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

Contravariant pairings between standard Whittaker modules and Verma modules