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We study the Brylinski filtration induced by a principal Heisenbergsubalgebra of an affine Kac-Moody algebra $\mathfrak{g}$, a notion firstintroduced by Slofstra. The associated graded space of this filtration ondominant weight spaces of integrable highest weight modules of $\mathfrak{g}$has Hilbert series coinciding with Lusztig's $t$-analogue of weightmultiplicities. For the level 1 vacuum module $L(\Lambda_0)$ of affineKac-Moody algebras of type $A$, we show that the Brylinski filtration may bemost naturally understood in terms of (vertex algebra) representations of thecorresponding $\mathcal{W}$-algebra. We show that the dominant weight spacestogether form an irreducible Verma module of $\mathcal{W}$ and that the naturalPBW basis of this module is compatible with the Brylinski filtration, therebydetermining explicitly the subspaces of the filtration. Our basis is theanalogue for the principal vertex operator realization of $L(\Lambda_0)$, ofFeigin-Frenkel's basis of $\mathcal{W}$.

The Brylinski filtration for affine Kac-Moody algebras and representations of $\mathcal{W}$-algebras