Jacquet modules for semisimple Lie groups having Verma module filtrations
Let G be a connected semisimple real matrix Lie group, F be a finite dimensional representation of G. Let \({\mathfrak g}\) be the complexification of Lie G and \({\mathfrak b}\) be an Iwasawa Borel subalgebra of \({\mathfrak g}\). We have two categories of representations: the category \({\mathcal H}{\mathcal C}_ F\) of Harish-Chandra modules with the same infinitesimal character as F and the category \({\mathcal O}'_ F\) of finitely generated U(\({\mathfrak g})\)-modules which are \({\mathfrak b}\)-locally finite and have the same infinitesimal character as F. There is a Jacquet functor J from the first category to the second one. The category \({\mathcal O}'_ F\) admits several basic families of indecomposable modules. For instance, the indecomposable projective modules and the indecomposable self-dual Verma flag modules. The paper reviewed determines when J(\(\pi\)) for irreducible \(\pi\) is in these families.
connected semisimple real matrix Lie group
indecomposable self-dual Verma flag modules
scientific article; zbMATH DE number 4179681
scientific article