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We introduce the notion of a favourable module for a complex unipotentalgebraic group, whose properties are governed by the combinatorics of anassociated polytope. We describe two filtrations of the module, one given bythe total degree on the PBW basis of the corresponding Lie algebra, the otherby fixing a homogeneous monomial order on the PBW basis.  In the favourable case a basis of the module is parameterized by the latticepoints of a normal polytope. The filtrations induce flat degenerations of thecorresponding flag variety to its abelianized version and to a toric variety,the special fibres of the degenerations being projectively normal andarithmetically Cohen-Macaulay. The polytope itself can be recovered as aNewton-Okounkov body. We conclude the paper by giving classes of examples forfavourable modules.

Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations