The fundamental items are the roots. We denote the set of roots by , the simple roots by and the positive roots by . Given a root , we can define the dual root where is the ambient inner product. When we start with a Lie algebra , the roots are the linear funtionals that arise as eigenvalues of the action of the maximal toral subalgebra on the rest of the Lie algebra via the adjoint action. This number is always an integer for roots.
Roots are nice.
And they span the space . The functionals which arise as eigenvalues of the action of on the representation are NOT necessarily integer linear combinations of the roots. BUT the true answer isn't that bad. The relation between the weight lattice and representation theory of runs as follows. The infinite-dimensional Verma module with highest weight has a maximal submodule, and thus has a unique irreducible quotient.