Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra g has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given g, there is a limit crystal, usually denoted by B(−∞), which contains all the other crystals. When g is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in B(−∞). This polytope sits in the dual space of a Cartan subalgebra of g, and its edges are parallel to the roots of g. In this paper, we generalize this construction to the case where g is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root δ. We prove that these decorated polytopes are characterized by conditions on their normal fans and on their 2faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for B(−∞) constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra Λ of the same type as g. The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott's tilting theory for the category Λ-mod. The partitions we need come from studying the category of semistable Λ-modules of dimension-vector a multiple of δ.One of these tools is Buan, Iyama, Reiten and Scott's tilting ideals for Λ [14]. Let S i be the simple Λ-module of dimension-vector α i and let I i be its annihilator, a one-codimensional two-sided ideal of Λ. The products of these ideals I i are known to satisfy the braid relations, so to each w in the Weyl group of g, we can attach a two-sided ideal I w of Λ by the rule I w = I i 1 · · · I i ℓ , where s i 1 · · · s i ℓ is any reduced decomposition of w. Given a finite-dimensional Λ-module T , we denote the image of the evaluation map I w ⊗ Λ Hom Λ (I w , T ) → T by T w .Recall that the dominant Weyl chamber C 0 and the Tits cone C T are the convex cones in the dual of RI defined asWe will show the equality T min θ = T max θ = T w for any finite dimensional Λ-module T , any w ∈ W and any linear form θ ∈ wC 0 . This implies that dim T w is a vertex of Pol(T ) and that the normal cone to Pol(T ) at this vertex contains wC 0 . This also implies that Pol(T ) is contained inWhen θ runs over the Tits cone, it generically belongs to a chamber, and we have just seen that in this case, the face P θ is a vertex. When θ lies on a facet, P θ is an edge (possibly degenerate). More precisely, if θ lies on the facet that separates the chambers wC 0 and ws i C 0 , with say ℓ(ws i ) > ℓ(w), then (T min θ , T max θ ) = (T ws i , T w ). Results in [1] and [27] moreover assert that T w /T ws i is the direct sum of a finite number of copies of the Λ-module I w ⊗ Λ S i .There is a similar description when θ is in −C T ; here the submodules T w of T that come into play are the kernels of the coevaluation maps T → ...
