#### 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).

Done