Category \({\mathcal O}\): gradings and translation functors.
The author presents a systematic approach to the construction of a graded version of the Bernstein-Gelfand-Gelfand category \(\mathcal{O}\) from [\textit{I. N. Bernshtein}, \textit{I. M. Gelfand} and \textit{S. I. Gelfand}, Category of \(\mathfrak{g}\) modules, Funct. Anal. Appl. 10, 87--92 (1976; Zbl 0353.18013)]. It has been shown in [\textit{A. Beilinson}, \textit{V. Ginzburg} and \textit{W. Soergel}, Koszul duality patterns in representation theory, J. Am. Math. Soc. 9, 473--527 (1996; Zbl 0864.17006)] that a block of the category \(\mathcal{O}\) is equivalent to the category of finite dimensional (not necessarily graded) modules over some graded (even Koszul) algebra. Thus it makes sense to speak about gradable and non-gradable modules in \(\mathcal{O}\), gradable and non-gradable endofunctors, etc. In the present paper it is shown that the usual duality on \(\mathcal{O}\) and the translation functors on \(\mathcal{O}\) are gradable. Using this a lot of gradable modules are constructed. Many classical exact sequences involving Verma modules or dual Verma modules are shown to be gradable, and many classical multiplicity formulae are shown to have graded analogues. At the same time, the author also constructs an example of a module in \(\mathcal{O}\) which does not admit a graded lift. The author also describes the relation between the graded combinatorics of \(\mathcal{O}\) and the Hecke algebra.
scientific article; zbMATH DE number 1995432
scientific article