Layer structure of irreducible Lie algebra modules

Let h be a Cartan subalgebra of the finite-dimensional simple complex Lie algebra g. It is argued that every finite-dimensional irreducible g-module admits a unique vector-space decomposition in terms of layers, where a layer is an h-module associated with the set of distinct weights appearing in an irreducible g-module. Ensuing results include a new approach to the computation of Weyl characters, and a closed-form expression for the number of distinct weights in a finite-dimensional irreducible g-module. The latter is given by a polynomial in the Dynkin labels, of degree given by the rank of g