For applications, one is often interested in a finite-dimensional simple formula_1-module (a finite-dimensional irreducible representation). This is especially the case when formula_1 is the Lie algebra of a Lie group (or complexification of such), since, via the Lie correspondence, a Lie algebra representation can be integrated to a Lie group representation when the obstructions are overcome. The next criterion then addresses this need: by the positive Weyl chamber formula_154, we mean the convex cone formula_155 where formula_156 is a unique vector such that formula_157. The criterion then reads: