Whittaker modules for Heisenberg and affine Lie algebras

In 1978, B. Kostant introduced a class of modules for finite-dimensional semisimple complex Lie algebras, which he termed Whittaker modules because of their connections with Whittaker equations in number theory. In this thesis we initiate the study of Whittaker modules for affine Lie algebras.;For an affine Lie algebra g , we construct Whittaker modules using parabolic induction. Given a triangular decomposition g=n-⊕h⊕n+ and a nonzero Lie homomorphism eta : n+→C such that eta is zero on at least one generator of n+ , we associate to eta a standard parabolic subalgebra p of g . We study Whittaker modules M(eta, chi) for g which are constructed by inducing over the subalgebra p starting from irreducible Whittaker modules with central character chi for the finite-dimensional reductive Levi factor of p . We show that M(eta, chi) has a unique irreducible quotient L(eta, chi). In the case g=sl2&d14; we prove that if eta is such a homomorphism and V is an irreducible Whittaker module of type eta, then V ≅ L(eta, chi) for some central character chi of the finite-dimensional reductive Levi factor of the standard parabolic subalgebra associated to eta. We also establish an irreducibility criterion for M(eta, chi) for g=sl2&d14; .;We classify irreducible Whittaker modules for Heisenberg Lie algebras and prove that their annihilators are centrally generated. We also determine the irreducible Whittaker modules for the Lie algebra obtained by adjoining a degree derivation to an infinite-dimensional Heisenberg Lie algebra. Using these results, we construct a new class of modules for non-twisted affine Lie algebras, which we call imaginary Whittaker modules, as they are constructed by inducing over the same parabolic subalgebra as imaginary Verma modules, but with the vectors corresponding to the imaginary roots acting in a non-zero fashion. We prove that the imaginary Whittaker modules of non-zero level are always irreducible. This is an analogue of a similar result for imaginary Verma modules due to V. Futorny.