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

Abstract

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