Characters of Representations of Affine Kac-moody Lie Algebras at the Critical Level

for X,Y ∈ ḡ, m,n ∈ Z, where X(m) = X⊗t with X ∈ ḡ and m ∈ Z and (·|·) is the normalized invariant inner product of ḡ. We identify ḡ with ḡ⊗C ⊂ g. Fix the triangular decomposition ḡ = n̄− ⊕ h̄⊕ n̄+, and the Cartan subalgebra of g as h = h̄⊕CK ⊕CD. We have h = h̄ ⊕CΛ0 ⊕Cδ, where Λ0 and δ are elements dual to K and D, respectively. Let L(λ) be the irreducible highest weight representation of g of highest weight λ ∈ h with respect to the standard triangular decomposition g = n− ⊕ h⊕ n+, where n− = n̄−⊕ ḡ⊗C[t ]t, n+ = n̄+ ⊕ ḡ⊗C[t]t.