  We study the PBW filtration on the highest weight representations $V(\la)$ of
$\msl_{n+1}$. This filtration is induced by the standard degree filtration on
$U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr
V(\la)$ in terms of generators and relations. We also construct a basis of $gr
V(\la)$. As an application we derive a graded combinatorial character formula
for $V(\la)$, and we obtain a new class of bases of the modules $V(\la)$
conjectured by Vinberg in 2005.
