Tensor product of holomorphic discrete series representations of U(p,q) and quivers

Abstract

As shown by P-E Paradan, the set of orbits contained in the sum of two holomorphic orbits in the Lie algebra of U(p,q) is determined by a set of inequalities similar to the Horn inequalities for the sum of conjugacy classes of two Hermitian matrices. We give another proof of these inequalities using representations of quivers. We also discuss the implications of these inequalities for the decomposition of tensor products of representations of the holomorphic discrete series.