{N}_{0},n\equiv m$ (mod 2).

######  Definition 10.

Consider $q^{\pm}_{m}$ and $q^{\pm}_{n}$ as in Prop. 3. We define the
polynomial $q_{n,m}\in\prescript{}{n}{\emph{\text{Pol}}}_{m}$ in
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ by

$q_{n,m}=\begin{cases}q^{+}_{m-2}\cdot q^{+}_{m-4}\cdots q^{+}_{n+2}\cdot
q^{+}_{n},&\text{if }n<m\\\ q^{-}_{m}\cdot q^{-}_{m+2}\cdots q^{-}_{n-4}\cdot
q^{-}_{n-2},&\text{if }n>m\\\ \emph{\text{Id}},&\text{if }n=m.\end{cases}$
(5.12)

Finally, the following theorem, gives explicitly the Paley-Wiener(-Schwartz)
spaces in (Level 3) for $G=\mathbf{SL}(2,\mathbb{C})$. Note that
$\prescript{}{n}{\text{Pol}}_{m}$ is isomorphic to the set of all invariant
differential operators $\mathcal{D}_{G}(\mathbb{E}_{n},\mathbb{E}_{m})$
([OlPa22-1], Sect. 7)..

######  Theorem 14 (Intertwining condition in (Level 3)).

Let $n,m\in\mathbb{N}_{0}$ be two $K$-types, which are not necessary distinct,
and let $l:=\min(n,m)$. Then, $\prescript{}{n}{\emph{\text{Pol}}}_{m}$ (resp.
$\prescript{}{n}{\mathcal{A}}_{m}$) is a free
$\prescript{}{l}{\emph{\text{Pol}}}_{l}$ (resp.
$\prescript{}{l}{\mathcal{A}}_{l}$)-module with generator $q_{n,m}$. This
means that there exists an unique function
$h\in\prescript{}{l}{\mathcal{A}}_{l}$ such that

$\prescript{}{n}{\mathcal{A}}_{m}\ni\varphi(\lambda)=\begin{cases}h(\lambda)q_{n,m}(\lambda),&\text{
if }m<n,\\\ q_{n,m}(\lambda)h(\lambda),&\text{ if }m>n,\\\
\end{cases}\;\;\;\;\text{for }\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}.$ (5.13)

Moreover, if $L=\max(n,m)$, then $\prescript{}{n}{\emph{\text{Pol}}}_{m}$
(resp. $\prescript{}{n}{\mathcal{A}}_{m}$) is a
$\prescript{}{L}{\emph{\text{Pol}}}_{L}$ (resp.
$\prescript{}{L}{\mathcal{A}}_{L}$)-module generated by $q_{n,m}$.

###### Proof.

Consider the case $m<n$, it suffices to prove that

  * (a)

there exists a unique $h\in\prescript{}{m}{\mathcal{A}}_{m}$ such that
$\varphi=h\cdot q_{n,m}\in\prescript{}{n}{\mathcal{A}}_{m}$,.

  * (b)

there exists a $\tilde{h}\in\prescript{}{n}{\mathcal{A}}_{n}$ such that
$\varphi=q_{n,m}\cdot\tilde{h}\in\prescript{}{n}{\mathcal{A}}_{m}$.

The polynomial case as well as $m>n$ can be proved in a similar way. Consider

$q_{m,n}\prescript{}{n}{\mathcal{A}}_{m}\subset\\{g\in\prescript{}{n}{\mathcal{A}}_{n}\;|\;g_{k}=0,\;\forall|k|>m\\}\subset\prescript{}{n}{\mathcal{A}}_{n}.$

We also have zeros between the lines $-m$ and $m$, this means
$g_{l}(k)=0,\forall k\equiv\pm(m+2),\pm(m+4),\dots,\pm n$. Let
$\varphi\in\prescript{}{n}{\mathcal{A}}_{m}$. Every component of
$g=q_{m,n}\cdot\varphi$ has zeros at described above and $q_{m,n}$ has zeros
only at negative $\lambda$, we have that every component $\varphi$ has zeros
at $m+2,\dots,n$. Hence, there exists a unique
$h\in\text{Hol}(\mathfrak{a}^{*}_{\mathbb{C}},\text{End}(E_{m}))$ so that

$\varphi=h\cdot q_{n,m}.$

In fact, since $g$ satisfies the intertwining condition $g_{k}(l)=g_{l}(k)$ as
well as the symmetry one $g_{k}(\lambda)=g_{-k}(-\lambda)$, and

$\displaystyle q_{m,n}(k,\lambda)q_{n,m}(k,\lambda)$ $\displaystyle=$
$\displaystyle q_{m,n}(-k,-\lambda)q_{n,m}(-k,-\lambda)$ $\displaystyle
q_{m,n}(k,l)q_{n,m}(k,l)$ $\displaystyle=$ $\displaystyle
q_{m,n}(l,k)q_{n,m}(l,k)$

we then have that $h_{l}(k)=h_{k}(l)$ and $h_{k}(\lambda)=h_{-k}(-\lambda)$
for $k\equiv l\equiv m$ (mod 2), $|k|\leq m$ and
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$. This proves (a).

For (b), we know from (a) that $\varphi=h\cdot q_{n,m}$ is in the ’small’
space, thus we need to find

$\tilde{h}=\text{diag}(\tilde{h}_{n},\dots,\tilde{h}_{m+2},\tilde{h}_{m},\dots,\tilde{h}_{-m},\tilde{h}_{-(m-2)},\dots,\tilde{h}_{-n})\in\prescript{}{n}{\mathcal{A}}_{n}$

with $\tilde{h}_{k}=h_{k}$ for $|k|\leq m$, so that
$\tilde{h}_{k}(l)=\tilde{h}_{l}(k)=h_{l}(k)$ and
$\tilde{h}_{k}(\lambda)=\tilde{h}_{-k}(-\lambda)$ for $\forall l$ and $k>m$
and $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$. Then
$\varphi=\prescript{}{n}{q}_{m}\cdot\tilde{h}$. By using the interpolation
polynomial for each $|k|>m$, we define recursively for
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$:

$\displaystyle\tilde{h}_{m+2}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}i=-m\\\ i\equiv m\text{ (mod
2)}\end{subarray}}^{m}h_{i}(m+2)\prod^{m}_{\begin{subarray}{c}l=-m\\\ l\neq
i\end{subarray}}\Big{(}\frac{\lambda-l}{i-l}\Big{)}$
$\displaystyle\tilde{h}_{m+4}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}i=-(m+2)\\\ i\equiv m+2\text{ (mod
2)}\end{subarray}}^{m+2}\tilde{h}_{i}(m+4)\prod^{m+2}_{\begin{subarray}{c}l=-(m+2)\\\
l\neq i\end{subarray}}\Big{(}\frac{\lambda-l}{i-l}\Big{)}$
$\displaystyle\vdots$ $\displaystyle\tilde{h}_{n-2}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}i=-(n-4)\\\ i\equiv n-4\text{ (mod
2)}\end{subarray}}^{n-4}\tilde{h}_{i}(n-2)\prod^{n-4}_{\begin{subarray}{c}l=-(n-4)\\\
l\neq i\end{subarray}}\Big{(}\frac{\lambda-l}{i-l}\Big{)}$
$\displaystyle\tilde{h}_{n}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}i=-(n-2)\\\ i\equiv n-2\text{ (mod
2)}\end{subarray}}^{n-2}\tilde{h}_{i}(n)\prod^{n-2}_{\begin{subarray}{c}l=-(n-2)\\\
l\neq i\end{subarray}}\Big{(}\frac{\lambda-l}{i-l}\Big{)},$

and $\tilde{h}_{-k}(\lambda):=\tilde{h}_{k}(\lambda)$. The intertwining
relations are also satisfies for $|l|,|k|>m$ due
$h\in\prescript{}{m}{\mathcal{A}}_{m}$. ∎

We observe an anologue of Thm. 13, in particular relation (5.7), is also true
in general for distinct $K$-type $n$ and $m$.

######  Corollary 1.

With the notations above,
$\prescript{}{n}{\mathcal{A}}_{m}\cong\emph{\text{Hol}}(\mathbb{C})\otimes_{\emph{\text{Pol}}(\mathbb{C})}\prescript{}{n}{\emph{\text{Pol}}}_{m}.$

###### Proof.

As a consequence of Thm. 14, the $\text{Hol}(\mathbb{C})$-module
$\prescript{}{n}{\mathcal{A}}_{m}$ is isomorphic to
$\prescript{}{l}{\mathcal{A}}_{l}$, and $\prescript{}{n}{\text{Pol}}_{m}$ is
isomorphic to $\prescript{}{l}{\text{Pol}}_{l}$ as
$\text{Pol}(\mathbb{C})$-module. Then, we can apply Thm. 13 and obtain the
desired result. ∎

### Acknowledgement

This research is supported by the research project PRIDE15/10949314/GSM of the
Luxembourg National Research Fund and by the University of Luxembourg.

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Université du Luxembourg,

Faculty of Science, Technology and Medicine,

Department of Mathematics

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