Definition of unitary representation of $\mathbf G(\mathbb A_k)$

Question

Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ is a Hilbert space $V$ together with a continuous homomorphism $\pi: G \rightarrow \operatorname{GL}(V)$ whose image is contained in the group $U(V)$ of unitary operators on $V$.

What is the topology on $\operatorname{GL}(V)$ (which I assume is the group of bounded linear operators on $V$) being considered here? Is it the induced topology coming from the norm topology?

I am trying to compare this definition with one given by Gerald Folland in A Course in Abstract Harmonic Analysis, which requires that for each $v \in V$ the map $g \mapsto \pi(g)v$ be continuous $G \rightarrow V$, where $V$ is taken in the norm topology. Are these two definitions of unitary representations different?

This matters because one later defines the Fell topology on the unitary dual $\hat{G}$ of $G$, and I want to know which representations are actually in $\hat{G}$.

Answer 1

It is absolutely essential that the space of (bounded/continuous) operators be given the "strong" operator topology (strictly weaker than the norm topology), and the map $G\times V\to V$ to be jointly continuous.

This is not a pathology: even in very simple cases, such as $G=\mathbb R$ acting on $V=L^2(\mathbb R)$, that joint continuity fails when operators are given the uniform norm topology, since there is no sufficient uniform bound on change in $g\in G$ so that $g\cdot v$ is close to $v$ for all $v\in V$. E.g., tent functions with ever-narrowing support illustrate this.

Answer 2

Considering a topological group $G$, a Hilbert space $V$ and a corresponding unitary representation, that is a homomorphism $\pi:G\to U(V)$, the following are equivalent:

1. $\pi$ is continuous when $U(V)$ is taken with the weak operator topology.

2. $\pi$ is continuous when $U(V)$ is taken with the strong operator topology.

3. For every $v\in V$, the orbit map $G\to V$ given by $g\mapsto gv$ is continuous.

4. The action map $G\times V\to V$ given by $(g,v)\mapsto \pi(g)(v)$ is continuous.

In fact, the implications $4 \Rightarrow 3\Rightarrow 2\Rightarrow 1$ are trivially true, while $1\Rightarrow 4$ follows from the uniform convexity of $V$ (thus an analogue is valid for any isometric representation on a uniformly convex space).

The standard terminology is to refer to $\pi$ as a continuous unitary representation if it satisfies the properties above.

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It should be mentioned that for non-discrete locally compact groups (eg for your $\mathbf{G}(\mathbb{A}_k)$) unitary representations are almost never continuous when $U(V)$ is endowed with the norm topology, so a careful writer is unlikely to make such an assumption without mentioning it explicitly.