Duality in Derived Category $\mathcal O^\infty$

Abstract

Let $\bf{G}$ be a split connected reductive group over a finite extension $F$ of $\mathbb Q_p$, and let $\bf{T} \subset \bf{B} \subset \bf{G}$ be a maximal split torus and a Borel subgroup, respectively. Denote by $G = {\bf{G}}(F)$ and $B= {\bf{B}}(F)$ their groups of $F$-valued points and by $\mathfrak g = \rm Lie(G)$ and $\mathfrak b = \rm Lie(B)$ their Lie algebras. Let $\mathcal O^\infty$ be the thick category $\mathcal O$ for $(\mathfrak g,\mathfrak b)$, and denote by $\mathcal{O}^\infty_{\rm alg} \subset \mathcal{O}^\infty$ the full subcategory consisting of objects whose weights are in $X^*(\bf{T})$. Both are Serre subcategories of the category of all $U$-modules, where $U = U(\mathfrak g)$. We show first that the functor $\mathbb D^\mathfrak g = \rm RHom_U(-,U)$ preserves $D^b(U)_{\mathcal{O}^\infty_{\rm alg}}$, and we deduce from a result of Coulembier-Mazorchuk that the latter category is equivalent to $D^b(\mathcal O^\infty_{\rm alg})$.