Oscillator Representations of Quantum Affine Orthosymplectic Superalgebras

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 \(X_n^{(1)}\) $X_n^{(1)}$ and the finite-dimensional irreducible representations of type \(Y_n^{(1)}\) $Y_n^{(1)}$ for \((X,Y)=(C,D),(D,C)\) $(X,Y)=(C,D),(D,C)$ 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 \((\mathfrak {g},G)\) $(\mathfrak{g},G)$ , where \(\mathfrak {g}=\mathfrak {sp}_{2n}, \mathfrak {so}_{2n}\) $\mathfrak{g}={\mathfrak{sp}}_{2n},{\mathfrak{so}}_{2n}$ and \(G=O_\ell , Sp_{2\ell }\) $G=O_{\ell },S{p}_{2\ell }$ .