Symplectic

In this talk, the Gelfand-Dikii Poisson structure is introduced as a Poisson structure on the space of $n$-th order Hill operators. For $n = 2$, the space of Hill operators is identified with the dual of the Virasoro algebra at level one, making the Poisson structure linear. For $n > 2$, the Poisson structure is quadratic. A coordinate-free construction of this Poisson structure is provided by the Drinfeld-Sokolov reduction. The symplectic leaves of this Poisson structure have been determined by Khesin and Ovsienko. In this talk, I will construct a symplectic groupoid integrating this Poisson structure and prove that it is Morita equivalent to a quasi-symplectic groupoid integrating the Cartan-Dirac structure on $\widetilde{PSL}(n,\mathbb{R})$.