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Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We give a descriptionof characters of irreducible Whittaker modules for $\mathfrak{g}$ with anyinfinitesimal character, along with a Kazhdan-Lusztig algorithm for computingthem. This generalizes Milicic-Soergel's and Romanov's results for integralinfinitesimal characters. As a special case, we recover the non-integralKazhdan-Lusztig conjecture for Verma modules.

Kazhdan-Lusztig Algorithm for Whittaker Modules with Arbitrary Infinitesimal Characters