Parametrizations of Flag Varieties

For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (set-theoretical) cross-section φ : G/B → G. The definition of φ depends only on a choice of reduced expression for the longest element w0 in the Weyl group W . It assigns to any gB a representative g ∈ G together with a factorization into simple root subgroups and simple reflections. The cross-section φ is continuous along the components of Deodhar’s decomposition of G/B [6]. We introduce a generalization of the Chamber Ansatz of [2] and give formulas for the factors of g = φ(gB). These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety (G/B)≥0 defined by Lusztig [9], giving a new proof of Lusztig’s conjectured cell decomposition of (G/B)≥0. We also give minimal sets of inequalities describing these cells.