We study Z-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac-Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function ϕ : N → {±} we associate a ϕ-highest weight module over the Heisenberg Lie algebra and a ϕ-imaginary Verma module over the affine Lie algebra. We show that any ϕ-imaginary Verma module of nonzero level is irreducible.

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Abstract. We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra P of an affine Lie algebra G, our main result establishes the equivalence between a certain category of P-induced G-modules and the category of weight P-modules with injective action of the central element of G. In particular, the induction functor preserves irreducible modules. If P is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra P ps , P ⊂ P ps . The structure of P-induced modules in this case is fully determined by the structure of P ps -induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S.

We develop and describe continuous and discrete transforms of class functions on compact simple Lie group G as their expansions into series of uncommon special functions, called here E-functions in recognition of the fact that the functions generalize common exponential functions. The rank of G is the number of variables in the E-functions. A uniform discretization of the decomposition problem is described on lattices of any density and symmetry admissible for the Lie group G.