Enright's completions and injectively copresented modules
Transactions of the American Mathematical Society
Let 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 O. Let A be the finite-dimensional algebra associated to a block of 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
... 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 O. Its simple objects, their characters and their multiplicities in projective or standard objects are determined. In particular, each block of the above categories of (relatively or absolutely) complete modules is equivalent to the module category of an algebra eAe, where A is a block of O and e is an idempotent, the primitive summands of which are naturally indexed by cosets of the Weyl group. Thus, these subcategories carry abelian structures, which are, in fact, not obtained by restriction from the abelian structure of O. From the abstract framework it is clear that the category of injectively copresented modules (for some choice of injectives) is equivalent to the category of projectively presented modules (for the corresponding choice of projectives). Explicit versions of the last categories occur in [BG, II 5.9] in the context of projective functors. There it is shown that certain translation functors are equivalences between categories of Harish-Chandra modules and categories of projectively presented modules in O. There is even another equivalent version of these categories, namely, a parabolic generalization of the category O, which has been introduced and studied in [FKM1, FKM2, FKM3] by Lie theoretic methods. For example, the abelian structure mentioned before was discovered in [FKM3] as a quite exotic looking property, which gets a natural explanation in the present framework. Altogether we get the following picture-it contains five categories, corresponding blocks of which are all equivalent (for suitable choices of the defining parameters): eAe-mod relatively) complete modules in O injectively copresented modules © parabolic category O(P, K) d d d d d s Having defined relative completions in [E], Enright posed the problem of showing that (on a certain subcategory) these relative completion functors satisfy the braid relations. This problem has been solved independently by Deodhar ([De]) and by Bouaziz ([Bo]). Later, Joseph ([Jo]) extended the result to the whole category O. It turns out that the subcategory considered by Deodhar is precisely the category of injectively cogenerated modules in our abstract setup, and this reformulation enables us to reprove the Bouaziz-Deodhar result in an easy way. As a by-product of the proof we get that complete modules have a Verma flag. This yields a lower bound for the representation type of the category of modules having a Verma flag (and, of course, also for all of O). See [BKM] for details of this application.