Toric degeneration of branching algebras

For each classical symmetric pair (G,H), there is a naturally defined multi-graded algebra A(G,H), called the branching algebra for (G,H), which encodes the branching rule from G to H. This algebra has a natural family of subalgebras, depending on integer parameters. For a certain range of the parameters, the subalgebras have a particularly simple structure and are called stable branching algebras.In this paper, we show that the stable branching algebras for eight out of the ten families of classical symmetric pairs are flat deformations of the semigroup algebras of explicitly described lattice cones.