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 \(G=O_\ell , Sp_{2\ell }\).