Whittaker modules, central characters, and tensor products for quantum enveloping algebras

Our focus is on the representation theory of the quantum enveloping algebra Uq( g ) of a finite-dimensional semisimple Lie algebra g . This algebra is a one-parameter deformation of the universal enveloping algebra of g .;We begin by defining and studying the class of Whittaker modules for the quantum enveloping algebra Uq( sl2 ) of sl2 . One of our main results describes an arbitrary Whittaker module as a quotient of Uq( sl2 ). From this description, we determine precise criteria for when a Whittaker module is simple as well as a decomposition of an arbitrary Whittaker module into indecomposable submodules. We also prove that the annihilator annUqsl2 (V) of a Whittaker module V is generated by its intersection with the center of Uq( sl2 ). This is the analogue of a classical result in the Lie algebra setting due to Kostant.;The tensor product of a Whittaker module with a highest weight module is a Whittaker module. To describe this tensor product, we study a related topic---central characters.;Often, the action of the center of Uq( g ) on a given module V is determined by a central character; that is, an algebra homomorphism from the center to the underlying field. In that case, V is said to admit a central character. We describe a process for constructing central characters of Uq ( g ) that involves linear characters on the group Q ∩ 2P, where Q is the root lattice and P is the weight lattice of g . By using the Weyl group of g , we determine when two characters are identical.;If V is a module for Uq( g ) admitting a central character, and if L a finite-dimensional simple module, then for any element z in the center of Uq( g ), we determine a polynomial in z that annihilates V ⊗ L. In the special case that V is a simple Whittaker module for Uq( sl2 ), this result leads to a complete description of the Whittaker module V ⊗ L. When V is a simple Whittaker module and M is a Verma module for U q( sl2 ) we show that the module V ⊗ M is the universal Whittaker module.