Consider the Hilbert space $V = L^2(Z(\mathbb{A})\mathrm{GL}_2(\mathbb{Q})\backslash \mathrm{GL}_2(\mathbb{A}))$. This is a unitary representation of $\mathrm{GL}_2(\mathbb{A})$, acting by right multiplication, so we can ask for the spectrum of unitary irreducible representations appearing in its decomposition.
$\mathrm{GL}_2(\mathbb{R})$ naturally embeds into $\mathrm{GL}_2(\mathbb{A})$, and let us define $V_0$ to be the subspace of $V$ consisting of vectors fixed by $\mathrm{GL}_2(\mathbb{R})$. The action of $\mathrm{GL}_2(\mathbb{A})$ on $V$ preserves $V_0$.
Question: What is the decomposition of $V_0$ into irreducible representations of $\mathrm{GL}_2(\mathbb{A})$? In other words, what are the automorphic representations which are trivial representations of $\mathrm{GL}_2(\mathbb{R})$?
Comment: I couldn't easily find this question answered in the literature, because there the focus is on cuspidal representations, and the trivial representation of $\mathrm{GL}_2(\mathbb{R})$ is not cuspidal. So the answer should somehow be encoded in the properties of the Eisenstein series, because in the classical theory the trivial representation = constant function, appears as a pole of the Eisenstein series.