Oscillator representations of quantum affine orthosymplectic superalgebras

Abstract: We introduce a category of $q$-oscillator representations over the quantum affine superalgebras of type $D$ and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible $q$-oscillator representations of type $X_n^{(1)}$ and the finite-dimensional irreducible representations of type $Y_n^{(1)}$ for $(X,Y)=(C,D),(D,C)$ under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finite-dimensional representations of classical Lie algebras arising from Howe's reductive dual pairs $(\mathfrak{g},G)$, where $\mathfrak{g}=\mathfrak{sp}{2n}, \mathfrak{so}{2n}$ and $G=O_\ell, Sp_{2\ell}$.