Each finite dimensional irreducible representation V of a simple Lie algebra L admits a filtration induced by a principal nilpotent element of L. This, so-called, Brylinski or Brylinski-Kostant filtration, can be restricted to the dominant weight spaces of V, and the resulting Hilbert series is very interesting q-analogs of weight multiplicity, first defined by Lusztig.

This picture can be extended to certain infinite-dimensional Lie algebras L and to irreducible highest weight, integrable representations V. We focus on the level 1 vacuum modules of special linear affine Lie algebras. In this case, we show how to produce a basis of the dominant weight spaces that is compatible with the Brylinski filtration. Our construction uses the so-called W-algebra, a natural vertex algebra associated to L.

This is joint work with Sachin Sharma (IIT Kanpur) and Suresh Govindarajan (IIT Madras).