ABSTRACT
The Building I (G) of a reductive algebraic group G, over an algebraically closed field k, is the simplicial complex whose simplices are the parabolic subgroups of G. The incidence relation is defined as the symmetric inclusion relation between parabolics subgroups. It thus extends the definition of the Flag Complex of Gl(kr+1) to the setting of k-reductive groups. The chapter introduces the Minimal Generalized Galleries in the Building I (G). The set of generalised galleries of type g, issued from a fixed facet F admits a canonical "Cell Decomposition" indexed by generalized galleries of type g in the Weyl complex, which generalizes Bruhat cell decomposition. The chapter describes the minimal generalized gallery block decomposition of a Schubert cell parametrizing subgroup. One explicates the parametrization of the big cell of the galleries of type and its relation with the corresponding Schubert cell.
