Contravariant form on tensor product of highest weight modules
journal contributionposted on 09.07.2019, 14:25 by Andrey I. Mudrov
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.