Combinatorics and topology of toric arrangements defined by root systems

Given the toric (or toral) arrangement defined by a root system Φ, we describe the poset of its layers (connected components of intersections) and we count its elements. Indeed we show how to reduce to zero-dimensional layers, and in this case we provide an explicit formula involving the maximal subdiagrams of the affine Dynkin diagram of Φ. Then we compute the Euler characteristic and the Poincare' polynomial of the complement of the arrangement, which is the set of regular points of the torus.