in later sections (independently of the present statements, of course).

As mentioned earlier, the decomposition is in general not orthogonal. However, by a Gram–Schmidt orthogonalization process based on below, we find for each pair of integral ideals $(\mathfrak{s}, {\mathfrak{t}})$ with $\mathfrak{s} \mid {\mathfrak{t}}\mid {\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1}$ complex numbers $\alpha_{{\mathfrak{t}}, \mathfrak{s}}$ such that $$\label{newR}
 R^{({\mathfrak{t}})} := \sum_{\mathfrak{s} \mid {\mathfrak{t}}} \alpha_{{\mathfrak{t}}, \mathfrak{s}} R_{\mathfrak{s}} :
 V_{\pi}({\mathfrak{c}}_{\pi}) \hookrightarrow V_{\pi}({\mathfrak{c}})
 ,\qquad {\mathfrak{t}}\mid{\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1},$$ are isometric embeddings with pairwise orthogonal images, and $R^{({\mathfrak{o}})}$ is the identical inclusion map. This yields an orthogonal decomposition $$\label{RTdecomp2}
V_\pi({\mathfrak{c}})=\bigoplus_{{\mathfrak{t}}\mid{\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1}}
R^{({\mathfrak{t}})}V_\pi({\mathfrak{c}}_\pi) \qquad\text{for any ${\mathfrak{c}}\subseteq{\mathfrak{c}}_\pi$,}$$ and an extension of the Kirillov map to each subspace $R^{({\mathfrak{t}})}V_\pi({\mathfrak{c}}_\pi)$. Namely, by every $\phi \in R^{({\mathfrak{t}})}V_{\pi}({\mathfrak{c}}_{\pi})$ has a Fourier expansion $$\label{Fourgen}
 \phi\left(\left(\begin{matrix} y & x\\ 0 & 1\end{matrix}\right)\right) =
 \sum_{r \in K^{\times}} \frac{\lambda^{({\mathfrak{t}})}_{\pi}(r y_{\text{fin}}) }{\sqrt{{\mathcal{N}}(r y_{\text{fin}})}} W_{\phi}(r y_{\infty}) \psi(rx)
 ,\qquad y\in \mathbb{A}^{\times},\ \ x \in \mathbb{A},$$ where $$\label{oldKir}
W_\phi:=W_{(R^{({\mathfrak{t}})})^{-1}\phi}\qquad\text{and}\qquad
\lambda^{({\mathfrak{t}})}_{\pi}(\mathfrak{m}) := \sum_{\mathfrak{s}
\mid\gcd({\mathfrak{t}},\mathfrak{m})} \alpha_{{\mathfrak{t}}, \mathfrak{s}}
({\mathcal{N}}\mathfrak{s})^{1/2} \lambda_{\pi}
(\mathfrak{m}\mathfrak{s}^{-1}).$$ It is clear that holds true when extended to $\phi_1,\phi_2\in
R^{({\mathfrak{t}})}V_\pi({\mathfrak{c}}_\pi)$, and Lemma \[lemma2\] below shows that $$\label{newWphibound3}C(\pi)^{-{\varepsilon}} \| \phi \| \ll_{K, {\varepsilon}} \| W_\phi \|
\ll_{K, {\varepsilon}} C(\pi)^{{\varepsilon}} \| \phi \|,\qquad\phi\in
R^{({\mathfrak{t}})}V_\pi({\mathfrak{c}}_\pi),$$ with implied constants depending only on $K$ and ${\varepsilon}$.

If ${\mathfrak{c}}$ is squarefree, then the orthogonalization can be 