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).