The algebraic structure of \alpha-stratified modules
by
Paolo Casati
Department of Mathematics, II University of Milano

A module V of a reductive Lie algebra g is said \alpha-stratified if there exists a simple root \alpha of g such that X _{+/- \alpha} act injective on V where X _{+/- \alpha} are elements of g spanning the +/- \alpha-root space g _{+/- \alpha} respectively. The most simple \alpha-stratified modules are those, which are not stratified for all but one simple root \alpha. These module can be seen as subquotiens of some universal modules (defined in the talk) which are natural generalization of the Verma modules. In the talk will be investigated the algebraic properties of these latter modules, which are likely to play a role in the classification of the unitary representations of the real reductive groups like the existence of a generalization of the Harish-Chandra map and of the Shapovalov bilinear invariant form. Their application in the study of the Jordan-Hölder series is explained, and finally a criterion of irreducibility for such modules is presented.

Date received: October 19, 2001