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 $\phi: G/B\to G$. The definition of $\phi$ depends only on a choice of reduced expression for the longest element $w_0$ in the Weyl group $W$. It assigns to any $gB$ a representative $g\in G$ together with a factorization into simple root subgroups and simple reflections. The cross-section $\phi$ is continuous along the components of Deodhar's decomposition of $G/B$. We introduce a generalization of the Chamber Ansatz and give formulas for the factors of $g=\phi(gB)$. These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety $(G/B)_{\ge 0}$ defined by Lusztig, giving a new proof of Lusztig's conjectured cell decomposition of $(G/B)_{\ge 0}$. We also give minimal sets of inequalities describing these cells.