Relative Weyl Character Formula, Relative Pieri Formulas and Branching Rules for Classical Groups

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 \((GL(n),GL(n-1))\) .