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 ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{2n\mathrm{\mid }2m}$ 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 ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{2\mathrm{\mid }2}$. In the Appendix, we construct an isomorphism between the Yangians associated with the Lie superalgebras ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{2\mathrm{\mid }2}$ and ${\mathfrak{g}\mathfrak{l}}_{1\mathrm{\mid }2}$.