Duality in derived category of 𝒪∞

Let Formula: see text be a reductive Lie algebra over a Formula: see text-adic field Formula: see text with a split Cartan algebra Formula: see text and a Borel subalgebra Formula: see text. In analogy with the classical category Formula: see text of Bernstein-Gelfand-Gelfand, we define category Formula: see text for Formula: see text, and the thick category Formula: see text, which is the smallest abelian subcategory of the category of all Formula: see text-modules which contains Formula: see text and is stable under extensions. We show that the functor Formula: see text preserves Formula: see text, which is the subcategory of complexes of Formula: see text-modules with cohomology modules in Formula: see text. From a result of Coulembier-Mazorchuk we deduce that this subcategory is equivalent to Formula: see text. We then introduce the duality functor Formula: see text on Formula: see text. Some examples are also provided for the connection between the dualities and locally analytic representations.