The symplectic groupoid for Adler-Gelfand-Dikii Poisson structure

The Adler-Gelfand-Dikii Poisson structure arises naturally in the study of $n$-th order differential operators on the circle and plays a central role in Poisson geometry and integrable systems. Let $G$ be one of the Lie groups $\mathrm{P}\mathrm{S}\mathrm{L}(n)$, $\mathrm{P}\mathrm{S}\mathrm{p}(n)$ (for even $n$), or $\mathrm{P}\mathrm{S}\mathrm{O}(n)$ (for odd $n$). In this paper, we construct the symplectic groupoid integrating the Adler-Gelfand-Dikii Poisson structure associated to $G$ and prove that it is Morita equivalent to the quasi-symplectic groupoid integrating the Dirac structure on $Y_n(\mathbf{C})$, where $Y_n(\mathbf{C})$ denotes the quotient of the space of quasi-periodic non-degenerate curves by homotopies preserving the monodromy.