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 \(\mathfrak {osp}_{2n|2m}\) ${\mathfrak{osp}}_{2n|2m}$ with \(n\geqslant 2\) $n⩾2$ . 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 \(\mathfrak {osp}_{2|2}\) ${\mathfrak{osp}}_{2|2}$ . In the Appendix, we construct an isomorphism between the Yangians associated with the Lie superalgebras \(\mathfrak {osp}_{2|2}\) ${\mathfrak{osp}}_{2|2}$ and \(\mathfrak {gl}_{1|2}\) ${\mathfrak{gl}}_{1|2}$ .