Invariants of the Cox rings of low-complexity double flag varieties for classical groups

We find the algebras of unipotent invariants of Cox rings for all double flag varieties of complexity 0 and 1 for the classical groups; namely, we obtain presentations of these algebras. It is well known that such an algebra is simple in the case of complexity 0. We show that, in the case of complexity 1, the algebra in question is either a free algebra or a hypersurface. Knowing the structure of this algebra permits one to effectively decompose tensor products of irreducible representations into direct sums of irreducible representations.