Collingwood, David H.; Irving, Ronald S.; Shelton, Brad Filtrations on generalized Verma modules for Hermitian symmetric pairs. (English) Zbl 0631.22014 J. Reine Angew. Math. 383, 54-86 (1988). Let \((G_ 0,K_ 0)\) be an irreducible Hermitian symmetric pair of noncompact type. Denote by \({\mathfrak g}\) and \({\mathfrak k}\) the complexified Lie algebras of \(G_ 0\) and \(K_ 0\), respectively. Let \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak k}\). Let \(\Phi^+_ n\) be the set of positive noncompact roots in \({\mathfrak h}^*\) determined by a choice of complex structure on \(G_ 0/K_ 0\). Then \({\mathfrak k}\) and the root subspaces \({\mathfrak g}_{\alpha}\), \(\alpha \in \Phi^+_ n\), span a maximal parabolic subalgebra \({\mathfrak p}\) of \({\mathfrak g}.\) The authors study the \({\mathfrak g}\)-module filtrations on the generalized Verma modules attached to the pairs (\({\mathfrak g},{\mathfrak p})\) with integral regular infinitesimal character. They prove that the socle and radical filtrations coincide with the geometric weight filtration. Reviewer: D.Miličić Cited in 13 Documents MSC: 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Keywords:socle filtration; irreducible Hermitian symmetric pair; noncompact roots; complex structure; maximal parabolic subalgebra; \({\mathfrak g}\)-module filtrations; generalized Verma modules; integral regular infinitesimal character; radical filtrations; geometric weight filtration PDFBibTeX XMLCite \textit{D. H. Collingwood} et al., J. Reine Angew. Math. 383, 54--86 (1988; Zbl 0631.22014) Full Text: DOI Crelle EuDML