Weyl’s character formula and Lie algebra homology

In addition, define certain lattices and subsets: L∨ = X∗(T ) = group of algebraic homomorphisms Gm → T L = Hom(X∗(T ),Z), the dual of L ∨ X(T ) = group of algebraic homomorphisms T → Gm, the weights of T . There is a canonical map from L toX(T )—to λ corresponds the multiplicative character e defined by the formula e(μ(x)) = x 〉 , which makes sense because all algebraic homomorphisms Gm → Gm are of the form x 7→ x . This map is an isomorphism, as one can verify by assigning coordinates to T , which then becomes a product of copies of C. The point of not simply defining L to be X(T ) is that I use additive notation for L but multiplicative forX(T ). Thus e = ee. There is a map α 7→ α from ∆ to L, such that the matrix (〈α, β〉) is the Cartan matrix associated to a positive definiteW -invariant metric.