On a class of tensor product representations for the orthosymplectic superalgebra

The spinor representations for osp(m∣2n) are introduced. These generalize the spinors for so(m) and the symplectic spinors for sp(2n) and correspond to representations of the supergroup with supergroup pair (Spin(m)×Mp(2n),osp(m∣2n)). These spinor spaces are proved to be uniquely characterized as the completely pointed osp(m∣2n)-modules. The main aim is to study the tensor product of these representations with irreducible finite dimensional osp(m∣2n)-modules. Therefore a criterion for complete reducibility of tensor product representations of semisimple Lie superalgebras is derived. Finally the decomposition into irreducible osp(m∣2n)-representations of the tensor product of the super spinor space with an extensive class of such representations is calculated and also cases where the tensor product is not completely reducible are studied.