Enright’s completions and injectively copresented modules

Abstract:

Let $\mathfrak {A}$ be a finite-dimensional simple Lie algebra over the complex numbers. It is shown that a module is complete (or relatively complete) in the sense of Enright if and only if it is injectively copresented by certain injective modules in the BGG-category ${\mathcal O}$. Let $A$ be the finite-dimensional algebra associated to a block of ${\mathcal O}$. Then the corresponding block of the category of complete modules is equivalent to the category of $eAe$-modules for a suitable choice of the idempotent $e$. Using this equivalence, a very easy proof is given for Deodhar’s theorem (also proved by Bouaziz) that completion functors satisfy braid relations. The algebra $eAe$ is left properly and standardly stratified. It satisfies a double centralizer property similar to Soergel’s “combinatorial description” of ${\mathcal O}$. Its simple objects, their characters and their multiplicities in projective or standard objects are determined.