Let $G$ be a locally compact group. Let $\pi$ be an orthogonal representation of $G$ on a real Hilbert space $H$. A continuous mapping $b \colon G \to H$ such that $b(gh) = b(g) + \pi(g)b(h)$ for all $g, h \in G$ is called a 1-cocycle with respect to $\pi$.
I am searching for interesting concrete examples of such cocycles which are not 1-coboundaries, on non-discrete groups (the existence of such cocycle implies that the group has not property (T)). More precisely, I am interested by:
1) non-amenable groups (for example on $\mathrm{SL}_2(\mathbb{R})$),
2) noncompact nilpotent groups,
3) the additive group of $p$-adic numbers $\mathbb{Q}_p$.
Any reference is welcome.