The affine Brylinski filtration and $\mathscr{W}$-algebras

The Brylinski-Kostant filtration on a representation of a finite-dimensional semisimple Lie algebra has interpretations in terms of the algebra, geometry and combinatorics of the representation. Its extension to affine Lie algebras was first studied by Slofstra. Recent work of the present authors constructed a Poincar\'{e}-Birkhoff-Witt type basis for the dominant weight spaces of the basic representation of affine Lie algebras of type $A$, which is compatible with the affine Brylinski filtration. In this paper, we overcome the constraint of type dependence, and furnish a new, uniform proof which holds for all simply-laced affine Lie algebras.