  Consider a locally compact quantum group $\mathbb{G}$ with a closed classical
abelian subgroup $\Gamma$ equipped with a $2$-cocycle
$\Psi:\hat{\Gamma}\times\hat{\Gamma}\to\mathbb{C}$. We study in detail the
associated Rieffel deformation $\mathbb{G}^{\Psi}$ and establish a canonical
correspondence between $\Gamma$-invariant convolution semigroups of states on
$\mathbb{G}$ and on $\mathbb{G}^{\Psi}$.
