Induced Modules for Affine Lie Algebras

We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra $\mathcal P$ of an affine Lie algeba $\mathfrak G$, our main result establishes the equivalence between a certain category of $\mathcal P$-induced $\mathfrak G$-modules and the category of weight $\mathcal P$-modules with injective action of the central element of $\mathfrak G$. In particular, the induction functor preserves irreducible modules. If $\mathcal P$ is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra $\mathcal P^{ps}$, $\mathcal P\subset\mathcal P^{ps}$. The structure of $\mathcal P$-induced modules in this case is fully determined by the structure of $\mathcal P^{ps}$-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. König, V. Mazorchuk [Forum Math. 13 (2001), 641–661], B. Cox [Pacific J. Math. 165 (1994), 269–294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47–63].