Sur le foncteur de Zuckerman. (On Zuckerman’s functor). (French) Zbl 0617.17007

Let G be a real Lie group with Lie algebra \({\mathfrak g}\), \(K\subset G\) be a maximal compact subgroup and \(H\subset K\) be a closed subgroup. To each (\({\mathfrak g}\)-H)-module V the authors associate a complex of (\({\mathfrak g}\)- H)-modules and prove that its cohomology spaces are exactly the Zuckerman functors \(\Gamma^ i(V)\), \(i\geq 0\). This enables them to give a simple description of the \({\mathfrak g}\)-module structure on \(\Gamma^ i(V)\) and to reprove an important duality result.
Reviewer: A.Verona


17B55 Homological methods in Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)