Let $ G$ be a connected, reductive group over $ \mathbb Q$ , with parabolic subgroup $ P = MN$ . Let $ \pi$ be a cuspidal automorphic representation of $ M(\mathbb A)$ . For a smooth, right $ K$ -finite function $ \phi$ in the induced space $ \operatorname{Ind}_{P(\mathbb A)}^{G(\mathbb A)} \pi$ (realized in a suitable way as a function $ \phi: G(\mathbb Q) \backslash G(\mathbb A )\rightarrow \mathbb C$ ), we can associate the Eisenstein series

$ $ E(g,\phi) = \sum\limits_{\delta \in P(\mathbb Q) \backslash G(\mathbb Q)} \phi(\delta g)$ $ Assuming $ \pi$ is chosen so that this series converges absolutely, one can define the constant term of the Eisenstein series along a parabolic subgroup $ P’$ with unipotent radical $ N’$ :

$ $ E_{P’}(g,\phi) = \int\limits_{N'(\mathbb Q) \backslash N'(\mathbb A)}E(n’g,\phi)dn’ \tag{0}$ $

I see the constant term defined in this way without reference to Fourier analysis. Is it possible to always realize this object as the constant term of an honest Fourier expansion on some product of copies of $ \mathbb A/\mathbb Q$ ?

This can be done when $ G = \operatorname{GL}_2$ and $ P = P’$ the usual Borel. The unipotent radical identifies with the additive group $ \mathbb G_a$ , and for fixed $ g \in G(\mathbb A)$ the function $ \mathbb A/\mathbb Q \rightarrow \mathbb C, n \mapsto \phi(ng)$ has an absolutely convergent Fourier expansion

$ $ E(ng,\phi) = \sum\limits_{\alpha \in \mathbb Q} \int\limits_{\mathbb A/\mathbb Q} E(n’ng,\phi) \psi(-\alpha n’)dn’ \tag{1}$ $ where $ \psi$ is a fixed nontrivial additive character of $ \mathbb A/\mathbb Q$ . The constant term is

$ $ \int\limits_{\mathbb A/\mathbb Q} E(n’ng,\phi) dn’$ $ Setting $ n = 1$ in (1) gives us a series expansion for $ E(g,\phi)$ and (0) is the constant term of this series.