Contravariant pairings between standard Whittaker modules and Verma modules

TL;DR

This paper classifies contravariant pairings between standard Whittaker modules and Verma modules, introduces generalized costandard modules, and demonstrates their role in the structure of category al N, extending classical techniques.

Contribution

It introduces a classification of contravariant pairings, defines generalized costandard modules, and establishes highest weight category structure and BGG reciprocity for category al N.

Findings

Contravariant pairings classified between Whittaker and Verma modules.

Costandard modules have unique irreducible submodules and match composition factors of standard Whittaker modules.

Category al N has a highest weight structure with BGG reciprocity.

Abstract

We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category $\mathcal{O}$ to the Mili\v{c}i\'{c}--Soergel category $\mathcal{N}$. We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of $\mathcal{D}$-modules. We prove that with these costandard modules, blocks of category $\mathcal{N}$ have the structure of highest weight categories and we establish a BGG reciprocity theorem for $\mathcal{N}$.