Representations of the Super-Yangian of Type D(n, m)

Abstract We consider the classification problem for finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras $$\mathfrak {osp}_{2n|2m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>osp</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:math> with $$n\geqslant 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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 $$\mathfrak {osp}_{2|2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>osp</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> . In the Appendix, we construct an isomorphism between the Yangians associated with the Lie superalgebras $$\mathfrak {osp}_{2|2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>osp</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> and $$\mathfrak {gl}_{1|2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>gl</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> .