Concept of Facets in the structure of reductive algebraic groups Where can I find a precise definition of Facet ? In some online notes it is stated that Facet is a maximal subset of co-characters having the same sign for every root. But shouldn't then every facet is a Weyl chamber? 
 A: Maybe it's helpful to expand on the comment made by Paul Broussous.
There may be some differences in the way "facets" are explained in the literature, but from any viewpoint these are not the same as Weyl chambers.   Instead, the Weyl chambers are the facets of maximal dimension (equal to the semisimple rank $n$ of the reductive group involved).   Starting with the root system and a corresponding system of reflecting hyperplanes (one for each pair of roots $\pm \alpha$), the Weyl chambers are the connected components of the complement of the union of hyperplanes in the ambient euclidean space $\mathbb{R}^n$.    But other facets occur in their closures, starting with the walls of each Weyl chamber (of codimension one in the closure of that chamber).   Eventually each closure of a Weyl chamber is a disjoint union of facets of different dimension, the extreme case being a point.   
This is the kind of description one finds in Bourbaki Lie Groups and Lie Algebras, Chap. 4-6 (especially section 5.1) or Jantzen Representations of Algebraic Groups (II.6).    The formal definition of "facet" is given just in terms of the root system.  Naturally it's also possible to recast definitions in terms of the coroots, depending on how the geometric ideas are being approached.   I guess this is where your reference to cocharacters comes into play. 
