What question did this study set out to answer?

This paper aims to explore duality concepts in the derived category associated with a reductive Lie algebra and its modules.

February 22, 2026

Duality in derived category of 𝒪∞

Key Points

This paper aims to explore duality concepts in the derived category associated with a reductive Lie algebra and its modules.
Defined category O∞ and its thick subcategory in relation to the Borel subalgebra and Cartan algebra.
Introduced the functor that preserves the subcategory of complexes of modules with specified cohomology.
Utilized results from Coulembier-Mazorchuk to establish equivalences between subcategories.
Demonstrated that the functor preserves cohomology-related structures within the derived category.
Established that the introduced duality functor yields significant connections to locally analytic representations.

Abstract

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.

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