Centers of deformed representation categories and translation functors (Q2716027)

Let \(\mathfrak g\) be a symmetrizable Kac-Moody algebra with Borel subalgebra \(\mathfrak b\) and Cartan subalgebra \({\mathfrak h}\subseteq {\mathfrak b}\). The goal of the paper under review is to study a deformed version of the category \(\mathcal O\) over \(\mathfrak g\) which is defined as follows. Let \(T\) be a commutative associative unitary algebra over the universal enveloping algebra of \(\mathfrak h\). Then the deformed category \({\mathcal O}_T\) is the subcategory of the category of all modules over the Lie algebra \({\mathfrak g}\otimes_\mathbb{C} T\) on which \(\mathfrak h\) acts semisimply and on which \({\mathfrak b}\otimes_\mathbb{C} T\) acts locally finite. As in the classical case one can define deformed Verma modules, which are parametrized by their highest weights.NEWLINENEWLINENEWLINEIt is clear that \({\mathcal O}_T\) is an abelian category but, in general, \({\mathcal O}_T\) has not enough projective objects. Nevertheless, in the truncated subcategory \({\mathcal O}^{\leq\lambda}_T\) of \({\mathcal O}_T\) where the weights of all modules are bounded above by \(\lambda\), this is the case. Moreover, if \(T\) is a complete local algebra, then every simple object in \({\mathcal O}^{\leq\lambda}_T\) has a projective cover with a Verma flag whose multiplicities coincide with the corresponding multiplicities for the residue class field of \(T\). In the latter case BGG reciprocity is well known and these multiplicities can also be employed to define a block decomposition for \({\mathcal O}_T\).NEWLINENEWLINENEWLINEA first step in determining the structure of the blocks of \({\mathcal O}_T\) is the computation of the center of \({\mathcal O}_T\). It is shown that the center of \({\mathcal O}_T\) is the inverse limit of the centers of the truncated subcategories \({\mathcal O}^{\leq\lambda}_T\), and the latter are explicitly determined in a direct product of endomorphisms of certain projective objects. Localization techniques allow for the calculation of the center of the blocks outside the critical hyperplane for the completion of \(U({\mathfrak h})\). The structure of these blocks is then either like the deformed block of an abelian Lie algebra or like the deformed principal block of \({\mathfrak s}{\mathfrak l}_2\).NEWLINENEWLINENEWLINEIn the second part of this paper the author considers translation functors between different blocks of \({\mathcal O}_T\). In the non-deformed case \(T = \mathbb{C}\) these were studied intensively by \textit{W. Neidhardt} [Pac. J. Math. 139, 107-153 (1989; Zbl 0694.17016)], who also proved that the translation functors in dominant direction are right adjoint to the corresponding translation functors in antidominant direction. The arguments carry over to the deformed case, and the author also shows the left adjointness of these functors. As a consequence, projectivity remains invariant under translation functors in both directions.