Convolution semigroups on Rieffel deformations of locally compact quantum groups

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