Note on some properties of generalized affine Grassmannian slices

Abstract

Braverman, Finkelberg and Nakajima introduced the generalized affine Grassmannian slices ${\overline{\mathcal{W}}}_{\mu }^{\lambda }$ 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 ${\mathcal{W}}_{\mu }^{\nu }$ in ${\overline{\mathcal{W}}}_{\mu }^{\lambda }$ is isomorphic to ${\overline{\mathcal{W}}}_{\nu }^{\lambda }$. 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 ${\overline{\mathcal{W}}}_{\mu }^{\lambda }$. Along the way we give a new proof of the smoothness of ${\mathcal{W}}_{\mu }^{\nu }$, which has been previously proven by Muthiah and Weekes using different method.

arXiv comment13 pages, an error in the proof of Proposition 4.12 corrected, comments welcome

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