The PBW filtration and convex polytopes in type B

We study the PBW filtration on irreducible finite-dimensional representations for the Lie algebra of type Bn. We prove in various cases, including all multiples of the adjoint representation and all irreducible finite-dimensional representations for B3, that there exists a normal polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence we obtain several classes of examples for favourable modules and graded combinatorial character formulas.