Mathematics Colloquia and Seminars

In the paper "Coulomb branches of $3d \, \mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian" authors defined so-called generalized transversal slices in the affine Grassmannian of a reductive group $G$. These varieties depend on a pair $\lambda, \mu$ of cocharacters of $G$ such that $\lambda$ is dominant and $\mu \leqslant \lambda$. In types $ADE$ these varieties are isomorphic to Coulomb branches of the corresponding framed quiver gauge theories. Symplectic duality predicts that these varieties should be isomorphic to affine spaces in the case when $\lambda$ is minuscule and $\mu$ lies in the $W$-orbit of $\lambda$. Hiraku Nakajima proved this fact in type $A$ using the identification of generalized slices with bow varieties (not published). We will give the proof for any reductive group $G$ and will also describe the standard Poisson structure on slices. Time permitting we will then discuss applications of this result including coverings of some convolution diagrams of generalized slices by affine spaces (trying to answer a question of Hiraku Nakajima and Michael Finkelberg). This is a work in progress with Ivan Perumov.