The Weyl character formula, the half-spin representations, and equal rank subgroups

Let B be a reductive Lie subalgebra of a semi-simple Lie algebra F of the same rank both over the complex numbers. To each finite dimensional irreducible representation V(λ) of F we assign a multiplet of irreducible representations of B with m elements in each multiplet, where m is the index of the Weyl group of B in the Weyl group of F. We obtain a generalization of the Weyl character formula; our formula gives the character of V(λ) as a quotient whose numerator is an alternating sum of the characters in the multiplet associated to V(λ) and whose denominator is an alternating sum of the characters of the multiplet associated to the trivial representation of F.