Dualities and vertex operator algebras of affine type

We notice that for any positive integer $k$, the set of $(1,2)$-specialized characters of level $k$ standard $A_1^{(1)}$-modules is the same as the set of rescaled graded dimensions of the subspaces of level $2k+1$ standard $A_2^{(2)}$-modules that are vacuum spaces for the action of the principal Heisenberg subalgebra of $A_2^{(2)}$. We conjecture the existence of a semisimple category induced by the “equal level” representations of some algebraic structure which would naturally explain this duality-like property, and we study potential such structures in the context of generalized vertex operator algebras.