Alladi Ramakrishnan Hall

The Brylinski filtration and W-algebras

S. Viswanath

IMSc

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-Kostant filtration, can be restricted to the dominant
weight spaces of V, and the resulting Hilbert series are very interesting
q-analogs of weight multiplicity, first defined by Lusztig.

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

The talk will be mostly self-contained. This is joint work with Sachin
Sharma (IIT Kanpur) and Suresh Govindarajan (IIT Madras).