A Kazhdan-Lusztig Algorithm for Whittaker Modules

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. Contents1 The formulation in [Soe97] is in terms of the antispherical module of the Hecke algebra. We prove in Section 6.3 that this formulation is equivalent to conditions (i) and (ii) in Theorem 1.3.

Kazhdan-Lusztig Algorithm for Whittaker Modules with Arbitrary Infinitesimal Characters

“…In this section, we briefly review basic facts about Whittaker modules and character theory without proof. References include [Kos78], [McD85], [MiS14], [Mil], and [Rom21].…”

“…In §6, we will see that this correspondence preserves Whittaker Kazhnda-Lusztig polynomials. By [Rom21], Whittaker Kazhdan-Lusztig polynomials for Whittaker modules for the data pW λ , Π λ , Θ u λ q are given by parabolic Kazhdan-Lusztig polynomials. In this way, we can describe characters of Lpw C λ, ηq, C P W Θ zW Θ uW λ using parabolic Kazhdan-Lusztig polynomials for pW λ , Π λ , Θ u λ q.…”

“…Whittaker modules for real groups have been studied, for example in the work of Casselman-Hecht-Milicic [CHM00]. Over a complex semisimple Lie algebra, previous contributors include Kostant [Kos78] who studied the "non-degenerate" case, McDowell [McD85] who constructed and studied standard Whittaker modules, Milicic-Soergel who reproved Kostant's results with geometry in [MiS14] and obtained a description of the composition factors of standard Whittaker modules and their multiplicities in the case of integral infinitesimal characters in [MiS97], and Romanov [Rom21] who established a Kazhdan-Lusztig algorithm for Whittaker modules with regular integral infinitesimal characters.…”

“…We will be using the character theory of Whittaker modules developed in [Rom21]. As a byproduct, we obtain a Kazhdan-Lusztig algorithm for Whittaker modules.…”

“…It is used in conjunction with the decomposition theorem of holonomic D-modules [Moc11] to increase dimension of the supports. This part is modified from the algorithm in [Mil, V.2] for proving Kazhdan-Lusztig conjecture for principal block of category O; it is also almost identical to the one in [Rom21]. The last ingredient of the proof is the non-integral intertwining functors.…”