A combinatorial description of blocks in \({\mathcal O}({\mathcal P},\Lambda)\) associated with \(sl(2)\)-induction
This paper is a continuation of a previous article by the same authors [Forum Math. 13, 641-661 (2001; Zbl 0991.17019)] and devoted to the study of categories \(\mathcal{O}(\mathcal{P},\Lambda)\) of Lie algebras having a triangular decomposition. In this work the category \(\mathcal{O}(\mathcal{P},\Lambda)\) for a parabolic subalgebra whose Levi part is \(\mathfrak{sl}(2,\mathbb{C})\) and for \(\Lambda\) an admissible category of \(\mathfrak{sl}(2,\mathbb{C})\)-modules is analyzed. The aim is to obtain a combinatorial description of \(\mathcal{O}(\mathcal{P},\Lambda)\) analogous to those introduced by \textit{W. Soergel} [J. Am. Math. Soc. 3, 421-445 (1990; Zbl 0747.17008)]. To this extent the authors consider a functor giving the equivalence of \(\mathcal{O}(\mathcal{P},\Lambda)\) with a full subcategory of \(\mathcal{O}\). It is shown that the combinatorial datum is the big projective module in the principal block and the coinvariant algebra. Further the tilting modules in \(\mathcal{O}(\mathcal{P},\Lambda)\) are investigated. Their existence is proven (in particular it does not follow from the results in [\textit{I. Ágoston, D. Happel, E. Lukács} and \textit{L. Unger}, J. Algebra 226, 144-160 (2000; Zbl 0949.16012)] due to a slight modification of the definition of tilting module). Multiplicities of generalized Verma modules occurring in a standard filtration are also determined.
multiplicities of generalized Verma modules
scientific article; zbMATH DE number 1525267
scientific article