| Description |
This dissertation develops the structure theory of the category Whittaker modules for a complex semisimple Lie algebra. We establish a character theory that distinguishes isomorphism classes of Whittaker modules in the Grothendieck group of the category, then use the localization functor of Beilinson and Bernstein to realize Whittaker modules geometrically as certain twisted D-modules on the associated flag variety (so called "twisted Harish-Chandra sheaves"). The main result of this document is an algorithm for computing the multiplicities of irreducible Whittaker modules in the composition series of standard Whittaker modules, which are generalizations of Verma modules. This algorithm establishes that the multiplicities are determined by a collection of polynomials we refer to as Whittaker Kazhdan--Lusztig polynomials. |
