Kazhdan-Lusztig Algorithm for Whittaker Modules with Arbitrary Infinitesimal Characters

TL;DR

This paper extends the Kazhdan-Lusztig algorithm to compute characters of irreducible Whittaker modules for complex semisimple Lie algebras with arbitrary infinitesimal characters, generalizing previous integral cases.

Contribution

It introduces a Kazhdan-Lusztig algorithm applicable to all infinitesimal characters for Whittaker modules, broadening the scope of character computations.

Findings

Provides a character formula for irreducible Whittaker modules with any infinitesimal character.

Generalizes existing algorithms from integral to arbitrary infinitesimal characters.

Recovers the non-integral Kazhdan-Lusztig conjecture for Verma modules.

Abstract

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