is an isometry by [@GJ Section 4, Part D]; in combination with the theory of Eisenstein series [@GJ Section 5] it yields a spectral decomposition of $L_\text{cont}$. For a section $\varphi\in H$ and for $g \in
\operatorname{GL}_2(\mathbb{A})$ we define the Eisenstein series $$\label{eisensteindef}
 E(\varphi(s), g) := \sum_{\gamma \in P(K)\backslash \operatorname{GL}_2(K)} \varphi(s,\gamma g)
,\qquad\Re s>1/2.$$ This is a holomorphic function which extends meromorphically to $s \in {\mathbb{C}}$ with no poles on $\Re s=0$. Moreover, for any $s\neq 0$ which is not a pole of $E(\varphi(s),
g)$, we can extract $\varphi(s)\in H(s)$ from the meromorphic continuation of the constant term as given by [@GJ (5.3)]. The above discussion suggests that for $y\in{\mathbb{R}}^\times$ we consider the complex vector space $$V(iy) := \{E(\varphi(iy)) \mid \varphi(iy)\in H(iy)\}$$ equipped with the inner product $$\label{eisensteininnerproduct}\langle
E(\varphi_1(iy)),E(\varphi_2(iy))\rangle:=
\langle\varphi_1(iy),\varphi_2(iy)\rangle.$$ By we have a $\operatorname{GL}_2(\mathbb{A})$-invariant (orthogonal) decomposition $$V(iy)=\bigoplus_{\substack{\chi_1\chi_2=\omega \\ \chi_1\chi_2^{-1} = |\cdot|^{2iy}
 \text{ on } K_{\infty, +}^{\text{diag}}}} V_{\chi_1, \chi_2},$$ where $$V_{\chi_1, \chi_2} := \left\{E(\varphi(iy)) \mid \varphi(iy)\in H(\chi_1, \chi_2)\right\}.$$ We note that $V_{\chi_1, \chi_2}=V_{\chi_2, \chi_1}$, in particular $V(iy)=V(-iy)$, by [@GJ (4.3), (4.24), (5.15)]. Now we have a $\operatorname{GL}_2(\mathbb{A})$-invariant decomposition $$L_\text{cont}=\int_0^\infty V(iy)\,dy=\int_0^\infty
 \bigoplus_{\substack{\chi_1\chi_2=\omega \\ \chi_1\chi_2^{-1} = |\cdot|^{2iy}
 \text{ on } K_{\infty, +}^{\text{diag}}}} V_{\chi_1, \chi_2}\ dy.$$ More precisely, by [@GJ (4.24), (5.15)–(5.17)] any 