Two proofs of a Jantzen Conjecture for Whittaker Modules

TL;DR

This paper introduces a filtration for standard Whittaker modules over complex semisimple Lie algebras, generalizing the Jantzen filtration and conjectures, with algebraic and geometric proofs.

Contribution

It defines a new filtration for Whittaker modules that generalizes the Jantzen filtration and proves key properties using algebraic and geometric methods.

Findings

Filtration specializes to the Jantzen filtration of Verma modules.

Embeddings of standard Whittaker modules are strict with respect to the filtration.

Filtration layers are semisimple.

Abstract

We define a filtration of a standard Whittaker module over a complex semisimple Lie algebra and and establish its fundamental properties. Our filtration specialises to the Jantzen filtration of a Verma module for a certain choice of parameter. We prove that embeddings of standard Whittaker modules are strict with respect to our filtration, and that the filtration layers are semisimple. This provides a generalisation of the Jantzen conjectures to Whittaker modules. We prove these statements in two ways. First, we give an algebraic proof which compares Whittaker modules to Verma modules using a functor introduced by Backelin. Second, we give a geometric proof using mixed twistor $\mathcal{D}$-modules.