The PBW filtration

Feigin, Evgeny

doi:10.1090/S1088-4165-09-00349-5

The PBW filtration
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by Evgeny Feigin

Represent. Theory 13 (2009), 165-181

DOI: https://doi.org/10.1090/S1088-4165-09-00349-5

Published electronically: May 1, 2009

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Abstract:

In this paper we study the PBW filtration on irreducible integrable highest weight representations of affine Kac-Moody algebras $\widehat {\mathfrak {g}}$. The $n$-th space of this filtration is spanned by the vectors $x_1\dots x_s v$, where $x_i\in \widehat {\mathfrak {g}}$, $s\le n$, and $v$ is a highest weight vector. For the vacuum module we give a conjectural description of the corresponding adjoint graded space in terms of generators and relations. For $\mathfrak {g}$ of the type $A_1$ we prove our conjecture and derive the fermionic formula for the graded character.

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