Relative Weyl Character formula, Relative Pieri formulas and Branching rules for Classical groups

Abstract: We give alternate proofs of the classical branching rules for highest weight representations of a complex reductive group $G$ restricted to a closed regular reductive subgroup $H$, where $(G,H)$ consist of the pairs $(GL(n+1),GL(n))$, $ (Spin(2n+1), Spin(2n)) $ and $(Sp(2n),Sp(2)\times Sp(2n-2))$. Our proof is essentially a long division. The starting point is a relative Weyl character formula and our method is an inductive application of a relative Pieri formula. We also give a proof of the branching rule for the case of $ (Spin(2n), Spin(2n-1))$, by a reduction to the case of $(GL(n),GL(n-1))$.