For two Hecke quasicharacters $\chi_1,
\chi_2:K^\times\backslash{\mathbb{A}}^\times\to{\mathbb{C}}^\times$ with $\chi_1\chi_2=\omega$ let $H(\chi_1, \chi_2)$ denote the space of functions $\varphi : \operatorname{GL}_2(\mathbb{A}) \to {\mathbb{C}}$ satisfying $$\int_{{\mathcal{K}}} |\varphi(k)|^2 dk < \infty$$ and $$\label{trafoH0}\varphi\left(\begin{pmatrix} a & x\\0 & b\end{pmatrix}g\right) = \chi_1(a)\chi_2(b) \left|\frac{a}{b}\right|^{1/2}\varphi(g), \qquad
x \in \mathbb{A},\ \ a, b \in \mathbb{A}^{\times}.$$ We can regard this as the space of functions $\varphi\in L^2({\mathcal{K}})$ satisfying $$\label{trafoH}
\varphi\left(\left(\begin{matrix} a & x\\0 &
b\end{matrix}\right)k\right) = \chi_1(a)\chi_2(b) \varphi(k), \qquad
\left(\begin{matrix} a & x\\0 & b\end{matrix}\right)\in{\mathcal{K}}.$$ There is a unique $s\in{\mathbb{C}}$ such that $\chi_1(a)=|a|^s$ and $\chi_2(a)=|a|^{-s}$ for all $a\in{K_{\infty,+}^{\text{diag}}}$. Accordingly, for $s \in {\mathbb{C}}$ we introduce[^5] $$\label{H(s)}
 H(s) := \bigoplus_{\substack{\chi_1\chi_2 = \omega\\\text{$\chi_1\chi_2^{-1} = |\cdot|^{2s}$ on ${K_{\infty,+}^{\text{diag}}}$}}} H(\chi_1,
 \chi_2),$$ and we view the space $H:=\int_{\mathbb{C}}H(s)\,ds$ as a holomorphic fibre bundle with base ${\mathbb{C}}$. For a section $\varphi \in H$ we use the obvious notations $\varphi(s)\in H(s)$ and $\varphi(s,g) \in {\mathbb{C}}$. The bundle $H$ is trivial, because any $\varphi(s_0)\in H(s_0)$ extends uniquely to a section $\varphi\in H$. There is a $\operatorname{GL}_2(\mathbb{A})$-equivariant isomorphism $$S:L_{\text{cont}} \to L'_{\text{cont}} := \int_0^\infty H(iy)\,dy,$$ given explicitly by [@GJ (4.23)] on a dense subspace. If we equip $L'_{\text{cont}}$ with the inner product $$\langle\varphi_1, \varphi_2\rangle := \frac{2}{\pi} \int_0^\infty \langle\varphi_1(iy), \varphi_2(iy)\rangle\,dy
 = \frac{2}{\pi} \int_0^\infty \int_{{\mathcal{K}}} \varphi_1(iy,k)\bar{\varphi}_2(iy,k) \,dk \, dy,$$ then this map 