A generalization of the category \(\mathcal O\) of Bernstein-Gelfand-Gelfand
Let \(\mathfrak{g}\) be a simple Lie algebra over the complex numbers, \(\Delta\) be the set of simple roots, and \(\theta \subseteq S \subseteq \Delta\) two subsets. In the paper under review, the author considers a certain full subcategory \(\mathcal{O}_{S,\theta}\) of the category of weight modules over \(\mathfrak{g}\). When \(S = \theta = \emptyset\) this is the classical BGG category \(\mathcal{O}\) [\textit{I. N. Bernstein, I. M. Gelfand} and \textit{S. I. Gelfand}, Lie Groups Represent., Proc. Summer Sch. Bolyai János Math. Soc., Budapest 1971, 21--64 (1975; Zbl 0338.58019)]. The main result is given in the case when \(S = \Delta\) and \( \theta \neq \emptyset\) or \(\Delta\). Ruling out \textit{a priori} a few exceptional cases: then \(\mathcal{O}_{\Delta,\theta}\) is non-empty only in type \(A\) or \(C\); in these cases, the simple modules in \(\mathcal{O}_{\Delta,\theta}\) are completely described. The last result highlights the difference with the classical category \(\mathcal{O}\). Let \(\mathfrak{g}\) be of type \(A\) or \(C\) (with a few exceptions for \(\theta\) in type \(A\)): then the category \(\mathcal{O}_{\Delta,\theta}\) is semisimple.
scientific article; zbMATH DE number 5712616
scientific article