Consider the convex cone of holomorphic orbits in the Lie algebra of \(U(p,q)\) . As in the classical case of Hermitian matrices, the set of elliptic orbits contained in the sum of two holomorphic orbits can be described by “Horn inequalities.” Using representations of quivers, we give another proof of these Horn recursion inequalities obtained by P-E Paradan. We recall the implications of these inequalities for decomposition of tensor product of two representations of the holomorphic discrete series of the group \(U(p,q)\) .