The Brylinski Filtration for Affine Kac-Moody Algebras and Representations of ${\mathscr{W}}$ -algebras

TLDR: In this article, the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra was studied, and it was shown that the corresponding Verma module of the affine kac-moody algebras is an irreducible Verma.

Abstract: We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra $\mathfrak {g}$ , a notion first introduced by Slofstra. The associated graded space of this filtration on dominant weight spaces of integrable highest weight modules of $\mathfrak {g}$ has Hilbert series coinciding with Lusztig’s t-analog of weight multiplicities. For the level 1 vacuum module L(Λ0) of affine Kac-Moody algebras of type A, we show that the Brylinski filtration may be most naturally understood in terms of representations of the corresponding ${\mathscr{W}}$ -algebra. We show that the sum of dominant weight spaces of L(Λ0) in the principal vertex operator realization forms an irreducible Verma module of ${\mathscr{W}}$ and that the Brylinski filtration is induced by the Poincare-Birkhoff-Witt basis of this module. This explicitly determines the subspaces of the Brylinski filtration. Our basis may be viewed as the analog of Feigin-Frenkel’s basis of ${\mathscr{W}}$ for the ${\mathscr{W}}$ -action on the principal rather than on the homogeneous realization of L(Λ0).