Steinberg's cross-sections over convex elements

The aim of this thesis is to prove the existence of Steinberg's cross-sections for convex elements. In the first part, we give some necessary preliminary knowledge about these objects, including reductive groups, tori, (twisted) Weyl groups, and some basic notions of algebraic geometry. In the second part, we introduce convex elements in the twisted Weyl group, show a way to build them, and associate a subvariety (the Steinberg's cross-section), which intersects any conjugacy class it meets transversely. Eventually, we discuss how this result generalizes previous works, state a conjecture about minimal lenght twisted Coxeter elements being convex, and introduce some applications of this work.