Coadjoint orbits of Lie groupoids

Abstract

For a Lie groupoid $\mathcal{G}$ with Lie algebroid $A$, we realize the symplectic leaves of the Lie-Poisson structure on $A^{\ast }$ as orbits of the affine coadjoint action of the Lie groupoid $\mathcal{J}\mathcal{G}⋉T^{\ast }M$ on $A^{\ast }$, which coincide with the groupoid orbits of the symplectic groupoid $T^{\ast }\mathcal{G}$ over $A^{\ast }$. It is also shown that there is a fiber bundle structure on each symplectic leaf. In the case of gauge groupoids, a symplectic leaf is the universal phase space for a classical particle in a Yang-Mills field.