$\phi\in
L_\text{cont}$ can be written as $$\phi(g)=\frac{1}{\pi}\int_0^\infty E(\varphi(iy),g)\,dy,\qquad\varphi:=S\phi\in L'_\text{cont},$$ and we have Plancherel’s identity $$\begin{aligned}
\langle \phi_1,\phi_2\rangle&=\frac{1}{\pi}\int_0^\infty \langle
E(\varphi_1(iy)),\phi_2\rangle\,dy =\frac{1}{\pi}\int_0^\infty
2\langle\varphi_1(iy),\varphi_2(iy)\rangle\,dy\\
&=\frac{2}{\pi}\int_0^\infty\langle
E(\varphi_1(iy)),E(\varphi_2(iy))\rangle\,dy.\end{aligned}$$

To summarize, we have a $\operatorname{GL}_2(\mathbb{A})$-invariant orthogonal decomposition $$\label{spectral}
 L^2(\operatorname{GL}_2(K)\backslash \operatorname{GL}_2(\mathbb{A}), \omega)= \bigoplus_{\pi\in \mathcal{C}_{\omega}} V_{\pi}
 \oplus \bigoplus_{\chi^2=\omega} V_\chi \oplus \int_0^\infty
 \bigoplus_{\substack{\chi_1\chi_2=\omega \\ \chi_1\chi_2^{-1} = |\cdot|^{2iy}
 \text{ on } K_{\infty, +}^{\text{diag}}}} V_{\chi_1, \chi_2}\ dy$$ in the sense that each function in the $L^2$-space decomposes into a convergent sum and integral of functions from each subspace, and a Plancherel formula holds. For notational simplicity we shall write the last term as $\int_{\mathcal{E}_\omega} V_{\varpi}\,d\varpi$, where $\mathcal{E}_\omega$ is the set of [*unordered pairs*]{} $\{\chi_1,\chi_2\}$ of Hecke characters with $\chi_1\chi_2 = \omega$ and nontrivial restrictions on ${K_{\infty,+}^{\text{diag}}}$.

Casimir eigenvalues and conductors
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Let $(\pi, V_{\pi})$ be an infinite-dimensional irreducible automorphic representation of $\operatorname{GL}_2(\mathbb{A})$ occurring in the spectral decomposition , i.e. one of $V_{\pi}$ with $\pi\in \mathcal{C}_{\omega}$, or $V_{\chi_1,\chi_2}$ with $\{\chi_1,\chi_2\}\in\mathcal{E}_\omega$, equipped with the right $\operatorname{GL}_2(\mathbb{A})$-action. By Flath’s Theorem [@Fl], $V_{\pi}$ decomposes as a restricted tensor product over the places of $K$, $$\label{flath}
 V_{\pi} = \bigotimes_v V_{\pi_v}.$$ For each $1{\leqslant}j{\leqslant}d$, the Laplace–Beltrami operator of the $j$-th component of $\operatorname{GL}_2(K_\infty)=\operatorname{GL}_2({\mathbb{R}})^d$, $$\label{LB}
\Delta_j:=-y_j^2(\partial^2_{x_j}+\partial^2_{y_j})+y_j\partial_{x_j}\partial_{\vartheta_j},$$ acts on the dense subset $V^{\infty}_{\pi}$ of smooth vectors by a scalar 