Braverman, Finkelberg and Nakajima introduced the generalized affine Grassmannian slices $\overline{ \mathcal W}^{ \lambda}_{ \mu}$ and showed that they are Coulomb branches of $3d$ $\mathcal N=4$ gauge theories. We prove a conjecture of theirs that a transversal slice of $\mathcal W^{ \nu}_{ \mu}$ in $\overline{ \mathcal W}^{ \lambda}_{ \mu}$ is isomorphic to $\overline{ \mathcal W}^{ \lambda}_{ \nu}$. In addition, they conjecture that Coulomb branches of $3d$ $\mathcal N=4$ gauge theories have symplectic singularities, and we confirm this conjecture for generalized affine Grassmannian slices $\overline{ \mathcal W}^{ \lambda}_{ \mu}$. Along the way we give a new proof of the smoothness of $\mathcal W^{ \nu}_{ \mu}$, which has been previously proven by Muthiah and Weekes using different method.
Categories: math.AG hep-th math-ph math.MP math.RT