Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras

We classify the irreducible Whittaker modules for finite- and infinite-dimensional Heisenberg algebras and for the Lie algebra obtained by adjoining a degree derivation to an infinite-dimensional Heisenberg algebra. We use these modules to construct a new class of modules for non-twisted affine Lie algebras, which we call imaginary Whittaker modules. We show that the imaginary Whittaker modules of non-zero level are irreducible.