Centers and translation functors for the category \({\mathcal O}\) over Kac-Moody algebras
In this very interesting paper the author studies the structure of the category \(\mathcal{O}\) associated with a triangular decomposition of a symmetrizable Kac-Moody Lie algebra \(\mathfrak{g}\). This category decomposes into a direct sum of indecomposable blocks, and the study reduces to the study of each block separately. In the case of a semi-simple finite-dimensional complex Lie algebra \textit{W. Soergel} [J. Am. Math. Soc. 3, 421-445 (1990; Zbl 0747.17008)] gave a combinatorial description for each block of the category \(\mathcal{O}\) via certain induced modules over the coinvariant algebra of the Weyl group of \(\mathfrak{g}\). The principal technical tools used by Soergel were translation functors and the action of the center of the universal enveloping algebra. Two main problems addressed in the paper under review are: to calculate the center of the blocks of the category \(\mathcal{O}\) for symmetrizable Kac-Moody Lie algebras (outside the critical hyperplanes, theory which does not have a reasonable finite-dimensional counterpart); and to study the properties of translation functors, in particular, to prove their biadjointness. For the first problem the author uses the approach based on Jantzen's deformation theory. The center of a block of the category \(\mathcal{O}\) outside the critical hyperplanes is described in terms of the evaluation of the action on deformed Verma modules. This is the first main result of the paper. The second main result of the paper states the biadjointness of the translation functors defined using the tensor product with a simple integrable highest weight module. The result the author obtains is quite far from being an easy generalization of the finite-dimensional analogue, as the theory of translation functors for Kac-Moody Lie algebras happens to be much more complicated than its finite-dimensional counterpart. Thus, for example, already the definition of translation functors is not absolutely straightforward as, for instance, to properly define translation to a wall one has to take some projective limits.
scientific article; zbMATH DE number 1944223