Symplectic leaves for generalized affine Grassmannian slices

Abstract

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.

9 pages, comments welcome