Singular Poisson reduction of cotangent bundles

We consider the Poisson reduced space $(T^{\ast }Q)\mathrm{/}K$ with respect to a cotangent lifted action. It is assumed that $K$ is a compact Lie group which acts by isometries on the Riemannian manifold $Q$ and that the action on $Q$ is of single isotropy type. Realizing $(T^{\ast }Q)\mathrm{/}K$ as a Weinstein space we determine the induced Poisson structure and its symplectic leaves. We thus extend the Weinstein construction for principal fiber bundles to the case of surjective Riemannian submersions $Q\to Q\mathrm{/}K$.