Root systems, affine subspaces, and projections

We tackle several problems related to a finite irreducible crystallographic root system Φ in the real vector space E. In particular, we study the combinatorial structure of the subsets of Φ cut by affine subspaces of E and their projections. As byproducts, we obtain easy algebraic combinatorial proofs of refinements of Oshima's Lemma and of a result by Kostant, a partial result towards the resolution of a problem by Hopkins and Postnikov, and new enumerative results on root systems