#  On the Schwartz space isomorphism theorem for the Riemannian symmetric
spaces

Joydip Jana  Syamaprasad College
Department of Mathematics
92, S. P. Mukherjee Road
Kolkata-700 026, India
E-mail<EMAIL_ADDRESS>
###### Abstract.

We deduce a proof of the isomorphism theorem for certain closed subspace
$\mathcal{S}^{p}_{\Gamma}(X)$ of the $L^{p}$-Schwartz class functions
$(0<p\leq 2)$ on a Riemannian symmetric space $X$ where $\Gamma$ is a finite
subset of $\widehat{K}_{M}$. The Fourier transform considered is the Helgason
Fourier transform. Our proof relies only on the Paley-Wiener theorem for the
corresponding class of functions and hence it does not use the complicated
higher asymptotics of the elementary spherical functions.

###### Key words and phrases:

Schwartz spaces, $\delta$-spherical transform, Helgason Fourier transform

_Mathematical Subject Classification_ : 43A80, 43A85, 43A90

##  Introduction

Let $X$ be a Riemannian symmetric space realized as $G/K$, where $G$ is a
connected, noncompact semisimple Lie group with finite center. Let us fix $K$
is a maximal compact subgroup of $G$. The $L^{p}$-Schwartz space isomorphism
theorem for bi-$K$-invariant functions on the group $G$ under the spherical
transform was first proved by Harish-Chandra [HC58a, HC58b, HC66] (for $p=2$),
Trombi and Varadarajan [TV71] (for $0<p<2$). Recently Anker [Ank91] gave a
remarkable short and elegant proof of the above theorem for $0<p\leq 2$.
Anker’s work does not involve the asymptotic expansion of the elementary
spherical functions which has a crucial role in the earlier works. The aim of
this paper is to extend Anker’s technique for the $L^{p}$-Schwartz class
functions (for $0<p\leq 2$) on $X$ and to establish an isomorphism theorem
under the Helgason fourier transform (HFT).
The main result of this paper is developed in two theorems Theorem 3.8 and
Theorem 5.3. In Theorem 3.8 the basic $L^{p}$-Schwartz space
$\mathcal{S}^{p}_{\delta}(X)$ is the space of operator valued
left-$\delta$-type $(\delta\in\widehat{K}_{M})$, smooth functions on $X$. HFT
when restricted to $\mathcal{S}^{p}_{\delta}(X)$ is identified with the
‘$\delta$-spherical transform’. In Theorem 3.8 we establish an isomorphism
between the Schwartz spaces $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ under the
$\delta$-spherical transform. The image
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ is a space of matrix
valued functions with certain decay defined on a closed complex tube
$\mathfrak{a}^{*}_{\varepsilon}=\mathfrak{a}^{*}+iC^{\varepsilon\rho}$.
Explicit definition of the space
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ and the domain
$\mathfrak{a}^{*}_{\varepsilon}$ will be given in the next section.
Restriction of Theorem 3.8 to the rank-one case is a part of the result of
Eguchi and Kowata [EK76]. Theorem 3.8 also relaxes the rank restriction of the
similar result obtained in [JS07].
Theorem 5.3 further extends the isomorphism obtained in Theorem 3.8 to the
space $\mathcal{S}^{p}_{\Gamma}(X)$ of scalar valued $K$-finite Schwartz class
functions on $X$ for which the left $K$-types lie in a fixed finite subset
$\Gamma\subset\widehat{K}_{M}$. The transform considered in Theorem 5.3 is the
HFT.
We shall closely follow the notations of [Hel, Hel94]. Some basic definitions
and results used in this paper are given in the following section.

##  1\. Notation and Preliminaries

Let $G$ be a connected, noncompact semisimple Lie group with finite center and
$K$ be a maximal compact subgroup of $G$. Let $\theta$ be the Cartan
involution corresponding to $K$. Let $X$ be a Riemannian symmetric space
realized as $G/K$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and
$\mathfrak{g}_{\mathbb{C}}$ be its complexification. The Hermitian norm of
both $\mathfrak{g}$ and $\mathfrak{g}_{\mathbb{C}}$ will be denoted by the
notation $\|\cdot\|$. We denote $\mathfrak{k}$ for the Lie algebra of the
maximal compact subgroup $K$ of $G$. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{s}$ be the Cartan decomposition of
the Lie algebra. We fix a maximal abelian subspace $\mathfrak{a}$ in
$\mathfrak{s}$. Let $A$ be analytic subgroup of $G$ with the Lie algebra
$\mathfrak{a}$. Let $\mathfrak{a}^{*}$ and $\mathfrak{a}^{*}_{\mathbb{C}}$
respectively be the real dual of $\mathfrak{a}$ and its complexification. The
Killing form induces a scalar product on $\mathfrak{a}$ and hence on
$\mathfrak{a}^{*}$. We shall denote $\langle\cdot,\cdot\rangle_{1}$ for the
$\mathbb{C}$-bilinear extension of that scalar product to
$\mathfrak{a}^{*}_{\mathbb{C}}$. A semisimple Lie group $G$ is said to be of
‘real rank-$n$’ if $dim\mathfrak{a}=n$ and the corresponding symmetric spaces
$X$ realized as $X=G/K$ are called ‘rank-$n$’ symmetric spaces.
Let $\Sigma$ be the root system associated with the pair
$(\mathfrak{g},\mathfrak{a})$. For each $\alpha\in\Sigma$ we write
$\mathfrak{g}_{\alpha}$ for the corresponding root space. Let $M^{\prime}$ and
$M$ respectively be the normalizer and the centralizer of $A$ in $K$. The
quotient $W=M^{\prime}/M$ be the Weyl group associated with the root system
$\Sigma$.

Let us choose and fix a system of positive roots which we denote by
$\Sigma^{+}$. Let $\mathfrak{a}^{+}$ be the corresponding positive Weyl
chamber and $\overline{\mathfrak{a}^{+}}$ be its closer. We denote
$\mathfrak{a}^{*+}$ and $\overline{\mathfrak{a}^{*+}}$ for the similar
chambers in $\mathfrak{a}^{*}$. We put $A^{+}=\exp\mathfrak{a}^{+}$ and
$\overline{A^{+}}=\exp\overline{\mathfrak{a}^{+}}$. The element
$\rho\in\mathfrak{a}^{*}$ is denoted by

(1.1)
$\rho(H)=\frac{1}{2}\sum_{\alpha\in\Sigma^{+}}m_{\alpha}\alpha(H),~{}~{}\mbox{
where}~{}m_{\alpha}=dim\mathfrak{g}_{\alpha}~{}\mbox{and}~{}H\in\mathfrak{a}.$

Let $\Sigma_{0}\subset\Sigma$ be the set of all indivisible roots and
$\Sigma_{0}^{+}=\Sigma_{0}\cap\Sigma^{+}$.
$\mathfrak{n}=\oplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{\alpha}$ is a
nilpotent subalgebra of $\mathfrak{g}$. Let $N$ be the nilpotent subgroup of
$G$ with the Lie algebra $\mathfrak{n}$. The Iwasawa decomposition of the
group $G$ is given as $G=KAN$. The map $(k,a,n)\mapsto kan$ is a
diffeomorphism from $K\times A\times N$ onto $G$. Let
$\mathcal{H}:G\rightarrow\mathfrak{a}$ and
$\mathcal{A}:G\rightarrow\mathfrak{a}$ are the $\mathfrak{a}$ projections of
$g\in G$ in Iwasawa $KAN$ and $NAK$ decompositions respectively. These two
projections are related by $\mathcal{A}(g)=-\mathcal{H}(g^{-1})$ for each
$g\in G$. Any element $g\in G$ can therefore be written as
$g=k\exp{\mathcal{H}(g)}n=n_{1}\exp{\mathcal{A}(g)}k_{1}$. In the Iwasawa
$KAN$ decomposition the Haar measure of the group $G$ is given by

(1.2)
$\int_{G}f(g)dg=const.\int_{K}dk\int_{\mathfrak{a}^{+}}e^{2\rho(\mathcal{H}(g))}d\mathcal{H}(g)\int_{N}dn~{}f(k\exp{\mathcal{H}(g)}n),$

where $const.$ is a normalizing constant. The ‘Cartan decomposition’ of the
group is $G=K\overline{A^{+}}K=K\exp\overline{\mathfrak{a}^{+}}K$. Let $g^{+}$
denote the $\mathfrak{a}^{+}$ component of $g\in G$, and we denote
$|g|=\|g^{+}\|$. We have a basic estimate: for some constant $c>0$

(1.3) $\|\mathcal{H}(g)\|\leq c|g|,\mbox{~{}for each~{} }g\in G.$

The Haar measure for the Cartan decomposition is given by

(1.4)
$\int_{G}f(g)dg=const.\int_{K}dk\int_{\mathfrak{a}^{+}}\Delta(H)dh\int_{K}dk^{\prime}f(k\exp
Hk^{\prime}),~{}~{}(H\in\overline{\mathfrak{a}^{+}})$

where $\Delta(H)=\prod_{\alpha\in\Sigma^{+}}\sinh^{m_{\alpha}}H$ and $const.$
is a positive normalizing constant. We shall be using the following estimate
for the density $\Delta(H)$:

(1.5) $0\leq\Delta(H)\leq ce^{2\rho(H)},\mbox{~{}for~{}}H\in\mathfrak{a}^{+}.$

A function $f$ on $G$ is said to be ‘bi-$K$-invariant’ if
$f(k_{1}gk_{2})=f(g)$ for all $k_{1},k_{2}\in K$ and $g\in G$. We refer a
function a function as ‘right-$K$-invariant’ if invariant under the right $K$
action on $G$ that is $f(gk)=f(g)$ for all $k\in K,g\in G$. Althrough in this
paper we shall consider a function on the symmetric space $X=G/K$ as a
right-$K$-invariant function on the group $G$. For any function space
$\mathfrak{F}(G)$ on $G$ or $\mathfrak{F}(G/K)$ on $X$, we shall denote
$\mathfrak{F}(G//K)$ for the corresponding subspace of bi-$K$-invariant
functions.
We denote $\mathcal{C}^{\infty}(G)$ for the set of all smooth functions on
$G$. We fix a basis $\\{X_{j}\\}$ for the Lie algebra $\mathfrak{g}$. Let
$\mathcal{U}(\mathfrak{g})$ be the ‘universal enveloping algebra’ over
$\mathfrak{g}$. Let $D_{1}\cdots D_{m},E_{1}\cdots
E_{n}\in\mathcal{U}(\mathfrak{g})$, then the action of
$\mathcal{U}(\mathfrak{g})$ on a function $f\in\mathcal{C}^{\infty}(G)$ is
defined as follows:

$\displaystyle f(D_{1}\cdots D_{m},x,E_{1}\cdots E_{n})=$
$\displaystyle\frac{d}{dt_{1}}\mid_{t_{1}=0}\cdots\frac{d}{dt_{m}}\mid_{t_{m}=0}~{}\frac{d}{ds_{1}}\mid_{s_{1}=0}\cdots\frac{d}{ds_{n}}\mid_{s_{n}=0}f(\exp{(t_{1}D_{1})}\cdots\exp{(t_{m}D_{m})}x\exp{(s_{1}E_{1})}\cdots\exp{(s_{n}E_{n})}).$

Let $b_{ij}=\mathfrak{B}(X_{i},X_{j})$ and $(b^{ij})$ be the inverse of the
matrix $(b_{ij})$. We now define a distinguished element, called the ‘Casimir
element’, of $\mathcal{U}(\mathfrak{g})$ by
$\Omega=\sum_{i,j}b^{ij}X_{i}X_{j}$. The differential operator $\Omega$ lies
in the center of $\mathcal{U}(\mathfrak{g})$. The action of the ‘Laplace-
Beltrami operator’ $\mathbf{L}$ on $X$ is defined by the action of $\Omega$:

(1.6) $\mathbf{L}f(xK)=f(x,\Omega),\hskip 36.135ptx\in G.$

Let us briefly describe the method of construction of a family of rank-one
symmetric spaces, the rank-one reductions, which are totally geodesic
submanifolds of the general rank symmetric space $G/K$ ( see [Hel01, Ch. IX,
§2], [Hel, Ch. IV, §6], [Kna03] ). Let $\beta$ be an indivisible root of the
system $\Sigma$ and $\mathfrak{g}_{(\beta)}$ be the Lie subalgebra of
$\mathfrak{g}$ generated by the root spaces
$\mathfrak{g}_{\beta},~{}\mathfrak{g}_{2\beta},~{}\mathfrak{g}_{-\beta}$ and
$\mathfrak{g}_{-2\beta}$. The subalgebra $\mathfrak{g}_{(\beta)}$ is stable
under the Cartan involution and it is simple. Let $G_{(\beta)}$ be the
analytic subgroup of $G$ corresponding to the Lie subalgebra
$\mathfrak{g}_{(\beta)}$. The Iwasawa decomposition of $G_{(\beta)}$ be
$G_{(\beta)}=K_{(\beta)}A_{(\beta)}N_{(\beta)}$ where $K_{(\beta)}=K\cap
G_{(\beta)}$, $A_{(\beta)}=A\cap G_{(\beta)}$ and $N_{(\beta)}=N\cap
G_{(\beta)}$. Also the centralizer of $A_{(\beta)}$ in $K_{(\beta)}$ is
$M_{(\beta)}=M\cap G_{(\beta)}$. The abelian subgroup $A_{(\beta)}$ is one-
dimensional and its Lie algebra $\mathfrak{a}_{(\beta)}$ is generated by the
element $H_{\beta}\in\mathfrak{a}$ determined by
$\lambda(H_{\beta})=\langle\lambda,\beta\rangle$. Hence $G_{(\beta)}$ is of
real-rank-one and consequently $G_{(\beta)}/K_{(\beta)}$ is a rank-one
symmetric space.
The restricted roots of $G_{(\beta)}$ are $\\{\beta,2\beta,-\beta,-2\beta\\}$
or $\\{\beta,-\beta\\}$ according as $2\beta\in\Sigma$ or not. Let us further
consider $\beta$ to be a positive root of $G_{(\beta)}$ thus the Lie algebra
$\mathfrak{n}_{\beta}$ of $N_{(\beta)}$ is the sum of the root spaces
$\mathfrak{g}_{\beta}$ and $\mathfrak{g}_{2\beta}$. We write $\rho_{(\beta)}$
for the $\rho$-function of $\mathfrak{g}_{(\beta)}$. It can be shown that
[Hel, B.3, page: 483] $\rho(H_{\beta})\geq\rho_{(\beta)}(H_{\beta})$ for all
$\beta\in\Sigma_{0}^{+}$. The equality holds only when $\beta$ is simple and
in that case $\rho_{(\beta)}$ is exactly the restriction of $\rho$ to
$\mathfrak{g}_{(\beta)}$. For each $\beta\in\Sigma_{0}$ the restriction
$\lambda_{\beta}$ of $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ to
$\mathfrak{a}_{(\beta)}$ is given by the expression
$\lambda_{\beta}=\frac{{\langle\lambda,\beta\rangle}_{1}}{\langle\beta,\beta\rangle}\beta$.

Let $\pi_{\lambda}$ ($\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$) be the
spherical principal series representations of $G$ realized on the Hilbert
space $L^{2}(K/M)$ and given by the following:

(1.7)
$\left\\{\pi_{\lambda}(g)\zeta\right\\}(kM)=e^{-(i\lambda+\rho)\mathcal{H}(g^{-1}k)}\zeta(\mathcal{K}(g^{-1}k)M),$

where $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},~{}\zeta\in L^{2}(K/M)$ and
$\mathcal{K}(g^{-1}k)$ denotes the $K$ part of $g^{-1}k$ in the Iwasawa $KAN$
decomposition. The spherical functions $\varphi_{\lambda}(\cdot)$, given by
the following formula

(1.8)
$\varphi_{\lambda}(g)=\int_{K}e^{(i\lambda-\rho)\mathcal{H}(g^{-1}k)}dk,~{}\mbox{where}~{}g\in
G,~{}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},$

are the matrix coefficient of the principal series representations. For each
$g\in G$ and $\omega\in W$ $\varphi_{\omega\lambda}(g)=\varphi_{\lambda}(g)$.
We shall use the following basic estimates for the elementary spherical
functions for our purpose.

  1. (i)

For each $H\in\overline{\mathfrak{a}^{+}}$ and
$\lambda\in\overline{{\mathfrak{a}^{*}}^{+}}$, we have

(1.9) $0<|\varphi_{-i\lambda}(\exp H)|\leq e^{\lambda(H)}\varphi_{0}(\exp H),$

here, $\varphi_{0}(\cdot)$ is the spherical function corresponding to
$\lambda=0$. For a proof of the above estimate see [GV88], Proposition 4.6.1.

  2. (ii)

For all $g\in G$, $0<\varphi_{0}(g)\leq 1$ [GV88, Proposition 4.6.3] . Also
for $\overline{\mathfrak{a}^{+}}$ we have the following estimate

(1.10) $\displaystyle e^{\rho(H)}\leq\varphi_{0}(\exp
H)\leq~{}{\ss}(1+\|H\|)^{c_{\mathfrak{a}}}~{}e^{\rho(H)},$

where, ${\ss},c_{\mathfrak{a}}>0$ are group dependent constants. This is an
work of Harish-Chandra, the above optimal estimate was obtained by Anker
[Ank87].

Let $\mathcal{D}(G)$ be the subspace of $\mathcal{C}^{\infty}(G)$ generated by
the compactly supported scalar valued smooth functions on $G$. For each
$f\in\mathcal{D}(G//K)$ the ‘spherical Fourier transform’ $\mathcal{S}f$ is
defined by

(1.11)
$\mathcal{S}f(\lambda)=\int_{G}f(x)\varphi_{-\lambda}(x)dx,~{}\mbox{where}~{}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}.$

The inversion of the spherical transform is given by

(1.12)
$f(x)=\frac{1}{|W|}\int_{\mathfrak{a}^{*}_{\mathbb{C}}}\varphi_{\lambda}(x)~{}\mathcal{S}f(\lambda)~{}|\mathbf{c}(\lambda)|^{-2}d\lambda,$

where $|W|$ is the cardinality of the group $W$ and $\mathbf{c}(\lambda)$ is
the ‘Harish-Chandra $\mathbf{c}$-function’. For our purpose we shall use the
following estimate [Ank92]: there exists constants $a,b>0$ such that

(1.13) $|\mathbf{c}(\lambda)|^{-2}\leq~{}a(\|\lambda\|+1)^{b},\mbox{ ~{}for
all~{}}\lambda\in\mathfrak{a}^{*}.$

For $f\in\mathcal{D}(X)$, the Helgason Fourier transform (HFT) $\mathcal{F}f$
is a function on $\mathfrak{a}^{*}_{\mathbb{C}}\times K/M$ and it is defined
by ( [Hel94], Ch. III, $\S$ 1)

(1.14)
$\mathcal{F}f(\lambda,kM)=\int_{G}f(x)~{}e^{(i\lambda-\rho)(H(x,kM))}dx$

where the function $H:G\times K/M\mapsto\mathfrak{a}$ is given by
$H(x,kM)=\mathcal{H}(x^{-1}k)$. For the sake of simplicity we fix the
notational convention $\mathcal{F}f(\lambda,kM)=\mathcal{F}f(\lambda,k)$. We
should note that, for a bi-$K$-invariant function (that is a
left-$K$-invariant function) $g$ on $X$, the HFT reduces to the spherical
transform:
$\mathcal{F}g(\lambda,k)=\mathcal{F}g(\lambda,e)=\mathcal{S}g(\lambda)$.
The inversion formula for HFT [Hel94, Ch.-III, Theorem 1.3] for
$f\in\mathcal{D}(X)$ is as follows:

(1.15)
$f(x)=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\int_{K}\mathcal{F}f(\lambda,k)~{}e^{-(i\lambda+\rho)(\mathcal{H}(x^{-1}k))}|\mathbf{c}(\lambda)|^{-2}d\lambda~{}dk.$

Let $\delta$ be a unitary irreducible representation of K realized on a finite
dimensional vector space $V_{\delta}$ with an inner product
$\langle\cdot,\cdot\rangle$. Let us denote $dimV_{\delta}=d_{\delta}$. We
denote by $\widehat{K}$ the set of equivalence classes of unitary irreducible
representations of $K$ and by customary abuse of notation regard each element
of $\widehat{K}$ as a representation from its equivalence class. For each
$\delta\in\widehat{K}$, let $\chi_{\delta}$ stand for the character of the
representation $\delta$ and $V_{\delta}^{M}=\\{v\in
V_{\delta}~{}|~{}\delta(m)v=v\mbox{~{}for all~{}}m\in M\\}$ is the subspace of
$V_{\delta}$ fixed under $\delta|_{M}$. Let $\widehat{K}_{M}$ stands for the
subset of $\widehat{K}$ consisting of $\delta$ for which
$V_{\delta}^{M}\neq\\{0\\}$ and we will mostly be interested in
representations $\delta\in\widehat{K}_{M}$. We set an orthogonal basis
$\\{v_{j}\\}_{1\leq j\leq d_{\delta}}$ of $V_{\delta}$ and we assume that
$\\{v_{1},\cdots v_{\ell_{\delta}}\\}$ generates $V_{\delta}^{M}$ where
$dimV_{\delta}^{M}=\ell_{\delta}$.

We also define a norm for each unitary irreducible representation of $K$. Let
$\Theta$ be the restriction of the Cartan-Killing form $\mathfrak{B}$ to
$\mathfrak{k}\times\mathfrak{k}$. Let $\mathcal{K}_{1},...,\mathcal{K}_{r}$ be
a basis for $\mathfrak{k}$ over $\mathbb{R}$ orthonormal with respect to
$\Theta$. Let
$\omega_{\mathfrak{k}}=-(\mathcal{K}_{1}^{2}+...+\mathcal{K}_{r}^{2})$ be the
Casimir element of $K$. Clearly $\omega_{\mathfrak{k}}$ is a differential
operator which commutes with both left and right translations of $K$. Thus
$\delta(\omega_{\mathfrak{k}})$ commutes with $\delta(k)$ for all $k\in K$.
Hence by Schur’s lemma [Sug90, Ch.I, Theorem 2.1]:
$\delta(\omega_{\mathfrak{k}})=c(\delta)\delta(e)$ where
$c(\delta)\in\mathbb{C}$. As $\delta(\mathcal{K}_{i})$ $(1\leq i\leq r)$ are
skew-adjoint operators, $c(\delta)$ is real and $c(\delta)\geq 0$. We define
$|\delta|^{2}=c(\delta)$, for $\delta\in\widehat{K}_{M}$. As,
$\delta\in\widehat{K}_{M}$, $\delta(k)$ is a unitary matrix of order
$d_{\delta}\times d_{\delta}$. So
$\|\delta(k)\|_{\mathbf{2}}=\sqrt{d_{\delta}}$ where $\|\cdot\|_{\mathbf{2}}$
denotes the Hilbert Schmidt norm. Also, from Weyl’s dimension formula we can
choose an $r\in\mathbb{Z}^{+}$ and a positive constant $c$ independent of
$\delta$ such that $\|\delta(k)\|_{\mathbf{2}}\leq c(1+|\delta|)^{r}$ for all
$k\in K$. Thus, $d_{\delta}\leq c^{\prime}(1+|\delta|)^{2r}$ with
$c^{\prime}>0$ independent of $\delta$.
For any $f\in\mathcal{C}^{\infty}(X)$ we put:

(1.16) $f^{\delta}(x)=d_{\delta}\int_{K}f(kx)\delta(k^{-1})dk.$

Clearly, $f^{\delta}$ is a $\mathcal{C}^{\infty}$ map from $X$ to
$Hom(V_{\delta},V_{\delta})$ satisfying

(1.17) $f^{\delta}(kx)=\delta(k)f^{\delta}(x),~{}~{}\mbox{for all~{}}x\in
X,k\in K.$

Any function satisfying the property (1.17) will be referred to as (a
$d_{\delta}\times d_{\delta}$ matrix valued) left $\delta$-type function. For
any function space $\mathcal{E}(X)\subseteq\mathcal{C}^{\infty}(X)$, we write
$\mathcal{E}_{\delta}(X)=\\{f^{\delta}~{}|~{}f\in\mathcal{E}(X)\\}$. We shall
denote by $\check{\delta}$ the contragradient representation of the
representation $\delta\in\widehat{K}_{M}$. and a function $f$ will be called a
scalar valued left $\check{\delta}$-type function if $f\equiv
d_{\delta}\chi_{\delta}\ast f$, where the operation $\ast$ is the convolution
over $K$. For any class of scalar valued functions $\mathcal{G}(X)$ we shall
denote

$\mathcal{G}(\check{\delta},X)=\\{g\in\mathcal{G}(X)~{}|~{}g\equiv
d_{\delta}\chi_{\delta}\ast g\\}.$

The following theorem, due to Helgason, identifies the two classes
$\mathcal{D}_{\delta}(X)$ and $\mathcal{D}(\check{\delta},X)$ corresponding to
each $\delta\in\widehat{K}_{M}$.

######  Proposition 1.1.

_[Helgason[Hel94, Ch.III, Proposition 5.10]]_
The map $\mathcal{Q}:f\mapsto g$, $g(x)=tr\left(f(x)\right)$ $(x\in X)$ is a
homeomorphism from $\mathcal{D}_{\delta}(X)$ onto
$\mathcal{D}(\check{\delta},X)$ and its inverse is given by $g\mapsto
f=g^{\delta}$.

######  Remark 1.2.

For each $\delta\in\widehat{K}_{M}$, the space
$\mathcal{D}(X,Hom(V_{\delta},V_{\delta}))$ of $\mathcal{C}^{\infty}$
functions on $X$ taking values in $Hom(V_{\delta},V_{\delta})$, carries the
inductive limit topology of the Fréchet spaces

$\mathcal{D}^{R}(X,Hom(V_{\delta},V_{\delta}))=\\{F\in\mathcal{D}(X,Hom(V_{\delta},V_{\delta}))~{}|~{}suppF\subseteq\overline{B^{R}(0)}\\},$

for $R=0,1,2,\cdots.$ As $\mathcal{D}(\check{\delta},X)\subset\mathcal{D}(X)$,
so the natural topology of $\mathcal{D}(\check{\delta},X)$ is the inherited
subspace topology.

A consequence of the Peter-Weyl theorem can be stated [Hel, Ch.IV, Corollary
3.4] in the form that any $f\in\mathcal{C}^{\infty}(X)$ has the decomposition

(1.18) $f(x)=\sum_{\delta\in\widehat{K}_{M}}tr(f^{\delta}(x)).$

A function $f\in\mathcal{C}^{\infty}(X)$ is said to be ‘left-$K$ finite’ if
there exists a finite subset $\Gamma(f)\subset\widehat{K}_{M}$ (depending on
the function $f$) such that $tr(f^{\gamma}(\cdot))\equiv 0$ for all
$\gamma\in\widehat{K}_{M}\setminus\Gamma(f)$. For any class
$\mathfrak{H}(X)\subseteq\mathcal{C}^{\infty}(X)$ of function we shall denote
$\mathfrak{H}(X)_{K}$ for its left $K$ finite subclass. Let $\Gamma$ be a
fixed subset (finite or infinite) of $\widehat{K}_{M}$. Then we shall use the
notation $\mathfrak{H}_{\Gamma}(X)$ for the subclass of $\mathfrak{H}(X)$

(1.19)
$\mathfrak{H}_{\Gamma}(X)=\\{g\in\mathfrak{H}(X)~{}|~{}g^{\delta}(\cdot)\equiv
0,\mbox{~{}for all~{}}\delta\in\widehat{K}_{M}\setminus\Gamma\\}.$

For each $f\in\mathcal{D}(X)$ and $\delta\in\widehat{K}_{M}$, we define the
$\delta$ projection of it’s HFT $\mathcal{F}f$ as follows:

(1.20)
$\left(\mathcal{F}f\right)^{\delta}(\lambda,k)=d_{\delta}\int_{K}\mathcal{F}f(\lambda,k_{1}k)\delta(k_{1}^{-1})dk_{1},~{}~{}\mbox{where}~{}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}~{}\mbox{and}~{}k\in
K.$

The HFT $\mathcal{F}(f^{\delta})$ of $f^{\delta}$ is also defined by the
formula (1.14), in this case the integration is taken over each matrix entry.

######  Lemma 1.3.

For each $f\in\mathcal{D}(X)$ and $\delta\in\widehat{K}_{M}$ the following are
true

  1. (i)

$\left(\mathcal{F}f\right)^{\delta}(\lambda,k)=\delta(k)\left(\mathcal{F}f\right)^{\delta}(\lambda,e),$

  2. (ii)

$\mathcal{F}(f^{\delta})(\lambda,k)=\left(\mathcal{F}f\right)^{\delta}(\lambda,k),\mbox{~{}for
all}~{}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}~{}\mbox{and}~{}k\in K.$

###### Proof.

Part (i) of the Lemma follows trivially from (1.20). Part (ii) can be deduced
from the following

(1.21) $\displaystyle\mathcal{F}(f^{\delta})(\lambda,kM)$ $\displaystyle=$
$\displaystyle\int_{X}f^{\delta}(x)e^{(i\lambda-1)\mathcal{H}(x^{-1}k)}dx,$
$\displaystyle=$ $\displaystyle
d_{\delta}\int_{X}\left\\{\int_{K}f(k_{1}x)\delta(k_{1}^{-1})dk_{1}\right\\}e^{(i\lambda-1)\mathcal{H}(x^{-1}k)}dx$

Now the desired result follows from (1.21) by a simple application of the
Fubini’s theorem.

##  2\.  The $\delta$-spherical transform

Let us now define the ‘$\delta$-spherical transform’ on
$\mathcal{D}_{\delta}(X)$. Most of the basic analysis was done by Helgason
[Hel94] on $\mathcal{D}({\check{\delta}},X)$, we shall follow those results
closely and prove them on $\mathcal{D}_{\delta}(X)$ using the homeomorphism
$\mathcal{Q}$, defined in Proposition 1.1.

######  Definition 2.1.

For $f\in\mathcal{D}_{\delta}(X)$ the $\delta$-spherical transform
$\widetilde{f}$ is an operator valued function on
$\mathfrak{a}^{*}_{\mathbb{C}}$ and is given by

(2.1)
$\widetilde{f}(\lambda)=d_{\delta}\int_{G}trf(x)\Phi_{\overline{\lambda},\delta}^{*}(x)dx$

where for each $\delta\in\widehat{K}_{M}$ and
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$, the function

(2.2)
$\Phi_{\lambda,\delta}(x)=\int_{K}e^{-(i\lambda+1)H(x^{-1}k)}\delta(k)dk,\hskip
14.45377ptx\in G,$

is called the ‘generalized spherical function’ of class $\delta$. For each
$x\in G$, $\Phi_{\lambda,\delta}(x)$ is an operator in
$Hom(V_{\delta},V_{\delta})$. Taking point-wise adjoint leads to the
expression

(2.3)
$\Phi_{\overline{\lambda},\delta}^{*}(x):=\Phi_{\overline{\lambda},\delta}^{*}(x)=\int_{K}e^{(i\lambda-1)H(x^{-1}k)}\delta(k^{-1})dk,\hskip
14.45377ptx\in G.$

######  Remark 2.2.

From the Iwasawa decomposition, if $x\in G$ and $\tau\in K$, $\mathcal{H}(\tau
x)=\mathcal{H}(x)$. Hence, the expressions (2.2) and (2.3) show that both
$\Phi_{\lambda,\delta}$ and $\Phi_{\overline{\lambda},\delta}^{*}$ can be
considered as functions on the space $X=G/K$.

In the following proposition we list out some basic properties of the
generalized spherical functions that we will be using.

######  Proposition 2.3.

  1. (i)

For each $x\in X$, the function $\lambda\mapsto\Phi_{\lambda,\delta}(x)$ is
holomorphic on $\mathfrak{a}^{*}_{\mathbb{C}}$.

  2. (ii)

Let $\delta\in\widehat{K}_{M}$ and $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$.
Then for each $x\in X$ and $k\in K$ we have

(2.4)
$\Phi_{{\lambda},\delta}(kx)=\delta(k)\Phi_{{\lambda},\delta}(x)\mbox{~{}~{}and~{}~{}}\Phi_{\overline{\lambda},\delta}^{*}(kx)=\Phi_{\overline{\lambda},\delta}^{*}(x)\delta(k^{-1}).$

Let $v\in V_{\delta}$ and $m\in M$ then

(2.5)
$\delta(m)\left(\Phi_{\overline{\lambda},\delta}^{*}(x)v\right)=\Phi_{\overline{\lambda},\delta}^{*}(x)v.$

  3. (iii)

_[Helgason[Hel94, Ch.III, Theorem 5.15 ]]_ For each
$\delta\in\widehat{K}_{M}$, $\omega\in W$ and for all
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$, the restrictions
$\Phi_{\lambda,\delta}|_{A}$ and $\Phi_{\overline{\lambda},\delta}^{*}|_{A}$
satisfy the relations

(2.6) $\displaystyle\Phi_{\lambda,\delta}|_{A}Q^{\delta}(\lambda)$
$\displaystyle=\Phi_{\omega\lambda,\delta}|_{A}Q^{\delta}(\omega\lambda),$
(2.7) $\displaystyle
Q^{\delta}(\lambda)^{-1}\Phi_{\overline{\lambda},\delta}^{*}|_{A}$
$\displaystyle=Q^{\delta}(\omega\lambda)^{-1}\Phi_{-\overline{\omega\lambda},\delta}^{*}|_{A}$

where $Q^{\delta}(\lambda)$ is a $(\ell_{\delta}\times\ell_{\delta})$ matrix
whose entries are certain constant coefficient polynomials in
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ (see [Hel94, Ch. III, §2] for
details ). Furthermore, both sides of (2.7) are holomorphic for all
$\lambda\in\mathbb{C}$, implying that
$\Phi_{\overline{\lambda},\delta}^{*}|_{A}$ is divisible by
$Q^{\delta}(\lambda)$ in the ring of entire functions.

  4. (iv)

For each fixed $\lambda$ and $\delta$, the function
$\Phi_{{\lambda},\delta}(x)$ and its adjoint are both joint eigenfunctions of
all $G$-invariant differential operators of $X$. Particularly, for the
Laplace-Beltrami operator $\mathbf{L}$, the eigenvalues are as follows:

(2.8)
$\left(\mathbf{L}\Phi_{\lambda,\delta}\right)(x)=-\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)\Phi_{\lambda,\delta}(x),~{}~{}x\in
X.$

  5. (v)

For each $\delta\in\widehat{K}_{M}$, the generalized spherical function
corresponding to $\delta$ is related with the elementary spherical function by
the following differential equation [Hel94, Ch.III, §5, Corollary 5.17]

(2.9)
$\Phi_{\lambda,\delta}(gK)|_{V_{\delta}^{M}}=\left(\mathbf{D}^{\delta}\varphi_{\lambda}\right)(g)Q^{\delta}(\lambda)^{-1}$

where $\mathbf{D}^{\delta}$ is a differential operator matrix of order
$(d_{\delta}\times\ell_{\delta})$. Individual matrix entries of
$\mathbf{D}^{\delta}$ are certain constant coefficient differential operators
on $G$.

  6. (vi)

For any
$\textbf{g}_{1},\textbf{g}_{2}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ there
exist constants $c=c(\textbf{g}_{1},\textbf{g}_{2})$,
$b=b(\textbf{g}_{1},\textbf{g}_{2})$ and $c_{0}>0$ so that

(2.10)
$\|\Phi_{\lambda,\delta}(\textbf{g}_{1},x,\textbf{g}_{2})\|_{\mathbf{2}}\leq
c(1+|\delta|)^{b}(1+\|\lambda\|)^{b}\varphi_{0}(x)e^{c_{0}\|\Im{\lambda}\|(1+|x|)},$

for all $x\in X$ and $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$.

###### Proof.

Property (2.4) follows trivially from the definition of the generalized
spherical function. (2.5) also follows from (2.2) as below:

$\displaystyle\delta(m)\left(\Phi_{\overline{\lambda},\delta}^{*}(x)v\right)$
$\displaystyle=\left\\{\int_{K}e^{(i\lambda-1)\mathcal{H}(x^{-1}k)}\delta(mk^{-1})dk\right\\}v$
$\displaystyle=\left\\{\int_{K}e^{(i\lambda-1)\mathcal{H}(x^{-1}k^{\prime}m)}\delta(k^{\prime-1})dk^{\prime}\right\\}v.$

The last line follows by a simple change of variable $mk^{-1}$ to
$k^{\prime-1}$. In the last expression above, let
$x^{-1}k^{\prime}=\mathcal{K}(x^{-1}k^{\prime})(\exp{\mathcal{H}(x^{-1}k^{\prime})})n^{\prime}$
for some $n^{\prime}\in N$. As $M$ normalizes $N$ and centralizes $A$ we have

$x^{-1}k^{\prime}m=\mathcal{K}(x^{-1}k^{\prime})m(\exp{\mathcal{H}(x^{-1}k^{\prime})})\mathcal{N}(x^{-1}k^{\prime}).$

This shows that
$\mathcal{H}(x^{-1}k^{\prime})=\mathcal{H}(x^{-1}k^{\prime}m)$. Thus

$\displaystyle\delta(m)\left(\Phi_{\overline{\lambda},\delta}^{*}(x)v\right)$
$\displaystyle=\left\\{\int_{K}e^{(i\lambda-1)\mathcal{H}(x^{-1}k^{\prime})}\delta(k^{\prime-1})dk^{\prime}\right\\}v=\Phi_{\overline{\lambda},\delta}^{*}(x)v.$

A proof of property (ii) may be found in [Hel94, Ch.III, §1 (6)] and [Hel,
Ch.II, Corollary 5.20]. The estimate (2.10) is a work of Arthur [Art79].

######  Remark 2.4.

The property (2.5) clearly shows that for each $x\in X$ the operator
$\Phi_{\overline{\lambda},\delta}^{*}(x)$ maps $V_{\delta}$ to
$V_{\delta}^{M}$. Hence $\Phi_{\overline{\lambda},\delta}^{*}(x)$ is a
$d_{\delta}\times d_{\delta}$ matrix whose only the first $\ell_{\delta}$ rows
can nonzero. Consequently, for each $x\in X$, $\Phi_{\lambda,\delta}(x)$ is a
$d_{\delta}\times d_{\delta}$ matrix of which only the first $\ell_{\delta}$
columns can be nonzero. In other words, the operator
$\Phi_{\lambda,\delta}(x)$ vanishes identically on the orthogonal complement
of the subspace $V_{\delta}^{M}$.

######  Remark 2.5.

In the case of the rank-one symmetric spaces the Kostant matrix $Q^{\delta}$
reduces to a constant coefficient polynomial (Kostant polynomial) [Hel94,
Theorem 11.2,S11, Ch.III]. The degree of the Kostant polynomials depends on
the choice of $\delta\in\widehat{K}_{M}$. Furthermore all the zeros of the
Kostant polynomials lie on the open lower half of the imaginary axis.
The general rank analogue of the above result states that $\det
Q^{\delta}(\lambda)\neq 0$ for all
$\lambda\in\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}$. This is an easy
consequence of Lemma 2.11 and Proposition 4.1 of [Hel94, Ch.III].

######  Lemma 2.6.

If $f\in\mathcal{D}_{\delta}(X)$, where $\delta\in\widehat{K}_{M}$, then
$\mathcal{F}f(\lambda,e)=\widetilde{f}(\lambda)$ for all
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$.

###### Proof.

For any $f\in\mathcal{D}_{\delta}(X)$, using the topological isomorphism
$\mathcal{Q}$ as described in Proposition 1.1, we get
$trf(\cdot)\in\mathcal{D}(\check{\delta},X)$ and also
$f(x)=d_{\delta}\int_{K}trf(kx)\delta(k^{-1})dk$. Now from the definition of
HFT (1.14) we get:

$\displaystyle\mathcal{F}f(\lambda,e)$
$\displaystyle=\int_{G}f(x)e^{(i\lambda-\rho)\mathcal{H}(x^{-1}e)}dx,$ (2.11)
$\displaystyle=d_{\delta}\int_{G}\int_{K}trf(kx)\delta(k^{-1})dke^{(i\lambda-\rho)\mathcal{H}(x^{-1}e)}dx.$

A simple application of the Fubini theorem and a substitution $kx=y$ in the
integrand of (2) gives the following.

$\displaystyle\mathcal{F}f(\lambda,e)$
$\displaystyle=d_{\delta}\int_{G}trf(y)\left\\{\int_{K}e^{(i\lambda-\rho)\mathcal{H}(y^{-1}k)}\delta(k^{-1})dk\right\\}dy,$
$\displaystyle=d_{\delta}\int_{G}trf(y)\Phi_{\overline{\lambda},\delta}^{*}(y)dy=\widetilde{f}(\lambda).$

######  Lemma 2.7.

Let $f\in\mathcal{D}_{\delta}(X)$, then the inversion formula for the
$\delta$-spherical transform (Definition 2.1) is given by:

(2.12)
$f(x)=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)\widetilde{f}(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda.$

Furthermore we get:
$\int_{G}{\|f(x)\|_{\mathbf{2}}}^{2}dx=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}{\|\widetilde{f}(\lambda)\|_{\mathbf{2}}}^{2}~{}|\mathbf{c}(\lambda)|^{-2}d\lambda.$

###### Proof.

The formula (2.12) is derived from the inversion formula (1.15) of the HFT.

$\displaystyle f(x)$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\int_{K}\mathcal{F}f(\lambda,k)e^{-(i\lambda+\rho)(\mathcal{H}(x^{-1}k))}|\mathbf{c}(\lambda)|^{-2}dk~{}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\int_{K}\delta(k)\mathcal{F}f(\lambda,e)e^{-(i\lambda+\rho)(\mathcal{H}(x^{-1}k))}|\mathbf{c}(\lambda)|^{-2}dk~{}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\left\\{\int_{K}e^{-(i\lambda+\rho)(\mathcal{H}(x^{-1}k))}\delta(k)dk\right\\}\widetilde{f}(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)\widetilde{f}(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda.$

The second and the third line of the above deduction are consequences of Lemma
1.3 and Lemma 2.6 respectively.
The second relation of this Lemma also follows from the Plancherel formula of
the HFT ([Hel94], Ch.-III, $\S$1, Theorem 1.5 ). For
$f\in\mathcal{D}_{\delta}(X)$, the HFT is defined matrix entry wise, hence in
this case the Plancherel formula is given by:

(2.13)
$\int_{G}{\|f(x)\|_{\mathbf{2}}}^{2}dx=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\int_{K}{\|\mathcal{F}f(\lambda,k)\|_{\mathbf{2}}}^{2}|\mathbf{c}(\lambda)|^{-2}dk~{}d\lambda,$

which follows easily from the classical Plancherel formula given in [Hel94].
The required Plancherel formula for the $\delta$-spherical transform follows
from (2.13) by using relation (i) of Lemma 1.3 and Schur’s Orthogonality
relation ([Sug90], Theorem 3.2).

We shall next deduce the deduce a topological Paley-Wiener (P-W) theorem for
the $\delta$-spherical transform (2.1).
A holomorphic function $\psi:\mathfrak{a}^{*}_{\mathbb{C}}\longrightarrow
Hom(V_{\delta},V_{\delta}^{M})$ is said to be of ‘exponential type-$R$’
($R\in\mathbb{R}^{+}$) if

$\sup_{\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}}e^{-R\|\Im\lambda\|}(1+\|\lambda\|)^{N}\|\psi(\lambda)\|_{\mathbf{2}}<+\infty\mbox{\hskip
36.135ptfor each~{} }N\in\mathbb{Z}^{+}.$

Let $\mathcal{H}_{\delta}^{R}(\mathfrak{a}^{*}_{\mathbb{C}})$ be the class of
all $Hom(V_{\delta},V_{\delta}^{M})$ valued exponential type-$R$ functions on
$\mathfrak{a}^{*}_{\mathbb{C}}$ and further let
$\mathcal{H}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})=\bigcup_{R>0}\mathcal{H}^{R}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$.

######  Theorem 2.8.

For each fixed $\delta\in\widehat{K}_{M}$, the $\delta$-spherical transform
given by (2.1) is a homeomorphism between the spaces $\mathcal{D}_{\delta}(X)$
and $\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$, where

(2.14)
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})=\left\\{\xi\in\mathcal{H}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})~{}|~{}\lambda\mapsto
Q^{\delta}(\lambda)^{-1}\xi(\lambda)~{}\mbox{is an}~{}W\mbox{-invariant entire
function}\right\\}.$

Here $Q^{\delta}(\lambda)$ is the matrix of constant coefficient polynomials
appeared in the expressions (2.6) and (2.7).

###### Proof.

Our proof solely relies on the proof of the topological Paley-Wiener theorem
given by Helgason ([Hel94], Ch.-III, Theorem 5.11), where he characterized the
image of the space $\mathcal{D}(\check{\delta},X)$ under the transform
$f\mapsto\widehat{f}$, where

(2.15)
$\widehat{f}(\lambda)=d_{\delta}\int_{G}f(x)~{}\Phi_{\overline{\lambda},\delta}^{*}(x)dx,~{}~{}~{}(\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}).$

Helgason proved that the above transform is a topological isomorphism between
the spaces $\mathcal{D}(\check{\delta},X)$ and
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$. From the Proposition
1.1 and the definition (2.1) of the $\delta$-spherical transform we get: for
each $f\in\mathcal{D}_{\delta}(X)$,
$\widehat{(\mathcal{Q}f)}(\lambda)=\widetilde{f}(\lambda)$,
$(\forall\lambda\in\mathfrak{a}^{*}_{\mathbb{C}})$. The proposition now
follows from the fact that both the maps $\mathcal{Q}$ and
$f\mapsto\widehat{f}$ are homeomorphisms.

Let us now consider the function space
$\mathcal{P}_{0}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})=\left\\{h\in\mathcal{H}(\mathfrak{a}^{*}_{\mathbb{C}})|~{}h~{}\mbox{is
}~{}W\mbox{-invariant}\right\\}$ with the relative topology. A function
$h\in\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$ can be written
as $h\equiv\left(h_{ij}\right)_{{\ell_{\delta}}\times d_{\delta}}$ where each
of the scalar valued component function $h_{ij}$ is entire, $W$-invariant and
of exponential type. Let $\mathcal{D}(G//K)$ and
$\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ are respectively be the
spaces of scalar valued and $Hom(V_{\delta},V_{\delta}^{M})$ valued
bi-$K$-invariant, compactly supported, $\mathcal{C}^{\infty}$ functions on
$G$. The Paley-Wiener theorem for the bi-$K$-invariant functions gives an
unique $f_{ij}\in\mathcal{D}(G//K)$ such that $\mathcal{S}f_{ij}=h_{ij}$. We
set $f\equiv\left(f_{ij}\right)_{{\ell_{\delta}}\times
d_{\delta}}\in\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ and define the
spherical transform $\mathcal{S}f$ matrix entry wise to get $\mathcal{S}f=h$.
Moreover by suitably modifying the proof of the P-W theorem for the
bi-$K$-invariant functions one can show that $\mathcal{S}$ is infact a
homeomorphism between the spaces
$\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ and
$\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$. The following Lemma
identifies the two PW-spaces
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ and
$\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$.

######  Lemma 2.9.

_[Helgason[Hel94, Ch.-III, $\S$5, Lemma 5.12]]_
The mapping $\psi(\lambda)\mapsto Q^{\delta}(\lambda)\psi(\lambda)$
($\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$) is a homeomorphism from
$\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$ onto
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$.

######  Lemma 2.10.

Any function $f\in\mathcal{D}_{\delta}(X)$ can be written as
$f(x)=\mathbf{D}^{\delta}\phi(x)$ ($\forall x\in G$), where
$\phi\in\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ and
$\mathbf{D}^{\delta}$ is the ${d_{\delta}}\times{\ell_{\delta}}$ matrix of
constant coefficient differential operators as mentioned in (2.9).

###### Proof.

Let $f\in\mathcal{D}_{\delta}(X)$, then by Theorem 2.8, its $\delta$-spherical
transform
$\widetilde{f}\in\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$. Using
the homeomorphism given in Lemma 2.9, we get an unique function
$\lambda\mapsto\Phi(\lambda)=Q^{\delta}(\lambda)^{-1}\widetilde{f}(\lambda)$
in $\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$. By the PW-
theorem for the bi-$K$-invariant functions we get a function
$\phi\in\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ such that:

(2.16)
$\phi(x)=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\varphi_{\lambda}(x)\Phi(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda.$

Now by applying the differential operator $\mathbf{D}^{\delta}$ on the both
sides of (2.16) and by using the absolute convergence of the integral in
(2.16) we get:

$\displaystyle\left(\mathbf{D}^{\delta}\phi\right)(x)$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\left(\mathbf{D}^{\delta}\varphi_{\lambda}(x)\right)\Phi(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)Q^{\delta}(\lambda)\Phi(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda,$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)\widetilde{f}(\lambda)|\mathbf{c}(\lambda)|^{-2}d\lambda=f(x).$

The second line of the above calculation is a consequence of (2.9) and the
last line is merely the inversion formula (2.12).

We conclude this section with a ‘product formula’, due to Helgason, for the
polynomial $\det Q^{\delta}(\lambda)$ with
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$. First we shall characterize the set
$\widehat{K_{\beta}}_{M_{\beta}}$ ( $\beta\in\Sigma^{+}_{0}$) of equivalence
classes of representations of $K_{\beta}$ as certain restrictions of the
representations $\delta\in\widehat{K}_{M}$. Let $\delta\in\widehat{K}_{M}$ and
$V_{\delta}$ and $V_{\delta}^{M}$ be as explained earlier. Let $V$ denote the
$K_{\beta}M$-invariant subspace of $V_{\delta}$ generated by $V_{\delta}^{M}$.
Then $V$ decomposes into $K_{\beta}M$-irreducible subspaces as
$V=\bigoplus_{i=1}^{\ell_{\delta}}V_{i}.$ Let $\delta(i,\beta)$ be the
representation of $K_{\beta}M$ on $V_{i}$ given by $\delta$. Then each
$\delta(i,\beta)$ is irreducible except when $dimK_{\beta}=1$ and
$mkm^{-1}=k^{-1}$ for all $k\in K_{\beta}$ and some $m\in M$ [Hel94, Ch. III,
Lemma 3.9 - 3.11 ]. In this case $\delta(i,\beta)$ brakes up into two
irreducible one dimensional representations as
$\delta(i,\beta)=\delta(i,\beta)_{0}\oplus\delta(i,\beta)_{0}\check{}$ where
$\delta(i,\beta)_{0}\check{}$ is the contragradient representation of
$\delta(i,\beta)_{0}$ and in this particular case we choose
$\delta(i,\beta)_{0}\check{}$ as $\delta(i,\beta)$.

######  Lemma 2.11.

_Helgason[Hel94, Ch. III, Prposition 4.3]_ For each $\beta\in\Sigma_{0}^{+}$,
as $\delta$ runs through $\widehat{K}_{M}$, the representations
$\delta(i,\beta)$ ($1\leq i\leq\ell$) runs through all of
$\widehat{K_{\beta}}_{M_{\beta}}$.

Let $Q^{\delta(i,\beta)}_{\beta}(\lambda_{\beta})$ be the Kostant polynomial
for the rank-one symmetric space $G_{\beta}/K_{\beta}$ corresponding to the
representation $\delta(i,\beta)\in\widehat{K_{\beta}}_{M_{\beta}}$. Then the
determinant of the Kostant matrix $Q^{\delta}(\lambda)$
($\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$) can be represented by a product
formula [Hel94, Ch. III, §3, (50) and §4, Theorem 4.2 ]

(2.17) $\det
Q^{\delta}(\lambda)=\mathcal{C}_{\delta}\prod_{\beta\in\Sigma_{0}^{+},~{}1\leq
i\leq\ell_{\delta}}Q^{\delta(i,\beta)}_{\beta}(\lambda_{\beta}),\mbox{~{}for
all~{}}\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$

where $\mathcal{C}_{\delta}$ is a nonzero constant depending on
$\delta\in\widehat{K}_{M}$.

##  3\. $L^{p}$-Schwartz spaces

######  Definition 3.1.

_[Classical $L^{p}$ Schwartz space]_
A $\mathcal{C}^{\infty}$ function $f$ on $X$ is said to be in the
$L^{p}$-Schwartz space ( $0<p\leq 2$ ) $\mathcal{S}^{p}(X)$ if for each
nonnegative integer $n$ and $D,E\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ the
function $f$ satisfies the following decay condition:

(3.1) $\mu_{D,E,n}(f)=\sup_{x\in
G}|f(D,x,E)|~{}\varphi_{0}^{-\frac{2}{p}}(x)~{}(1+|x|)^{n}~{}<+\infty.$

The topology induced by the countable family of seminorms $\mu_{D,E,n}(\cdot)$
makes $\mathcal{S}^{p}(X)$ a Fréchet space. It can be shown that
$\mathcal{D}(X)$ is a dense subspace of $\mathcal{S}^{p}(X)$ for $0<p\leq 2$.
Let $f\in\mathcal{S}^{p}(X)$, we take its $\delta$-projection $f^{\delta}$ as
defined in (1.16). Then $f^{\delta}$ is a left $\delta$-type
$Hom(V_{\delta},V_{\delta})$ valued function with a decay

(3.2) $\mu_{D,E,n}(f^{\delta})=\sup_{x\in
G}\|f^{\delta}(D,x,E)\|_{\mathbf{2}}~{}\varphi_{0}^{-\frac{2}{p}}(x)~{}(1+|x|)^{n}~{}<+\infty,$

where $\|\cdot\|_{\mathbf{2}}$ denotes the Hilbert Schmidt norm. Let us denote
$\mathcal{S}^{p}_{\delta}(X)$ for the class of left $\delta$-type
$Hom(V_{\delta},V_{\delta})$ valued functions with the decay (3.2). For each
$h\in\mathcal{S}^{p}_{\delta}(X)$, the scalar valued function $trh(\cdot)$
satisfies $trh\equiv d_{\check{\delta}}(\chi_{\check{\delta}}\ast trh)$ and
the decay (3.1). Let
$\mathcal{S}^{p}(\check{\delta},X)\subset\mathcal{S}^{p}(X)$ be the class of
all scalar valued left-$\check{\delta}$ type Schwartz class functions. Both
the spaces $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}^{p}(\check{\delta},X)$ becomes Frèchet spaces with the topologies
induced by the family of seminorms given in (3.2) and (3.1) respectively.
Moreover, $\mathcal{D}_{\delta}(X)$ and $\mathcal{D}(\check{\delta},X)$ are
respectively dense subspaces of $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}^{p}(\check{\delta},X)$ in the respective Schwartz space
topologies. The topological isomorphism $\mathcal{Q}$ in Proposition 1.1 can
be extended between the Schwartz spaces $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}^{p}(\check{\delta},X)$.

Next we shall to extend the definition of the $\delta$-spherical transform to
the Schwartz space $\mathcal{S}^{p}_{\delta}(X)$ ($0<p\leq 2$). The spherical
transform (1.11) defined on $\mathcal{D}(G//K)$ can be extended to the
$L^{p}$-Schwartz spaces $\mathcal{S}^{p}(G//K)$ of bi-$K$-invariant functions
on the group $G$. The image $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$
(defined below) of $\mathcal{S}^{p}(G//K)$ under the spherical transform is
again a Schwartz class of functions defined o the complex tube
$\mathfrak{a}^{*}_{\varepsilon}=\mathfrak{a}^{*}+iC^{\varepsilon\rho}$ where
$\varepsilon=\left(\frac{2}{p}-1\right)$ and

$C^{\varepsilon\rho}=\\{\lambda\in\mathfrak{a}^{*}~{}|~{}\omega\lambda(H)\leq\varepsilon\rho(H)\mbox{~{}for
all~{}}H\in\overline{\mathfrak{a}^{+}}\mbox{~{}and~{}}\omega\in W\\}.$

######  Definition 3.2.

The space $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$ consists of the
complex valued functions $h$ on $\mathfrak{a}^{*}_{\varepsilon}$ such that:

  1. (i)

$h$ is holomorphic in the interior of the tube
$\mathfrak{a}^{*}_{\varepsilon}$ and continues on the closed tube,

  2. (ii)

$h$ is $W$-invariant,

  3. (iii)

for any polynomial $P$ in the algebra $S(\mathfrak{a})$ of the symmetric
polynomials on $\mathfrak{a}^{*}$ and any positive integer $r$

(3.3) $\tau_{P,r}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}(1+\|\lambda\|)^{r}\left|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right|<+\infty.$

The countable family $\left\\{\tau_{P,r}~{}|~{}P\in
S(\mathfrak{a}),r\in\mathbb{Z}^{+}\cup\\{0\\}\right\\}$ gives a Frèchet norm
on the space $S(\mathfrak{a}^{*}_{\varepsilon})$.

######  Remark 3.3.

The topology of $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$ can also be
given by the following equivalent family of seminorms.

(3.4) $\tau_{P,r}^{+}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}\cap\left(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}\right)}(1+\|\lambda\|)^{r}\left|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right|<+\infty.$

This is an easy consequence of (ii) of Definition 3.2 and the fact
$\|\omega\lambda\|=\|\lambda\|$ for all
$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ and $\omega\in W$.

We now state the $L^{p}$-Schwartz space isomorphism theorem for the
bi-$K$-invariant functions.

######  Theorem 3.4.

The spherical transform $f\mapsto\mathcal{S}f$ defined in (1.11) is
topological isomorphism between the spaces $\mathcal{S}^{p}(G//K)$ ($0<p\leq
2$) and $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$.

For a proof of this theorem one can see [GV88]. We are mainly interested in
Anker’s proof [Ank91] of the above theorem, which relies on the PW theorem for
the spherical transform.
Let $\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ be the space of all
$Hom(V_{\delta},V_{\delta}^{M})$ valued functions $h$ on
$\mathfrak{a}^{*}_{\varepsilon}$ satisfying (i), (ii) of Definition 3.2 along
with the decay: for each polynomial $P\in S(\mathfrak{a})$ and integer $n\geq
0$

(3.5) $\tau_{P,n}^{+}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}\cap\left(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}\right)}(1+\|\lambda\|)^{n}\left\|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right\|_{\mathbf{2}}<+\infty.$

The countable family $\\{\tau_{P,n}^{+}\\}$ induces a a Frèchet structure on
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$. The space
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ can also be viewed as a
space of $Hom(V_{\delta},V_{\delta}^{M})$ valued functions with each of its
matrix entry function in $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$. For
our purpose we shall use the following equivalent (inducing the same topology)
family of seminorms on $\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$:

(3.6) $\tau^{+*}_{P,r}(h)=\sup_{\lambda\in
Int(\mathfrak{a}^{*}_{\varepsilon}\cap(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}))}\left\|P\left(\frac{\partial}{\partial\lambda}\right)\left\\{h(\lambda)(\|\rho\|^{2}+\langle\lambda,\lambda\rangle_{1})^{r}\right\\}\right\|_{\mathbf{2}},~{}P\in
S(\mathfrak{a}),r\in\mathbb{Z}^{+}\cup\\{0\\}.$

Let $f=(f_{ij})\in\mathcal{S}^{p}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$. Then
by defining $\mathcal{S}f=(\mathcal{S}f_{ij})$ we can the following extension
of the isomorphism of Theorem 3.4.

######  Lemma 3.5.

The spherical transform is a topological isomorphism between the spaces
$\mathcal{S}^{p}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$ and
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$.

This Lemma can be proved easily by using the conclusion of the Theorem 3.4 for
each matrix entry of the functions of
$\mathcal{S}^{p}(G//K,Hom(V_{\delta},V_{\delta}^{M}))$.
We now define the ambient space for the image of the Schwartz space
$\mathcal{S}^{p}_{\delta}(X)$ under the $\delta$-spherical transform.

######  Definition 3.6.

Let $\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ be the set of all
$Hom(V_{\delta},V_{\delta}^{M})$ valued functions $h$ on the complex tube
$\mathfrak{a}^{*}_{\varepsilon}$ such that

  1. (i)

h is holomorphic in the interior of the tube $\mathfrak{a}^{*}_{\varepsilon}$
and extends as a continuous function to the closed tube.

  2. (ii)

$\lambda\mapsto Q^{\delta}(\lambda)^{-1}~{}h(\lambda)$ is $W$-invariant
holomorphic function the interior of the complex tube
$\mathfrak{a}^{*}_{\varepsilon}$.

  3. (iii)

for each polynomial $P\in S(\mathfrak{a})$ and integer $n\geq 0$

(3.7) $\nu_{P,n}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}(1+\|\lambda\|)\left\|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right\|_{\mathbf{2}}<+\infty.$

Clearly, $\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ becomes a
Frèchet space with the topology induced by the countable family
$\left\\{\nu_{P,n}\right\\}$ of seminorms.

######  Remark 3.7.

It is easy to observe that the topology of the space
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ can also be determined
by the following equivalent family of seminorms: for each polynomial $P\in
S(\mathfrak{a})$ and each nonnegative integer $n$

(3.8) $h\mapsto{\nu}^{\\#}_{P,n}(h)=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\left|P\left(\frac{\partial}{\partial\lambda}\right)\left\\{(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2})^{n}h(\lambda)\right\\}\right|.$

Let us now state the first main theorem of this paper.

######  Theorem 3.8.

The $\delta$-spherical transform (2.1) is a topological isomorphism between
the Schwartz spaces $\mathcal{S}^{p}_{\delta}(X)$ and
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ with $0<p\leq 2$ and
$\varepsilon=\left(\frac{2}{p}-1\right)$.

The following proposition is a key step to prove Theorem 3.8.

######  Proposition 3.9.

For each $\delta\in\widehat{K}_{M}$, there exists a $\delta$-dependent
constant $c_{\delta}>0$ such that

(3.9) $\inf_{\lambda\in(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}})}|\det
Q^{\delta}(\lambda)|\geq c_{\delta}.$

###### Proof.

To prove this we use the product formula (2.17) of $\det Q^{\delta}(\lambda)$.
Each of the factors $Q^{\delta(i,\beta)}_{\beta}$ is the Kostant polynomials
of he rank-one restrictions $G_{\beta}/K_{\beta}$ and hence it is a polynomial
in one complex variable
$\lambda_{\beta}\in{\mathfrak{a}^{*}_{(\beta)}}_{\mathbb{C}}\cong\mathbb{C}$.
It is easy to check that
$\lambda\in\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}$ if and only if
$\Re{\langle i\lambda,\alpha\rangle}\leq 0$ for all $\alpha\in\Sigma^{+}$.
Using the definition of the restriction
$\lambda_{\beta}=\frac{\langle\lambda,\beta\rangle_{1}}{\langle\beta,\beta\rangle}\beta$
we get

(3.10) $\displaystyle\langle i\lambda_{\beta},\beta\rangle_{\beta}$
$\displaystyle=\left\langle
i\frac{\langle\lambda,\beta\rangle_{1}}{\langle\beta,\beta\rangle}\beta,\beta\right\rangle_{\beta}=\langle
i\lambda,\beta\rangle$

where $\langle\cdot,\cdot\rangle_{\beta}$ denotes the inner product on
$\mathfrak{a}^{*}_{(\beta)}$ as well as its $\mathbb{C}$-bilinear extension to
the complexification ${\mathfrak{a}^{*}_{(\beta)}}_{\mathbb{C}}$. Now (3.10)
clearly suggests that if
$\lambda\in\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}$ then for each
$\beta\in\Sigma_{0}^{+}$ the restriction
$\lambda_{\beta}\in\mathfrak{a}_{(\beta)}^{*}+i\overline{\mathfrak{a}_{(\beta)}^{*+}}$.
As the polynomial $Q^{\delta(i,\beta)}_{\beta}(\lambda_{\beta})\neq 0$ for all
$\lambda_{\beta}\in\mathfrak{a}_{(\beta)}^{*}+i\overline{\mathfrak{a}_{(\beta)}^{*+}}$
and $\mathfrak{a}_{(\beta)}^{*}+i\overline{\mathfrak{a}_{(\beta)}^{*+}}$ being
a closed subset of $\mathbb{C}$ so we get a positive constant
$c_{\delta(i,\beta)}$ such that

$\inf_{\lambda_{\beta}\in\mathfrak{a}_{(\beta)}^{*}+i\overline{\mathfrak{a}_{(\beta)}^{*+}}}|Q^{\delta(i,\beta)}_{\beta}(\lambda_{\beta})|\geq
c_{\delta(i,\beta)}.$

Since corresponding to each $\delta\in\widehat{K}_{M}$ the product formula
(2.17) of $\det Q^{\delta}(\lambda)$ has only finitely many factors hence we
get the desired conclusion of the proposition.

The next lemma extends the homeomorphism given in Lemma 2.9 between the P-W
spaces to the corresponding Schwartz classes.

######  Lemma 3.10.

The map

(3.11) $g(\lambda)\mapsto Q^{\delta}(\lambda)g(\lambda),~{}~{}\mbox{~{}~{}for
all ~{}}\lambda\in\mathfrak{a}^{*}_{\varepsilon},$

is a homeomorphism from the space
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ onto
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.

###### Proof.

Let us first take $g\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$. We
denote $h(\cdot)=Q^{\delta}(\cdot)g(\cdot)$. We shall show that
$h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. As
$Q^{\delta}(\lambda)$ is an ${\ell_{\delta}}\times{\ell_{\delta}}$ matrix of
polynomials in $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ so $\lambda\mapsto
Q^{\delta}(\lambda)$ is a holomorphic function on
$\mathfrak{a}^{*}_{\mathbb{C}}$. Hence the function $h$ satisfies condition
(i) and (ii) of Definition 3.6 which easily follows from the similar
properties of $g\in\mathcal{S}_{0}(\mathfrak{a}_{\varepsilon})$ and the
construction (3.11) of the function $h$.
To establish the decay condition (3.7) for $h$ let us take a polynomial $P\in
S(\mathfrak{a})$ and $m\in\mathbb{Z}^{+}\cup\\{0\\}$. Then

$\displaystyle\sup_{\lambda\in Int\mathfrak{a}^{*}_{\varepsilon}}$
$\displaystyle\left\|P\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m}$
$\displaystyle\leq\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\sum_{\kappa}c_{\kappa}\left\|\left\\{P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}(\lambda)\right\\}\left\\{P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\\}\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m}$
$\displaystyle\leq\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\sum_{\kappa}c_{\kappa}\left\|\left\\{P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}(\lambda)\right\\}\right\|_{\mathbf{2}}\left\|\left\\{P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\\}\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m}$
(3.12) $\displaystyle\leq\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\sum_{\kappa}c_{\kappa}^{\delta}\left\|\left\\{P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)g\right\\}(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m_{\kappa}^{\delta}}$

where, $m_{\kappa}^{\delta}$ are nonnegative integers and
$c_{\kappa}^{\delta}$ are positive constants both depending on
$\delta\in\widehat{K}_{M}$. As
$g\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$, the right hand side of
(3) is clearly finite. Moreover, (3) shows that the map (3.11) is a continuous
function from $\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ into
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.
Now let $\psi\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ and
define $g(\cdot):=Q^{\delta}(\cdot)^{-1}\psi(\cdot)$. As, $\psi\in
S_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$, by Definition 3.6 the function

$\lambda\mapsto g(\lambda)=\frac{1}{\det
Q^{\delta}(\lambda)}Q^{\delta}_{c}(\lambda)\psi(\lambda)$

(here, $Q^{\delta}_{c}(\lambda)$ is the cofactor matrix of
$Q^{\delta}(\lambda)$) is $W$-invariant and it is holomorphic in the interior
of the tube $\mathfrak{a}^{*}_{\varepsilon}$. To infer
$g\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ all we have to show is
that the function $g$ has certain decay. Let $P\in S(\mathfrak{a})$ and $t$ be
any nonnegative integer, then

$\displaystyle\left\\{P\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\\}$
$\displaystyle=\left\\{P\left(\frac{\partial}{\partial\lambda}\right)\frac{1}{\det
Q^{\delta}(\lambda)}Q^{\delta}_{c}(\lambda)\psi(\lambda)\right\\}$ (3.13)
$\displaystyle=\sum_{\kappa}\frac{P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}_{c}(\lambda)P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)}{\left(\det
Q^{\delta}(\lambda)\right)^{m_{\kappa}}}.$

The last line of (3) follows by an easy application of the Leibniz rule. Here
$P_{\kappa},P^{\prime}_{\kappa}$ are finite degree polynomials, $m_{\kappa}$
is a positive integer depending on $\kappa$ and the sum is over a finite set.
From (3) we get:

$\displaystyle\left\|P\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\|_{\mathbf{2}}$
$\displaystyle\leq\sum_{\kappa}\frac{\|P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}_{c}(\lambda)\|_{\mathbf{2}}\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\|_{\mathbf{2}}}{|\det
Q^{\delta}(\lambda)|^{m_{\kappa}}}$ (3.14) $\displaystyle\leq
c(\delta)\sum_{\kappa}\frac{\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\|_{\mathbf{2}}}{|\det
Q^{\delta}(\lambda)|^{m_{\kappa}}}(1+\|\lambda\|)^{n_{\kappa}}.$

The above inequality is obtained by using the fact that:
$\|P_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)Q^{\delta}_{c}(\lambda)\|_{\mathbf{2}}\leq
c(\delta)(1+\|\lambda\|)^{n_{\kappa}}$ where $c(\delta)>0$ is a
$\delta$-dependent constant and $n_{\kappa}$ is a positive integer depending
on the degree of $P_{\kappa}$ (it may also depend on $\delta$). Now from (3)
we get the following inequality for any nonnegative integer $t$.

$\displaystyle\sup_{Int(\mathfrak{a}^{*}_{\varepsilon}\cap(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}))}\left\|P\left(\frac{\partial}{\partial\lambda}\right)g(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t}$
$\displaystyle\leq\sum_{\kappa}\sup_{Int(\mathfrak{a}^{*}_{\varepsilon}\cap(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}))}\frac{\left\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t+n_{\kappa}}}{|\det
Q^{\delta}(\lambda)|^{m_{\kappa}}}$
$\displaystyle\leq\sum_{\kappa}\frac{\sup_{\lambda\in\mathfrak{a}^{*}_{\varepsilon}}\left\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t+n_{\kappa}}}{\inf_{\lambda\in
Int(\mathfrak{a}^{*}_{\varepsilon}\cap(\mathfrak{a}^{*}+i\overline{\mathfrak{a}^{*+}}))}|\det
Q^{\delta}(\lambda)|^{m_{\kappa}}}$ (3.15)
$\displaystyle\leq\sum_{\kappa}\frac{1}{c_{\delta}^{\kappa}}\sup_{\lambda\in\mathfrak{a}^{*}_{\varepsilon}}\left\|P^{\prime}_{\kappa}\left(\frac{\partial}{\partial\lambda}\right)\psi(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t+n_{\kappa}}.$

The last line of the above successive inequalities is a consequence of
Proposition 3.9. As
$\psi\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ so each term of
the finite summation on the right hand side of (3) is finite. Hence we
conclude that $g$ satisfies the decay (3.5) of the space
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$. The inequality (3) further
concludes that the continuous map (3.11) from
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ onto
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ is injective and also
its inverse map is continuous. As both the spaces
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ and
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ are Frèchet spaces so
the map (3.11) is a homeomorphism.

######  Lemma 3.11.

The Paley-Wiener space $\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$,
defined in (2.14), is a dense subspace of the Schwartz space
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.

###### Proof.

Let us take any $H\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$, it
is enough to show that, there is a sequence $\\{G_{n}\\}$
($G_{n}\in\mathcal{P}(\mathfrak{a}^{*}_{\mathbb{C}})$) converging to $H$ in
the topology of the space
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. Let
$H(\lambda)=\left(H_{ij}(\lambda)\right)_{{\ell_{\delta}}\times d_{\delta}}$.
By the isomorphism obtained in Lemma 3.10, we get one unique
$G\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ such that

$\displaystyle H(\lambda)=\left(H_{ij}(\lambda)\right)_{{\ell_{\delta}}\times
d_{\delta}}$ $\displaystyle=Q^{\delta}(\lambda)G(\lambda),$
$\displaystyle=Q^{\delta}(\lambda)\left(G_{ij}(\lambda)\right)_{{\ell_{\delta}}\times
d_{\delta}},$ (3.16)
$\displaystyle=\left(\sum_{k=1}^{{\ell_{\delta}}}Q^{\delta}(\lambda)_{ik}G_{kj}(\lambda)\right)_{{\ell_{\delta}}\times
d_{\delta}}.$

As $G\in\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$, so from the
definition of the Schwartz space
$\mathcal{S}_{0}(\mathfrak{a}^{*}_{\varepsilon})$ it follows that the matrix
entry functions $G_{ij}\in\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$ for
each $1\leq i\leq{\ell_{\delta}}$ and $1\leq j\leq d_{\delta}$. We know that
the Paley-Wiener space $\mathcal{P}(\mathfrak{a}^{*}_{\mathbb{C}})$ under the
spherical transform is dense in the Schwartz class
$\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$ [GV88]. Therefore we can get a
sequence
$\left\\{{g_{ij}}_{n}\right\\}_{n}\subset\mathcal{P}(\mathfrak{a}^{*}_{\mathbb{C}})$
converging to $G_{ij}$ in $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$. As,
each $Q^{\delta}_{ik}(\lambda)$ ($1\leq i\leq{\ell_{\delta}},1\leq k\leq
d_{\delta}$) is a polynomial in $\lambda$, so the sequence
$\left\\{\sum_{k=1}^{\ell_{\delta}}Q^{\delta}_{ik}(\lambda){g_{ij}}_{n}(\lambda)\right\\}_{n}$
converges to
$\sum_{k=1}^{\ell_{\delta}}Q^{\delta}_{ik}(\lambda)G_{kj}(\lambda)$ (for each
$\lambda$) in $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon})$.
Let $g_{n}(\lambda)=\left({g_{ij}}_{n}(\lambda)\right)_{{\ell_{\delta}}\times
d_{\delta}}$. As each
${g_{ij}}_{n}\in\mathcal{P}(\mathfrak{a}^{*}_{\mathbb{C}})$, so from the
definition it follows that, the matrix valued function
$g_{n}\in\mathcal{P}^{\delta}_{0}(\mathfrak{a}^{*}_{\mathbb{C}})$. Clearly by
Lemma 2.9 for each natural number $n$,
$Q^{\delta}(\cdot)g_{n}(\cdot)\in\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$.
Let $P$ be any polynomial in $S(\mathfrak{a})$ and $t$ be any nonnegative
integer then:

$\displaystyle\tau_{P,t}\left(Q^{\delta}(\cdot)g_{n}(\cdot)-Q^{\delta}(\cdot)G(\cdot)\right)$
$\displaystyle=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\left\|P\left(\frac{\partial}{\partial\lambda}\right)\left\\{Q^{\delta}(\lambda)g_{n}(\lambda)-Q^{\delta}(\lambda)G(\lambda)\right\\}\right\|_{\mathbf{2}}(1+\|\lambda\|)^{t},$
(3.17) $\displaystyle=\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\sum_{i=1}^{\ell_{\delta}}\sum_{j=1}^{d_{\delta}}\left\|P\left(\frac{\partial}{\partial\lambda}\right)\sum_{k=1}^{{{\ell_{\delta}}}}\left\\{Q^{\delta}(\lambda)_{ik}{g_{ik}}_{n}(\lambda)-Q^{\delta}(\lambda)_{ik}G_{ik}(\lambda)\right\\}\right\|^{2}(1+\|\lambda\|)^{t}.$

A suitable choice of $n$ can made the right hand side of (3) arbitrarily
small. Hence we get the sequence $\left\\{Q^{\delta}(\cdot)g_{n}\right\\}_{n}$
in $\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ converging to $H$ in
the topology of $\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. This
completes the proof of the Lemma.

Next we shall try to extend the definition of the $\delta$-spherical transform
(2.1) to the Schwartz class $\mathcal{S}^{p}_{\delta}(X)$ where $0<p\leq 2$.

######  Lemma 3.12.

For each $f\in\mathcal{S}^{p}_{\delta}(X)$, the function
$\lambda\mapsto\widetilde{f}(\lambda)$, where $\widetilde{f}$ is given by
(2.1), is a holomorphic function in the interior of the complex tube
$\mathfrak{a}^{*}_{\varepsilon}$.

###### Proof.

For each $f\in\mathcal{S}^{p}_{\delta}(X)$ the function $x\mapsto trf(x)$ has
the following decay: for each $D,E\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$
and integer $n\geq 0$

(3.18) $\sup_{x\in
G}|trf(D,x,E)|(1+|x|)^{n}\varphi_{0}^{-\frac{2}{p}}(x)<+\infty,$

which follows easily from (3.2) and the fact that
$|trf(x)|\leq\|f(x)\|_{\mathbf{2}}$ ( $x\in X$). Using (3.18) and the estimate
(2.10) of the generalized spherical functions one can show, by following a
standard argument (see [GV88, $\S$6.2], [EK76]), that the $\delta$-spherical
transform, defined by the integral (2.1), exists for
$\lambda\in\mathfrak{a}^{*}_{\varepsilon}$.
Let $\gamma$ be a closed curve in the interior of the tube
$\mathfrak{a}^{*}_{\varepsilon}$. Then for $f\in\mathcal{S}^{p}_{\delta}(X)$
we get

$\displaystyle\int_{\gamma}\widetilde{f}(\lambda)d\lambda$
$\displaystyle=d_{\delta}\int_{\gamma}\left\\{\int_{G}trf(x)\Phi_{\overline{\lambda},\delta}(x)dx\right\\}d\lambda.$

As the integral within braces exists absolutely for
$\lambda\in\mathfrak{a}^{*}_{\varepsilon}$, so we apply Fubini’s theorem to
get:
$\int_{\gamma}\widetilde{f}(\lambda)d\lambda=d_{\delta}\int_{G}trf(x)\left\\{\int_{\gamma}\Phi_{\overline{\lambda},\delta}(x)d\lambda\right\\}dx$.
We also recall that the functions
$\lambda\mapsto\Phi_{\overline{\lambda},\delta}(\cdot)$ are entire. Hence by
an application of Morera’s theorem the desired conclusion of the lemma
follows.

##  4\.  Proof of Theorem 3.8

To show that the $\delta$-spherical transform is a topological isomorphism it
is enough to show that it is a continuous surjection from
$\mathcal{S}^{p}_{\delta}(X)$ onto
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.

######  Lemma 4.1.

The $\delta$-spherical transform $f\mapsto\widetilde{f}$ is a continuous map
from the Schwartz space $\mathcal{S}^{p}_{\delta}(X)$ into
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.

###### Proof.

It is enough to show that given any seminorm $\nu$ (or equivalently
$\nu^{\\#}$) on $\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ one can
find a seminorm $\mu$ on $\mathcal{S}^{p}_{\delta}(X)$ such that

$\nu(\widetilde{f})\leq c\mu(f)\mbox{~{}for
all~{}}f\in\mathcal{S}^{p}_{\delta}(X).$

With $\texttt{P}\in S(\mathfrak{a})$ and $t\in\mathbb{N}\cup\\{0\\}$ we get
the following by using the integral expression (2.1) of the $\delta$-spherical
transform.

$\displaystyle\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\left\\{\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{t}\widetilde{f}(\lambda)\right\\}$
$\displaystyle=\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}trf(x)~{}\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{t}\Phi_{\overline{\lambda},\delta}^{*}(x)dx$
(4.1)
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}trf(x)~{}~{}\mathbf{L}^{t}\Phi_{\overline{\lambda},\delta}^{*}(x)dx.$

The last equality is a consequence of the property (2.8) of the generalized
spherical function. Now a simple application of integration by parts gives:

$\displaystyle(\ref{ali:10})$
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}\mathbf{L}^{t}trf(x)~{}~{}\Phi_{\overline{\lambda},\delta}^{*}(x)dx$
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}tr~{}\mathbf{L}^{t}f(x)~{}~{}\Phi_{\overline{\lambda},\delta}^{*}(x)dx$
(4.2)
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}tr~{}\mathbf{L}^{t}f(x)\left\\{\int_{K}e^{(i\lambda-\rho)(\mathcal{H}(x^{-1}k))}\delta(k^{-1})dk\right\\}dx.$

The second line in the above chain of equalities uses the fact
$\mathbf{L}trf(\cdot)=tr\mathbf{L}f(\cdot)$ which is clear as the differential
operator $\mathbf{L}$ acts entry-wise to the operator valued function $f$.
As $f\in\mathcal{S}^{p}_{\delta}(X)$ so it can also be considered as a
right-$K$-invariant function on the group $G$. The action of the Laplace
Beltrami operator $\mathbf{L}$ on $f$ is the same as the action of the Casimir
operator on $f$ considering as a function on $G$. Therefore, by the property
of the Casimir operator, the action of $\mathbf{L}$ does not change the
left-$K$-type of the function $f$, i. e the function $\mathbf{L}^{t}f(\cdot)$
is again of left-$\delta$-type. Moreover, for each nonnegative integer $n$ the
function $\mathbf{L}^{t}f(\cdot)\in\mathcal{S}^{p}_{\delta}(X)$. Hence by
Lemma 3.12 the integral on the right hand side of (4) exists absolutely. We
apply Fubini’s theorem to interchange the integrals and then we put
$x^{-1}k=y^{-1}$ to get:

$\displaystyle(\ref{ali:101})$
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{K}\int_{G}tr\mathbf{L}^{t}f(ky)~{}~{}e^{(i\lambda-\rho)\mathcal{H}(y^{-1})}\delta(k^{-1})dydk$
$\displaystyle=(-1)^{n}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}\left\\{\int_{K}tr\mathbf{L}^{t}f(ky)~{}~{}\delta(k^{-1})dk\right\\}~{}~{}e^{(i\lambda-\rho)\mathcal{H}(y^{-1})}dy$
$\displaystyle=\frac{(-1)^{t}}{d_{\delta}}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}\mathbf{L}^{t}f(y)~{}~{}e^{(i\lambda-\rho)\mathcal{H}(y^{-1})}dy$
(4.3)
$\displaystyle=\frac{(-1)^{t}}{d_{\delta}}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{G}\mathbf{L}^{t}f(y^{-1})~{}~{}e^{(i\lambda-\rho)\mathcal{H}(y)}dy$

The third lie follows by using the Schwartz space extension of the isomorphism
$\mathcal{Q}$ of Proposition 1.1 and the last line uses the invariance of the
Haar measure under the transformation $g\mapsto g^{-1}$. Let us now break up
the group $G$ as well as the Haar measure using the Iwasawa decomposition
$KAN$ decomposition to get.

$\displaystyle(\ref{ali:102})=$
$\displaystyle\frac{(-1)^{t}}{d_{\delta}}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{K}\int_{\mathfrak{a}^{+}}\int_{N}\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1}k^{-1})e^{(i\lambda-\rho)(\mathcal{H}(k(\exp H)n)}dke^{2\rho(H)}dHdn$
(4.4)
$\displaystyle=\frac{(-1)^{t}}{d_{\delta}}\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\int_{\mathfrak{a}^{+}}\int_{N}\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1})e^{(i\lambda+\rho)(H)}dHdn.$

Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{r}$ be the set of all positive
restricted roots. Let $\varepsilon_{i}\in\mathfrak{a}^{*}$ ($1\leq i\leq r$)
be such that $\langle\alpha_{i},\varepsilon_{j}\rangle=\delta_{ij}$. Then
clearly $\\{\varepsilon_{i}\\}_{1\leq i\leq r}$ forms a basis of
$\mathfrak{a}^{*}$ and thus we introduce a global coordinate on
$\mathfrak{a}^{*}$ by $\lambda=\sum_{i=1}^{r}\lambda_{i}\varepsilon_{i}$
($\forall\lambda\in\mathfrak{a}^{*}$).
Let
$\texttt{P}(\lambda)=\sum_{\theta=0}^{\beta}\sum_{\beta_{1}+\cdots+\beta_{r}=\theta}\alpha_{\beta_{1}+\cdots+\beta_{r}}\lambda_{1}^{\beta_{1}}\lambda_{2}^{\beta_{2}}\cdots\lambda_{r}^{\beta_{r}}$.
Thus,

$\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)=\sum_{\theta=0}^{\beta}\sum_{\beta_{1}+\cdots+\beta_{r}=\theta}\alpha_{\beta_{1}+\cdots+\beta_{r}}\left(\frac{\partial}{\partial\lambda_{1}}\right)^{\beta_{1}}\left(\frac{\partial}{\partial\lambda_{2}}\right)^{\beta_{2}}\cdots\left(\frac{\partial}{\partial\lambda_{r}}\right)^{\beta_{r}}.$

There fore it is easy to check that:

(4.5)
$\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)e^{(i\lambda+\rho)(H)}=\texttt{P}(H)~{}e^{(i\lambda+\rho)(H)},$

where
$\texttt{P}(H)=\sum_{\theta=0}^{\beta}\sum_{\beta_{1}+\cdots+\beta_{r}=\theta}\alpha_{\beta_{1}+\cdots+\beta_{r}}\varepsilon_{1}^{\beta_{1}}(iH))\cdots\varepsilon_{r}^{\beta_{r}}(iH)$.
From (4) it follows that:

$\displaystyle\left\|\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\left\\{\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{n}\widetilde{f}(\lambda)\right\\}\right\|_{\mathbf{2}}$
$\displaystyle\hskip
72.26999pt\leq\frac{1}{d_{\delta}}\int_{\mathfrak{a}^{+}}\int_{N}\|\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1})\|_{\mathbf{2}}~{}\|\texttt{P}(H)\|\left|e^{(i\lambda+\rho)(H)}\right|dHdn$
(4.6) $\displaystyle\hskip
72.26999pt\leq\frac{1}{d_{\delta}}\int_{\mathfrak{a}^{+}}\int_{A}\|\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1})\|_{\mathbf{2}}~{}\|\texttt{P}(H)\|~{}e^{(|\Im\lambda|+\rho)(H)}dHdn.$

Some basic estimates gives the following:

$\displaystyle\|\varepsilon_{j}^{\beta_{j}}(iH)\|$
$\displaystyle\leq\|\varepsilon_{j}\|^{\beta_{j}}\|H\|$ $\displaystyle\leq
c\|\varepsilon_{j}\|^{\beta_{j}}|(\exp H)n|$ (4.7) $\displaystyle\leq
c\|\varepsilon_{i}\|^{\beta_{i}}(1+|(\exp H)n|).$

The above estimate is a consequence of (1.3) and it is true for all $n\in N$.
Using (4) one can find $d_{\texttt{P}}\in\mathbb{Z}^{+}$ such that

(4.8) $\|\texttt{P}(H)\|\leq c_{1}(1+|(\exp H)n|)^{d_{\texttt{P}}}.$

As $f\in\mathcal{S}^{p}_{\delta}(X)$ so for each $m\in\mathbb{Z}^{+}$ we have:

(4.9) $\|\mathbf{L}^{t}f(n^{-1}(\exp
H)^{-1})\|_{\mathbf{2}}\leq\mu_{\mathbf{L}^{t},m}(f)~{}(1+|(\exp
H)n|)^{-m}~{}\varphi_{0}^{\frac{2}{p}}(n^{-1}(\exp H)^{-1}).$

The above inequality also uses the fact that $|g|=|g^{-1}|$ for all $g\in G$.
The estimates (4) and (4.9) reduce the inequation (4) to the following:

$\displaystyle\left\|\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\left\\{\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{t}\widetilde{f}(\lambda)\right\\}\right\|_{\mathbf{2}}$
$\displaystyle\leq
c_{1}\frac{\mu_{\mathbf{L}^{t},m}(f)}{d_{\delta}}\int_{\mathfrak{a}^{+}}\int_{N}\varphi_{0}^{\frac{2}{p}}(n^{-1}(\exp
H)^{-1})~{}(1+|(\exp
H)n|)^{-m+d_{\texttt{P}}}~{}e^{(|\Im\lambda|+\rho)(H)}dHdn$
$\displaystyle=c_{1}\frac{\mu_{\mathbf{L}^{t},m}(f)}{d_{\delta}}\int_{K}\int_{\mathfrak{a}^{+}}\int_{N}\varphi_{0}^{\frac{2}{p}}(n^{-1}(\exp
H)^{-1}k^{-1})(1+|k(\exp H)n|)^{-m+d_{\texttt{P}}}$ $\displaystyle\hskip
224.03743pte^{(|\Im\lambda|-\rho)(\mathcal{H}(k(\exp H)n))}dke^{2\rho(H)}dHdn$
(4.10)
$\displaystyle=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{G}\varphi_{0}^{\frac{2}{p}}(g^{-1})(1+|g|)^{-m+d_{\texttt{P}}}e^{(|\Im\lambda|-\rho)(\mathcal{H}(g))}dg$

where $c_{\delta}=c_{1}\frac{1}{d_{\delta}}$. Now we use the Cartan
decomposition i.e $g=k_{1}\exp{|g|}k_{2}$ and appropriate form of the Haar
measure (1.4) to get:

$\displaystyle(\ref{ali:14})$
$\displaystyle=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\int_{K}\varphi_{0}^{\frac{2}{p}}(\exp{|g^{-1}|})(1+|g|)^{-m+d_{\texttt{P}}}e^{(|\Im\lambda|-\rho)(\mathcal{H}(\exp{|g|}k_{2}))}\Delta(|g|)d|g|~{}dk_{2},$
$\displaystyle=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\varphi_{0}^{\frac{2}{p}}(\exp{|g|})(1+|g|)^{-m+d_{\texttt{P}}}\left\\{\int_{K}e^{(i(-i|\Im\lambda|)-\rho)(\mathcal{H}(\exp{|g|}k_{2}))}dk_{2}\right\\}$
$\displaystyle\hskip 325.215pt\Delta(|g|)d|g|$
$\displaystyle=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\varphi_{0}^{\frac{2}{p}}(\exp{|g|})(1+|g|)^{-m+d_{\texttt{P}}}~{}\varphi_{-i|\Im\lambda|}(\exp|g^{-1}|)\Delta(|g|)d|g|$
(4.11) $\displaystyle\leq
c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\varphi_{0}^{\frac{2}{p}+1}(\exp{|g|})(1+|g|)^{-m+d_{\texttt{P}}}~{}e^{|\Im\lambda(|g|)|}\Delta(|g|)d|g|$

where the last inequality in this chain follows by using the estimate (1.9) of
the elementary spherical function. We take
$\lambda\in\mathfrak{a}^{*}_{\varepsilon}$, therefore
$|\Im\lambda(|g|)|\leq\varepsilon\rho(|g|)$ where
$\varepsilon=\left(\frac{2}{p}-1\right)$. Now by using the another fundamental
estimate (1.10) we further reduce the inequality (4) to the following

$\displaystyle\left\|\texttt{P}\left(\frac{\partial}{\partial\lambda}\right)\left\\{\left(\langle\lambda,\lambda\rangle_{1}+\|\rho\|^{2}\right)^{t}\widetilde{f}(\lambda)\right\\}\right\|_{\mathbf{2}}$
$\displaystyle\hskip 108.405pt\leq
c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{\mathfrak{a}^{+}}\varphi_{0}^{2}(\exp{|g|})(1+|g|)^{-m+d_{\texttt{P}}+\varepsilon\theta}\Delta(|g|)~{}d|g|,$
(4.12) $\displaystyle\hskip
108.405pt=c_{\delta}\mu_{\mathbf{L}^{t},m}(f)\int_{G}\varphi_{0}^{2}(g)~{}(1+|g|)^{-m+d_{\texttt{P}}+\varepsilon\theta}~{}dg.$

We choose a suitably large $m\in\mathbb{Z}^{+}$ so the integral (4) converges
([HC66, Lemma 7]). This completes the proof of the Lemma.

We now take up the extension of the inversion formula (2.12) of the
$\delta$-spherical transform.

######  Lemma 4.2.

For each $h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ the
following integral

(4.13)
$\frac{1}{\omega}\int_{\mathfrak{a}^{*}}\Phi_{\lambda,\delta}(x)~{}h(\lambda)~{}|\mathbf{c}(\lambda)|^{-2}d\lambda,~{}~{}\mbox{~{}~{}~{}~{}for
}x\in X,$

gives a $Hom(V_{\delta},V_{\delta})$ valued, left-$\delta$-type
$\mathcal{C}^{\infty}$ function on $X$. (From now on we shall denote this
function by $\mathcal{I}h(\cdot)$.)

###### Proof.

Let us take any $D\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$. Then,

(4.14)
$\displaystyle\frac{1}{\omega}\int_{\mathfrak{a}^{*}}\left\|\Phi_{\lambda,\delta}(D,x)\right\|_{\mathbf{2}}$
$\displaystyle\|h(\lambda)\|_{\mathbf{2}}|\mathbf{c}(\lambda)|^{-2}d\lambda\leq
c_{\delta}~{}\varphi_{0}(x)\int_{\mathfrak{a}^{*}}(1+\|\lambda\|)^{b_{\textbf{D}}+b-n}d\lambda.$

The above inequality follows from the fact that
$h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ and by using the
decay (3.7), the estimate (1.13) and the estimate (2.10) for the generalized
spherical functions. One can choose a suitably large $n$ so that the integral
in the right hand side of (4.14) converges. This proves $\mathcal{I}h$ is a
function on $X$ and $\textbf{D}\mathcal{I}h$ exists for all
$\textbf{D}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$. Hence
$\mathcal{I}h\in\mathcal{C}^{\infty}(X,Hom(V_{\delta},V_{\delta}))$. As,
$\Phi_{\lambda,\delta}(\cdot)$ is of left-$\delta$-type ( (2.4) of Proposition
2.3), so is $\mathcal{I}h$.

######  Lemma 4.3.

If $h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ then the inverse
$\mathcal{I}h\in\mathcal{S}^{p}_{\delta}(X)$.

###### Proof.

To prove this Lemma we shall first consider the spaces
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ and
$\mathcal{D}_{\delta}(X)$ equipped with the topologies of the respective
Schwartz spaces containing them. We have already noticed that
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ and
$\mathcal{D}_{\delta}(X)$ are dense subspaces of
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ and
$\mathcal{S}^{p}_{\delta}(X)$ respectively.
We shall show that $\mathcal{I}$ is a continuous map from
$\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ onto (by Theorem 2.8)
$\mathcal{D}_{\delta}(X)$ with respective to the Schwartz space topologies.
That is for $h\in\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ and for
each seminorm $\mu$ on $\mathcal{D}_{\delta}(X)$ , there exists a seminorm
$\nu$ on $\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ such that
$\mu(f)\leq c_{\delta}\nu(h)$, where
$f=\mathcal{I}h\in\mathcal{D}_{\delta}(X)$ and $c_{\delta}$ is a positive
constant depending on $\delta\in\widehat{K}_{M}$.
As, $f\in\mathcal{D}_{\delta}(X)$, by Lemma 2.10, we get a function
$\phi\in\mathcal{D}(G//K,Hom(V_{\delta},V_{\delta}))$ such that
$f\equiv\textbf{D}^{\delta}\phi$. If $\Phi$ be the image of $\phi$ under the
spherical transform then it follows easily that $h=Q^{\delta}\Phi$. Let
$\textbf{D},\textbf{E}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ and $n$ be
any nonnegative integer, then

$\displaystyle\mu_{\textbf{D},\textbf{E},n}(f)$ $\displaystyle=\sup_{x\in
G}\|f(\textbf{D},x,\textbf{E})\|_{\mathbf{2}}(1+|x|)^{n}\varphi^{-\frac{2}{p}}_{0}(x),$
$\displaystyle=\sup_{x\in
G}\|\textbf{D}^{\delta}\phi(\textbf{D},x,\textbf{E})\|_{\mathbf{2}}(1+|x|)^{n}\varphi^{-\frac{2}{p}}_{0}(x),$
(4.15)
$\displaystyle={\mu_{0}}_{\textbf{D}^{\delta}\textbf{D},\textbf{E},n}(\phi).$

(  _Here $\mu_{0}$ denote the seminorms on the Fréchet space
$\mathcal{S}^{p}(G//K,Hom(V_{\delta},V_{\delta}))$_. ) At this point we use
Anker’s [Ank91] proof of the Schwartz space isomorphism theorem for
bi-$K$-invariant functions. For each
$\textbf{D},\textbf{E}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ and
$n\in\mathbb{Z}^{+}$ one can find a polynomial $P\in S(\mathfrak{a})$ and
$m_{\delta}\in\mathbb{Z}^{+}$ (depending on $d_{\delta}$) such that,

$\displaystyle{\mu_{0}}_{\textbf{D}^{\delta}\textbf{D},\textbf{E},n}(\phi)$
$\displaystyle\leq c_{\delta}\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\left\|P\left(\frac{\partial}{\partial\lambda}\right)\Phi(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m_{\delta}},$
(4.16) $\displaystyle\leq c_{\delta}\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon}}\left\|P_{1}\left(\frac{\partial}{\partial\lambda}\right)h(\lambda)\right\|_{\mathbf{2}}(1+\|\lambda\|)^{m^{\prime}_{\delta}}.$

The last line in (4) follows by using the isomorphism, proved in Lemma 3.10,
between the Schwartz spaces $S_{0}(\mathfrak{a}^{*}_{\varepsilon})$ and
$S_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. Hence (4) and (4) togather gives
$\mu_{\textbf{D},\textbf{E},n}(f)\leq
c_{\delta}\nu_{P_{1},m^{\prime}_{\delta}}(h)$. As we have started with an
$h\in\mathcal{P}^{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})\subset\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$,
the right hand side of (4) is clearly finite. Hence
$\mathcal{I}h=f\in\mathcal{S}^{p}_{\delta}(X)$.
Now we apply the density argument to conclude the Lemma. Let us now take
$h\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$. As,
$\mathcal{P}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$ is dense in
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$, there exists a Cauchy
sequence
$\left\\{h_{n}\right\\}\subset\mathcal{P}_{\delta}(\mathfrak{a}^{*}_{\mathbb{C}})$
converging to $h$. Then, by what we have proved above, we can get a Cauchy
sequence $\left\\{f_{n}\right\\}\subset\mathcal{D}_{\delta}(X)$ such that
$\widetilde{f_{n}}=h_{n}$. As $\mathcal{S}^{p}_{\delta}(X)$ is a Fréchet space
the sequence must converge to some $f\in\mathcal{S}^{p}_{\delta}(X)$. Clearly,
$f=\mathcal{I}h$. This completes the proof of the Lemma.

We note that, the Lemma 4.3 also implies the fact that the $\delta$-spherical
transform is an injection in the corresponding Schwartz space level.
Finally, Lemma 4.1 and Lemma 4.3 together shows that the $\delta$-spherical
transform is a continuous surjection of $\mathcal{S}_{\delta}^{p}(X)$ onto
$\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$ for $(0<p\leq 2)$. A
simple application of the open mapping theorem concludes that the
$\delta$-spherical transform is a topological isomorphism between the
corresponding Schwartz spaces. This proves the Theorem 3.8.
In the next section we shall extend this result to a slightly larger class of
functions.

##  5\.  Finite $K$-type functions

Let choose and fix a finite subset $\Gamma\subset\widehat{K}_{M}$. We denote
$\mathcal{D}_{\Gamma}(X)$ for the space of all compactly supported
$\mathcal{C}^{\infty}$ functions on $X$ with the property that: for
$f\in\mathcal{D}_{\Gamma}(X)$ $f^{\delta}\equiv 0$ for $\delta\notin\Gamma$.
We take the subclass $\mathcal{S}_{\Gamma}^{p}(X)$ of the Schwartz class
$\mathcal{S}^{p}(X)$ (for $0<p\leq 2$) defined by

(5.1)
$\mathcal{S}^{p}_{\Gamma}(X)=\\{f\in\mathcal{S}^{p}(X)~{}|~{}f(X)=\sum_{\delta\in\Gamma}trf^{\delta}(x)~{}\mbox{~{}for
all~{}}x\in X\\}.$

The seminorms on $\mathcal{S}^{p}_{\Gamma}(X)$ are as follows: for each
$\textbf{D},\textbf{E}\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ and
$n\in\mathbb{Z}^{+}$,

(5.2) ${\mu_{\Gamma}}_{\textbf{D},\textbf{E},n}(f)=\sup_{\delta\in\Gamma,x\in
X}\left\|f^{\delta}(\textbf{D},x,\textbf{E})\right\|_{\mathbf{2}}~{}(1+|x|)^{n}~{}\varphi_{0}^{-\frac{2}{p}}(x)~{}<+\infty.$

Clearly $\mathcal{D}_{\Gamma}(X)$ is a dense subset of the Schwartz space
$\mathcal{S}^{p}_{\Gamma}(X)$ with respect to the Fréchet topology induced by
the countable family of seminorms
$\\{{\mu_{\Gamma}}_{\textbf{D},\textbf{E},n}\\}$. It also follows easily from
the definition 3.1 and (3.2 ) that, if $f\in\mathcal{S}^{p}_{\Gamma}(X)$ then
for each $\delta\in\Gamma$ the projection
$f^{\delta}\in\mathcal{S}^{p}_{\delta}(X)$. For these classes of functions the
transform we shall mainly consider is the Helgason Fourier transform.
Let us now define the Schwartz class functions on the domain
$\mathfrak{a}^{*}_{\varepsilon}\times K/M$.

######  Definition 5.1.

Let $\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ denotes
the class of functions $h$ on $\mathfrak{a}^{*}_{\varepsilon}\times K/M$
satisfying the following properties:

  1. (i)

For each $kM\in K/M$, the function $\lambda\mapsto h(\lambda,kM)$ is
holomorphic on $Int\mathfrak{a}^{*}_{\varepsilon}$, and it extends as a
continuous function on the closed complex tube
$\mathfrak{a}^{*}_{\varepsilon}$. The function $h$ is a smooth function in the
$k\in K/M$ variable.

  2. (ii)

For all $\lambda\in\mathfrak{a}^{*}_{\varepsilon}$, $\omega\in W$ and $x\in G$

(5.3) $\check{h}(\lambda,x)=\check{h}(\omega\lambda,x),$

where
$\check{h}(\lambda,x)=\int_{K}h(\lambda,k)e^{-(i\lambda+\rho)H(x^{-1}k)}dk$.

  3. (iii)

For each $P\in S(\mathfrak{a})$ and for integers $n,m>0$ the function $h$
satisfies the following decay condition

(5.4) $\sup_{(\lambda,k)\in Int\mathfrak{a}^{*}_{\varepsilon}\times
K/M}\left|P\left(\frac{d}{d\lambda}\right)h(\lambda,k,\omega^{m}_{\mathfrak{k}})\right|~{}(1+|\lambda|)^{n}<~{}+\infty.$

  4. (iv)

For each $\delta\in\widehat{K}_{M}\setminus\Gamma$ the
left-$\delta$-projection $h^{\delta}$ defined by

(5.5)
$h^{\delta}(\lambda,k)=d_{\delta}\int_{K}h(\lambda,k_{1}k)\delta(k_{1}^{-1})dk_{1},$

is identically a zero function on $\mathfrak{a}^{*}_{\varepsilon}\times K/M$.

The space $\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ is
a Fréchet space with the topology induced by the seminorms (5.4). By the
theory of smooth functions on compact groups [Sug71], the topology of the
space $\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ can be
given by the following equivalent family of seminorms, for each $P\in
S(\mathfrak{a})$ and $m\in\mathbb{Z}^{+}$ we have

(5.6) $\sup_{\lambda\in Int\mathfrak{a}^{*}_{\varepsilon},\delta\in
F}\left\|P\left(\frac{d}{d\lambda}\right)h^{\delta}(\lambda,eM)\right\|_{\textbf{2}}(1+|\lambda|)^{m}<+\infty,\mbox{~{}for~{}}h\in\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times
K/M).$

We denote by $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ the
Fréchet space satisfying all the conditions of the Definition 5.1 except
condition (iv). The space
$\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ is a closed
subspace of $\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$. We know
that the HFT can extended to the Schwartz class $\mathcal{S}^{p}(X)$ [EK76],
furthermore the HFT is a continuous map from $\mathcal{S}^{p}(X)$ into
$\mathcal{S}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$. Hence the HFT is a
continuous map from $\mathcal{S}^{p}(F,X)$ into
$\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$.

######  Lemma 5.2.

Let $h\in\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$, then
for each $\delta\in F$, the left-$\delta$-projection
$h^{\delta}\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.

###### Proof.

The function $\lambda\mapsto h^{\delta}(\lambda,eM)$ trivially satisfies
condition (i) of Definition 3.6. The required decay (3.3) is also an easy
consequence of (5.4). It can be shown that the $\delta$-projection
$h^{\delta}$ also satisfies the condition

$\check{(h^{\delta})}(\lambda,x)=\check{(h^{\delta})}(\omega\lambda,x)\mbox{~{}for
all~{}}\omega\in W.$

It is easy to check that
$h^{\delta}(\lambda,kM)=\delta(k)h^{\delta}(\lambda,eM)$, hence for each
$(\lambda,a)\in\mathfrak{a}^{*}_{\varepsilon}\times A$ we write
$\check{(h^{\delta})}(\lambda,a)$ as follows

$\check{(h^{\delta})}(\lambda,a)=\Phi_{\lambda,\delta}(a)h^{\delta}(\lambda,eM).$

By the property (2.6) of the generalized spherical functions it follows that
the function $\lambda\mapsto Q^{\delta}(\lambda)^{-1}h^{\delta}(\lambda,eM)$
is $W$-invariant. Hence we conclude that
$h^{\delta}(\cdot,eM)\in\mathcal{S}_{\delta}(\mathfrak{a}^{*}_{\varepsilon})$.

By using Theorem 3.8, for each
$h\in\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$ we get an
unique finite sequence $\\{f^{\delta}\\}_{\delta\in\Gamma}$ of
$\mathcal{C}^{\infty}$ functions on $X$ such that each member
$f^{\delta}\in\mathcal{S}^{p}_{\delta}(X)$. We consider the following scalar
valued function

(5.7) $f(x)=\sum_{\delta\in F}trf^{\delta}(x),~{}~{}~{}x\in X.$

For each $\delta\in\Gamma$,
$\left(\mathcal{F}^{-1}h\right)^{\delta}(x)=\mathcal{I}(h^{\delta})(x)=f^{\delta}(x)$.
Hence, we get $\mathcal{F}^{-1}h(x)=f(x)$ for all $x\in X$. The function
$f\in\mathcal{S}^{p}_{\Gamma}(X)$. Furthermore for each
$D,E\in\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ and $n\in\mathbb{Z}^{+}$ we
have

$\displaystyle\sup_{x\in G}|f(D,x,E)|$
$\displaystyle(1+|x|)^{n}\varphi_{0}^{-\frac{2}{p}}(x)$ $\displaystyle\leq
c\sup_{x\in
G,\delta\in\Gamma}\|f^{\delta}(D,x,E)\|_{\textbf{2}}(1+|x|)^{n}\varphi_{0}^{-\frac{2}{p}}(x)$
$\displaystyle\leq c_{1}\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon},\delta\in\Gamma}\left\|P\left(\frac{d}{d\lambda}\right)h^{\delta}(\lambda,eM)\right\|_{\textbf{2}}(1+|\lambda|)^{m}$
$\displaystyle\leq c_{2}\sup_{\lambda\in
Int\mathfrak{a}^{*}_{\varepsilon},k\in
K}\left|P_{1}\left(\frac{d}{d\lambda}\right)h(\lambda,k,\omega_{\mathfrak{k}}^{r})\right|(1+|\lambda|)^{m_{1}},$

for some $P_{1}\in\mathcal{S}(\mathfrak{a}^{*})$ and
$r,m_{1}\in\mathbb{Z}^{+}$. Thus, the HFT is a bijective map from
$\mathcal{S}^{p}_{\Gamma}(X)$ to
$\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times K/M)$. Once again,
by the open mapping theorem we conclude the following.

######  Theorem 5.3.

Let $\Gamma$ be a finite subset of $\widehat{K}_{M}$, then the HFT is a
topological isomorphism of the space $\mathcal{S}^{p}_{\Gamma}(X)$ onto the
Fréchet space $\mathcal{S}_{\Gamma}(\mathfrak{a}^{*}_{\varepsilon}\times
K/M)$.

Acknowledgments: The author is thankful to Prof. Angela Pasquale of Université
de Metz, France for her valuable suggestions.

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