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

TL;DR

This paper extends the affine Brylinski-Kostant filtration to all simply-laced affine Lie algebras, providing a uniform proof and exploring its connections with $\\mathscr{W}$-algebras, representation theory, and algebraic combinatorics.

Contribution

It offers a type-independent, uniform proof of the affine Brylinski filtration for all simply-laced affine Lie algebras, generalizing previous type-specific results.

Findings

Constructed a uniform proof for all simply-laced affine Lie algebras.

Extended the affine Brylinski-Kostant filtration beyond type A.

Connected the filtration with $\\mathscr{W}$-algebras and representation theory.

Abstract

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.