On the coadjoint Virasoro action

The set of coadjoint orbits of the Virasoro algebra at level 1 is in bijection with the set of conjugacy classes in a certain open subset $\widetilde{\rm SL}(2,\mathbb{R})_+$ of the universal cover of ${\rm SL}(2,\mathbb{R})$. We strengthen this bijection to a Morita equivalence of quasi-symplectic groupoids, integrating the Poisson structure on $\mathfrak{vir}^*_\mathsf{1}(S^1)$ and the Cartan-Dirac structure on $\widetilde{\rm SL}(2,\mathbb{R})_+$, respectively.