Generalized Verma modules (Q2723184)

The theory of Verma modules has had an enormous and integral role in the development of representation theory of Lie algebras. One of the most natural generalizations of the theory of Verma modules is the so called theory of generalized Verma modules. In this monograph, the author gathers together much of the theory of generalized Verma modules that has been developed by many individuals over the last 14 years. The author played a large role in this development. NEWLINENEWLINENEWLINETo define a generalized Verma module, let \({\mathfrak g}\) be a simple complex finite dimensional Lie algebra and \({\mathfrak p}={\mathfrak l}\oplus{\mathfrak n}\) the Levi decomposition of a parabolic subalgebra with Levi factor \({\mathfrak l}\) and nilradical \({\mathfrak n}\). Decompose \({\mathfrak l}={\mathfrak l}_{ss}\oplus{\mathfrak z}\) where \({\mathfrak l}_{ss}\) is semisimple and \({\mathfrak z}\) is Abelian and central in \({\mathfrak l}\). Let \(V\) be a simple \({\mathfrak z}\)-diagonalizable \({\mathfrak l}\)-module viewed as a \({\mathfrak p}\)-module with trivial \({\mathfrak n}\)-action. The generalized Verma module \(M_{\mathfrak p}(V)\) is defined as \(U({\mathfrak g})\otimes_{U({\mathfrak p})}V\) where \(U({\mathfrak g})\) and \(U({\mathfrak p})\) are the universal enveloping algebras for \({\mathfrak g}\) and \({\mathfrak p}\), respectively. NEWLINENEWLINENEWLINEThe monograph encompasses many topics. Some of the highlights include an analogue of the BGG theorem, an analogue of the BGG resolution for simply laced algebras, an analogue of the Kazhdan-Lusztig theorem, and an analogue of category \({\mathcal O}\), including a Soergel type description for its blocks. For the most part, proofs are sketched and the reader is referred to the original papers for technical details.