$t\in\mathbb{A}_\text{fin}^\times$ is any finite idele representing ${\mathfrak{t}}$. It follows from and the local result of Casselman [@Ca] that the spaces $V_\pi({\mathfrak{c}})$ decompose (in general not orthogonally) as $$\label{RTdecomp}
V_\pi({\mathfrak{c}})=\bigoplus_{{\mathfrak{t}}\mid{\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1}} R_{\mathfrak{t}}V_\pi({\mathfrak{c}}_\pi)
\qquad\text{for any ${\mathfrak{c}}\subseteq{\mathfrak{c}}_\pi$,}$$ and $V_{\pi_{{\mathfrak{p}}}}(c_{\pi_{{\mathfrak{p}}}})$ is one-dimensional for each prime ${\mathfrak{p}}$. For each character $k(\vartheta)\mapsto\exp(i q \cdot
\vartheta)$, $q \in {\mathbb{Z}}^d$, of $\operatorname{SO}_2(K_{\infty})$ we define $$V_{\pi,q} := \left\{\phi \in V_{\pi} \mid \text{$\phi(gk(\vartheta)) = \exp(i q\cdot \vartheta) \phi(g)$ for all
 $\vartheta\in({\mathbb{R}}/2\pi{\mathbb{Z}})^d$}\right\},$$ and correspondingly we write $$V_{\pi, q}({\mathfrak{c}}):=V_{\pi,q}\cap V_{\pi}({\mathfrak{c}}).$$ This gives an orthogonal decomposition (in a Hilbert space sense) $$\label{weightdecomp}
 V_{\pi}({\mathfrak{c}}) = \bigoplus_{q \in {\mathbb{Z}}^d}V_{\pi, q}({\mathfrak{c}}) \qquad\text{for any ${\mathfrak{c}}\subseteq{\mathfrak{c}}_\pi$},$$ and also a decomposition of vector spaces $$\label{oldforms}
V_{\pi,q}({\mathfrak{c}})=\bigoplus_{{\mathfrak{t}}\mid{\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1}} R_{\mathfrak{t}}V_{\pi,q}({\mathfrak{c}}_\pi) \qquad\text{for any ${\mathfrak{c}}\subseteq{\mathfrak{c}}_\pi$},$$ where $V_{\pi, q}({\mathfrak{c}}_{\pi})$ is at most one-dimensional. Alternately, and global multiplicity-one were also established by Miyake [@Mi] which then imply the above local results.

In the case of $V_\pi=V_{\chi_1,\chi_2}$ consisting of Eisenstein series we replace the subscript $\pi$ by $\chi_1,\chi_2$ for convenience, e.g. we write ${\mathfrak{c}}_{\chi_1,\chi_2}:={\mathfrak{c}}_\pi$. For each $1{\leqslant}j{\leqslant}d$ we have $$\label{eiseneig}
\lambda_{\chi_1,\chi_2,j}=\frac{1}{4}-s_j^2,\qquad
\nu_{\chi_1,\chi_2,j}=\pm s_j,$$ where $s_j \in i{\mathbb{R}}$ denotes the unique exponent such that $\chi_1\chi_2^{-1} =
|\cdot|^{2s_j}$ on the $j$-th component of ${K_{\infty,+}^{\times}}$. We note that $\chi_1\chi_2^{-1}=|\cdot|^{2iy}$ on ${K_{\infty,+}^{\text{diag}}}$, where $$\label{ydef}
 y:= \frac{s_1+s_2+\cdots+s_d}{i d}\in{\mathbb{R}}^\times.$$ It follows from the discussion 