A Kazhdan-Lusztig algorithm for Whittaker modules

TL;DR

This paper develops a geometric algorithm to compute composition multiplicities of Whittaker modules over semisimple Lie algebras, generalizing Kazhdan-Lusztig theory through Whittaker Kazhdan-Lusztig polynomials.

Contribution

It introduces a novel geometric algorithm for Whittaker modules, extending Kazhdan-Lusztig theory to this broader context.

Findings

The algorithm computes multiplicities via Whittaker Kazhdan-Lusztig polynomials.

Specializes to classical Kazhdan-Lusztig polynomials for trivial nilpotent character.

Provides a geometric realization of Whittaker modules using D-modules and localization.

Abstract

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.