Abstract. We provide N-filtrations on the negative part U q (n − ) of the quantum group associated to a finite-dimensional simple Lie algebra g, such that the associated graded algebra is a skew-polynomial algebra on n − . The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an N-filtration on any finite dimensional simple g-module. We prove for type A n , C n , B 3 , D 4 and G 2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any g.

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Abstract. In this note, we study the Hilbert-Poincaré polynomials for the associated PBW-graded modules of simple modules for a simple complex Lie algebra. The computation of their degree can be reduced to modules of fundamental highest weight. We provide these degrees explicitly.Nousétudions les polynômes de Hilbert-Poincaré pour les modules PBWgradués associés aux modules simples d'une algèbre de Lie simple complexe. Le calcul de leur degré peutêtre restreint aux modules de plus haut poids fondamental. Nous donnons une formule explicite pour ces degrés.

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