Convolution semigroups on Rieffel deformations of locally compact quantum groups

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