Contravariant pairings between standard Whittaker modules and Verma modules

DOI10.1016/J.JALGEBRA.2022.06.017arXiv2106.10029MaRDI QIDQ6370651

Anna Romanov, Adam Brown

Publication date: 18 June 2021

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 $mathcalO$ to the Miliv{c}i'{c}--Soergel category $mathcalN$. 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 $mathcalD$-modules. We prove that with these costandard modules, blocks of category $mathcalN$ have the structure of highest weight categories and we establish a BGG reciprocity theorem for $mathcalN$.

Mathematics Subject Classification ID

Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) (17B10) Semisimple Lie groups and their representations (22E46) Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) (22E47)

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