Representations of the super-Yangian of type $D(n,m)$

We consider the classification problem for finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras osp2n|2m with n 2. We give necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient and prove the conjecture for a class of representations with linear highest weights. The arguments are based on a new type of odd reflections for the Yangian associated with osp2|2. In the Appendix, we construct an isomorphism between the Yangians associated with the Lie superalgebras osp2|2 and gl1|2.