Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $G(k_v)$ which is smooth (the stabilizer of any vector is open in $G(k_v)$) and such that the $K$-fixed vectors of any open compact subgroup $K$ of $G(k_v)$ is finite dimensional.
If $v$ is infinite (real or complex), then $G(k_v)$ has the structure of a real Lie group. An admissible representation of $G(k_v)$ is defined here (https://en.wikipedia.org/wiki/Admissible_representation). It is define to be a representation $\pi: G(k_v) \rightarrow \textrm{GL}(\mathscr H)$ on a Hilbert space $\mathscr H$, together with a maximal compact subgroup $K$ of $G(k_v)$, with the following properties:
1 . $G(k_v) \times \mathscr H \rightarrow \mathscr H$ is continuous.
2 . The restriction of $\pi$ to $K$ is unitary.
3 . Each irreducible unitary representation of $K$ occurs with finite multiplicity.
I have a few questions about this:
1 . When $v$ is infinite, does the definition depend on the choice of maximal compact subgroup $K$ of $G(k_v)$? I don't know if they are all conjugate as in the case where $G$ is semisimple.
2 . When $v$ is infinite, does the notion of admissible representation require a Hilbert space? Is there such a thing as, say, an admissible Banach representation? (dropping the unitary assumptions)
3 . What is the definition of an admissible representation of $G(\mathbb{A}_k)$, where $\mathbb{A}_k$ is the ring of adeles of $k$?
Thank you.
