Reduction for Branching Multiplicities

Abstract

Abstract A reduction formula for the branching coefficients of the restrictions of representations of a semisimple group to a semisimple subgroup is proved in [9, 17, 33]. This formula holds when the highest weights of the representations belong to a codimension $1$ face of the Horn cone, which by [32] corresponds to some Schubert coefficient equal to $1$. We prove a similar reduction formula when this Schubert coefficient is equal to $2$ and show some properties of the class of the branch divisor corresponding to a generically finite morphism naturally defined in this context.