of generality[^3] we shall assume that $\omega$, viewed as a character of $\mathbb{A}^{\times}$, is trivial on ${K_{\infty,+}^{\text{diag}}}$. The group $\operatorname{GL}_2(\mathbb{A})$ acts by right translation on the Hilbert space $$L^2(\operatorname{GL}_2(K)\backslash \operatorname{GL}_2(\mathbb{A}), \omega)$$ of measurable functions $\phi:\operatorname{GL}_2(\mathbb{A})\to{\mathbb{C}}$ satisfying $$\begin{gathered}
\phi(\gamma zg) = \omega(z) \phi(g), \qquad
\gamma \in \operatorname{GL}_2(K), \ \ z \in Z(\mathbb{A}), \ \
g \in \operatorname{GL}_2(\mathbb{A}),\\
\langle g, g\rangle := \int_{\operatorname{GL}_2(K)Z(\mathbb{A})\backslash
\operatorname{GL}_2(\mathbb{A})} |\phi(g)|^2 \,dg < \infty.\end{gathered}$$ A function $\phi \in L^2(\operatorname{GL}_2(K)\backslash \operatorname{GL}_2(\mathbb{A}),
\omega)$ is called cuspidal if $$\int_{K\backslash \mathbb{A}}\phi\left(\left(\begin{matrix}1 & x\\0 & 1\end{matrix}\right) g\right) dx = 0 \quad \text{ for almost all } g \in \operatorname{GL}_2(\mathbb{A}).$$ We have a $\operatorname{GL}_2(\mathbb{A})$-invariant decomposition $$L^2(\operatorname{GL}_2(K)\backslash \operatorname{GL}_2(\mathbb{A}), \omega) = L_{\text{cusp}} \oplus L^{\perp}_{\text{cusp}}$$ into the space of cuspidal functions and its orthogonal complement. The cuspidal space decomposes into a Hilbert space direct sum[^4] of irreducible automorphic representations: $$L_{\text{cusp}} = \bigoplus_{\pi\in \mathcal{C}_\omega} V_{\pi}.$$ The orthogonal complement $L^{\perp}_{\text{cusp}} $ is described in detail in [@GJ Sections 3–5], see also [@Bu Section 3.7]: For any Hecke character $\chi$ satisfying $\chi^2 = \omega$ (which is necessarily trivial on ${K_{\infty,+}^{\text{diag}}}$) let $V_\chi$ be the subspace generated by the function $g\mapsto\chi(\det g)$, then we have a $\operatorname{GL}_2(\mathbb{A})$-invariant (orthogonal) decomposition $$L^{\perp}_{\text{cusp}}=L_{\text{sp}} \oplus L_{\text{cont}},\qquad
L_{\text{sp}}:=\bigoplus_{\chi^2=\omega} V_\chi,$$ where $L_{\text{cont}}$ can be described as follows.

