Symplectic leaves for generalized affine Grassmannian slices

The generalized affine Grassmannian slices $\overline{\mathcal{W}}_\mu^\lambda$ are algebraic varieties introduced by Braverman, Finkelberg, and Nakajima in their study of Coulomb branches $3d$ $\mathcal{N}=4$ quiver gauge theories. We prove a conjecture theirs showing that the dense open subset $\mathcal{W}_\mu^\lambda \subseteq \overline{\mathcal{W}}_\mu^\lambda$ is smooth. An explicit decomposition into symplectic leaves follows as corollary. Our argument works over an arbitrary ring particular implies complex points $\mathcal{W}_\mu^\lambda(\mathbb{C})$ smooth holomorphic manifold.