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 $G=O_{\ell },S{p}_{2\ell }$.

52 pages, v2: Appendix A added to give details of proof of Proposition 3.1, together with minor revisions