  We study a category of Whittaker modules over a complex semisimple Lie
algebra by realizing it as a category of twisted D-modules on the associated
flag variety using Beilinson-Bernstein localization. The main result of this
paper is the development of a geometric algorithm for computing the composition
multiplicities of standard Whittaker modules. This algorithm establishes that
these multiplicities are determined by a collection of polynomials we refer to
as Whittaker Kazhdan-Lusztig polynomials. In the case of trivial nilpotent
character, this algorithm specializes to the usual algorithm for computing
multiplicities of composition factors of Verma modules using Kazhdan-Lusztig
polynomials.
