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