Contravariant Form on Tensor Product of Highest Weight Modules

Abstract

We give a criterion for complete reducibility of tensor product V⊗Z of two irreducible highest weight modules V and Z over a classical or quantum semi-simple group in terms of a contravariant symmetric bilinear form on V⊗Z. This form is the product of the canonical contravariant forms on V and Z. Then V⊗Z is completely reducible if and only if the form is non-degenerate when restricted to the sum of all highest weight submodules in V⊗Z or equivalently to the span of singular vectors.