# Some questions about cuspidal representations and automorphic representations

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?

• 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. – Marty Dec 31 '18 at 22:25