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My reference is Daniel Bump's book, Automorphic Forms and Representations. $G$ is a connected reductive group over a number field $k$ (in Bump's book he takes $G = \operatorname{GL}_n$). Let $K = K_{\infty}K_f$ be a usual compact subgroup of $G(\mathbb A)$, and let $\mathfrak g_{\infty}$ be the Lie algebra of the real Lie group $G(k_{\infty}) = \prod\limits_{v \mid \infty} G(k_v)$.

Definition 1: A representation of $G(\mathbb A)$ is a vector space $V$ which is both a $(\mathfrak g_{\infty},K_{\infty})$-module as well as a smooth representation of the totally disconnected group $G(\mathbb A_f) = \prod\limits_{v < \infty}'G(k_v)$. The actions of $\mathfrak g_{\infty}$ and $K_{\infty}$ must each commute with the action of $G(\mathbb A_f)$. Thus a "representation of $G(\mathbb A)$" is not, strictly speaking, actually a representation of the topological group $G(\mathbb A)$. It is called irreducible (resp. admissible) if the both underlying $(\mathfrak g_{\infty}, K_{\infty})$-modules and $G(\mathbb A_f)$-representations are so.

Let $\omega$ be a unitary character of $Z(\mathbb A)/Z(k)$. Let $\mathscr H = L_0^2(G(k)\backslash G(\mathbb A),\omega)$ be the Hilbert space of cusp forms with central character $\omega$. The definition of $\mathscr H$ is given here. This is a continuous unitary representation of the topological group $G(\mathbb A)$, which acts by right translation.

Definition 2: A cuspidal representation of $G(\mathbb A)$ is a closed irreducible subrepresentation of $\mathscr H$. So a cuspidal representation is an honest group representation of $G(\mathbb A$).

If $(\pi,V)$ is a cuspidal representation, then the space $V_{\textrm{fin}}$ of $K$-finite vectors in $V$ is dense and an irreducible, admissible representation of $G(\mathbb A)$ in the sense of Definition 1 (Theorem 3.3.4, Bump). Since $\pi$ is unitary, we may uniquely decompose $\pi = \pi_{\infty} \otimes \pi_f$, with $\pi_{\infty}$ an irreducible admissible unitary representation of the real group $G(k_{\infty})$, and $\pi_f$ an irreducible admissible unitary representation of $G(\mathbb A_f)$, and $\pi_{\infty}$ is uniquely determined by its underlying $(\mathfrak g_{\infty},K_{\infty})$-module structure.

Definition 3: Let $\omega$ be a character of $Z(\mathbb A)/Z(k)$ (no longer assumped unitary). An automorphic form on $G(\mathbb A)$ with central character $\omega$ is a smooth (in the usual sense) function $f: G(\mathbb A) \rightarrow \mathbb C$ which satisfying $f(z\alpha g) = \omega(z)f(g)$ for $z \in Z(\mathbb A), \alpha \in G(k), g \in G(\mathbb A)$ which is $K$-finite, $\mathcal Z$-finite (where $\mathcal Z$ is the center of the complexified universal enveloping algebra of $\mathfrak g_{\infty}$), and is of "moderate growth." (Reference, pg. 300, Bump)

Let $\mathcal A(G(k) \backslash G(\mathbb A),\omega)$ be the space of automorphic forms of $G(\mathbb A)$ with central character $\omega$. We call $f \in \mathcal A(G(k) \backslash G(\mathbb A),\omega)$ a cusp form if for all unipotent radical $N(k)$ of proper parabolic $k$-subgroups, we have $$\int\limits_{N(k) \backslash N(\mathbb A)} f(ng)\, dn =0$$ for all $g \in G(\mathbb A)$. Let $\mathcal A_0(G(k) \backslash G(\mathbb A),\omega)$ be the space of cusp forms.

Then $\mathcal A$ and $\mathcal A_0$ are representations of $G(\mathbb A)$ in the sense of Definition 1.

Definition 4: An automorphic representation of $G(\mathbb A)$ is an irreducible representation (in the sense of Definition 1) which is isomorphic to a subquotient of $\mathcal A$.

Okay, here are my questions:

  1. Did I state anything incorrectly?

  2. Is an automorphic representation automatically admissible?

  3. If $V$ is an automorphic representation which is isomorphic to a subquotient of cusp forms $\mathcal A_0$, is $V$ actually isomorphic to a subrepresentation of $\mathcal A_0$?

  4. Assume $\omega$ is unitary. If $(\pi,V)$ is a cuspidal representation of $G(\mathbb A)$ in the sense of Definition 2, is $V_{\textrm{fin}}$ an automorphic representation of $G(\mathbb A)$ in the sense of Definition 4? If so, is it isomorphic to a subrepresentation of $\mathcal A_0$?

  5. Assume $\omega$ is unitary. If $V$ is an automorphic representation of $G(\mathbb A)$ in the sense of Definition 4, which is isomorphic to a subquotient of $\mathcal A_0 = \mathcal A_0(G(k) \backslash G(\mathbb A),\omega)$, is $V$ is the space of $K$-finite vectors of a cuspidal representation of $G(\mathbb A)$ in the sense of Definition 2?

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    $\begingroup$ It looks like you've got everything right, and the answers to your questions are uniformly YES. A standard reference is Borel and Jacquet's article in Corvallis (Proc. of Symposia in Pure Mathematics, vol 33 (1979) part 1, pp.189--202), and they give references to earlier work of Harish-Chandra and others along the way. See, for example, section 1.8 (working with real groups) and 4.4-4.8 for the adelic formulation. $\endgroup$ – Marty Dec 31 '18 at 22:25

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