A new formula for weight multiplicities and characters

Let V0, (, ) be the real Euclidean space spanned by the root system R0 of g and let V be the space of affine linear functions on V0. We shall identify V with Rδ ⊕ V0 via the pairing (rδ + x, y) = r + (x, y) for r ∈ R, x, y ∈ V0. The dual affine root system is R = {mδ + α | m ∈ Z, α ∈ R0} ⊆ V where α ∨ means 2α (α,α) as usual. Fix a positive subsystem R + 0 ⊆ R0 with base {α1, · · · , αn} and let β be the highest short root. Then a base for R is given by a0 = δ − β , a1 = α ∨ 1 , · · · , an = α ∨ n , and we write si for the (affine) reflection about the hyperplane {x | (ai, x) = 0} ⊆ V0. The dual affine Weyl group is the Coxeter group W generated by s0, · · · , sn, and the finite Weyl group is the subgroup W0 generated by s1, · · · , sn. For w ∈ W , its length is the length of a reduced (i.e. shortest) expression of w in terms of the si. The group W acts on the weight lattice P of g, and each orbit contains a unique (minuscule) weight from the set: O := {λ ∈ P | (α, λ) = 0 or 1 ∀α ∈ R}.