Symplectic leaves for generalized affine Grassmannian slices

Abstract

The generalized affine Grassmannian slices $\overline{\mathcal{W}}_μ^λ$ 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 $\mathcal{W}_μ^λ\subseteq \overline{\mathcal{W}}_μ^λ$ is smooth. An explicit decomposition of $\overline{\mathcal{W}}_μ^λ$ into symplectic leaves follows as a corollary. Our argument works over an arbitrary ring and in particular implies that the complex points $\mathcal{W}_μ^λ(\mathbb{C})$ are a smooth holomorphic symplectic manifold.

9 pages, comments welcome