Equivariant symplectic geometry of cotangent bundles

It is proved that, for any action of a reductive algebraic group $G$ on a quasiaffine algebraic variety $X$, there is a canonical $G$-equivariant symplectic rational Galois covering $f\colon T^*\mathrm{Hor}X\to T^*X$, where $\mathrm{Hor}X$ is the variety of horospheres (orbits of maximal unipotent subgroups of $G$) in $X$.