Affine vertex algebras and an affine analog of Barbasch-Vogan's construction

This is an expository paper based on the authors’ joint works. The goal is to explain the statements and the ideas behind two conjectures on associated varieties and simple modules of simple affine vertex algebras $L_k(\mathfrak{g})$ for a simple and simply-laced Lie algebra $\mathfrak{g}$ and a integer level $k$ above the critical level.