The Brylinski Filtration for Affine Kac-Moody Algebras and Representations of ð'²-algebras

We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra g, a notion first introduced by Slofstra. The associated graded space of this filtration on dominant weight spaces of integrable highest weight modules of 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 ð'²-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 ð'² and that the Brylinski filtration is induced by the Poincaré-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 ð'² for the ð'²-action on the principal rather than on the homogeneous realization of L(Λ0).