When trying to understand the Plancherel formula of reductive symmetric space of Harish-Chandra class, I get confused on the convention of Fourier and related transforms. $\newcommand{\H}{\mathcal{H}} \newcommand{\RR}{\mathbb{R}} $
Background and convention
Firstly let me set up the conventions. The inner product is linear with respect to the first argument. The reductive symmetric space $G/H$ is regarded as the left coset space, and the (quasi-)regular representation on $L^{2}(G/H)$ is the left one
\begin{equation} g\cdot f(x)=f(g^{-1}x) \quad \text{for } g\in G, x\in G/H. \end{equation}
For a unitary irrep $\pi$ on $\H$, the smooth vectors and distributions are summarized in the Gelfand triple $\H^{\infty}\to \H \to \H^{-\infty}$. Then the $H$-fixed distributions $v\in\H^{-\infty}$ are left-invariant under the action of $H$, the space of which is denoted as $\H^{-\infty,H}$.
It is this $\H^{-\infty,H}$ that determines the irreps in the direct integral decomposition of $L^{2}(G/H)$ by the following Poisson transform
\begin{equation} P_v: \, w\in \H^{\infty}\, \mapsto (w,\pi(x)v), \end{equation}
and the Fourier transform
\begin{equation} F_v:\, f(x)\in L^2(G/H)\, \mapsto \int_{{G/H}}{dx} f(x)\pi(x)v. \end{equation}
They are adjoint to each other and the composition $PF_{v}$ extracts the $\H$-part of $f(x)\in L^2(G/H)$
\begin{equation} PF_v: \, f(x)\, \mapsto \int_{{G/H}}{dy} f(y)(\pi(y)v,\pi(x)v). \end{equation}
Then the Plancherel formula provides the information how to reconstruct $f(x)$ from $PF_v[f](x)$.
This convention is the one used in Helgason's book Groups and Geometric Analysis and van den Ban's lecture The Plancherel formula for a reductive symmetric space and other materials.
Confusion
Taking $G=\mathbb{R}$ and trivial $H$, the Fourier transform is \begin{equation} F_v:\, f(x)\, \mapsto \int_{\RR}{dx} f(x)e^{ipx} \end{equation} which is basically the conjugate of the usual convention.
In the classic theory of Gelfand pair, when $G$ and $H$ are compact or $H$ is maximally compact in $G$ (or for generalized Gelfand pairs by Dijk), the (elementary/zonal) spherical function is defined as
\begin{equation} \phi_{v}(x)=(v,\pi(x)v), \end{equation}
and $PF_v$ can be rewritten as a convolution with the spherical function on $G/H$
\begin{equation} PF_v: \, f(x)\, \mapsto \int_{{G/H}}{dy} f(y)\phi_{v}(y^{-1}x). \end{equation}
The spherical function indeed matches the convention of the Fourier transform on $\RR$, $\phi_{v}(x)=e^{-ipx}$. Then it seems that there are two conventions of spherical transforms.
Supposing $f$ is $H$-left invariant on $G/H$, then the spherical transform is defined either by
\begin{equation} S[f](v)=\int_{{G/H}}{dy} f(y)\phi_{v}(y^{-1}), \end{equation}
which is consistent with $PF_{v}$, or by
\begin{equation} S[f](v)=\int_{{G/H}}{dy} f(y)\phi_{v}(y), \end{equation}
which is consistent with the Fourier transform on locally compact abelian groups.
In Dieudonne's book Treatise on Analysis, the two versions of the spherical transform are called Fourier transform/co-transform.
Question
My question is as follows.
Which convention is more natural? The Fourier and Poisson transforms are quite natural since they are intertwining operators in certain sense. If using the spherical transform from $PF_{v}$, for spherical functions, $(v,\pi(x)v)$ is directly related to the Poisson transform, while the conjugate $(\pi(x)v,v)$ is compatible with usual Fourier analysis.
Is this issue only a matter of conventions, or have I missed something really important? I have noticed that in the latter sections of van den Ban's lecture The Plancherel formula for a reductive symmetric space, he introduces some Weyl reflection in the Eisenstein integrals. But the discussion is much more involved, and I'm still trying to understand the relations between the $(K,\tau)$-spherical functions and the Eisenstein integrals.
For the rather simple and physical example, $\operatorname{SO}(d,1)/{\operatorname{SO}(d)}$, there are only continuous series in the Plancherel measure, and the convention doesn't matter by the Knapp-Stein intertwining operator on the principal series (coming from the Weyl reflection).
But for noncompact $H$, there are discrete series in $L^{2}(G/H)$, and in this case the Knapp–Stein intertwining operator is not an isomorphism, and I suspect the convention should matter.
Any answer or suggestion would be helpful to me.
