Let $G$ be a locally compact unimodular group with center $Z$ and let $\omega$ be a unitary character of $Z$. To fix the discussion, an **irreducible** unitary representation $\pi$ with central character $\omega$ is

**square integrable**if for all $u,v\in V_\pi$, the matrix coefficient $g\mapsto \left< gu,v\right>$ is square integrable when viewed as a function on $G/Z$, and**tempered**if $\pi$ is weakly contained in the right regular representation $\mathrm{L}^2(G)$. That is, for any compact $C\subseteq G$, $\varepsilon>0$ and unit vector $v\in V_\pi$, there is a unit vector $\psi\in \mathrm{L}^2(G)$ such that $|\left<gv,v\right>-\left<g\psi,\psi\right>|<\varepsilon$ for all $g\in C$.

(There are other equivalent definitions for temperedness when $G$ is a real or $p$-adic reductive Lie group.)

It is not immediately obvious from the definitions that irreducible square-integrable representations are tempered, but it is known to be true in the following cases:

- $Z=1$ (since then $\pi$ above can be embedded in $\mathrm{L}^2(G)$),
- $G$ is a real or $p$-adic reductive Lie group.
- $G=Z$, or $G$ is compact, or more generally $G$ is amenable (since any irreducible representation is tempered in the previous sense).

My question is whether this is true for general locally compact groups. (Assume $G$ is unimdular and all irreducible unitary representations are admissible if this simplifies things.)

The question can be generalized even further as follows: Suppose that $Z$ is just a normal amenable subgroup of $G$ and let $\mathrm{L}^2_\omega(G/Z)$ denote the set of $(Z,\omega)$-equivariant functions $\psi:G\to \mathbb{C}$ such that $|\psi|$ is square integrable when viewed as a function on $G/Z$. Is it true that any irreducible representation of $G$ that is weakly contained in $\mathrm{L}^2_\omega(G/Z)$ ($G$ acts by right translations) is weakly contained in $\mathrm{L}^2(G)$?