A polytope calculus for semisimple groups

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Affine Mirković-Vilonen polytopes (2011, methodology): “In this model, proposed by Anderson [2], one associates a convex polytope Pol(b) ⊆ RI to each element b ∈ B(−∞).”
The decomposition theorem, perverse sheaves and the topology of algebraic maps (2007, background): “This formula implies that: i) if j : U → Y is an open immersion of irreducible varieties and U is nonsingular of dimension d, then j!∗L[d] is canonically isomorphic to ICY (L); ii) if Y is a nonsingular curve, then j!∗L[1] = ICY (L) = R j∗L[1].”
Preprojective algebras and MV polytopes (2010, background): “On the combinatorial side, a crystal can be realized by several models, each of which provides a concrete access to it; as an example, Anderson [1] and the second author’s [10, 11] MV (Mirković, Vilonen) polytopes form a model for the crystal B(−∞) of the positive part U(n) of U(g).”
Mirkovic-Vilonen cycles and polytopes (2005, methodology): “Lemma 2.1. If Sµ w ∩ Sν v 6= ∅ then ν≥w µ. Proof. Let L∈ Sµ w ∩ Svν. Then by (2), tν = lims→∞ L· (v· ρ∨)(s). Since Sµ w is T-invariant, this shows that tν ∈ Sµ w. So by (3), ν≥w µ. Following Anderson [A2], for µ≥ 0 we deﬁne an MV cycle of coweight µto be a component of X(µ) where X(µ) is the interse”