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Extensions between unitary irreducible representations of nilpotent Lie groups. (Extensions entre représentations unitaires irréductibles des groupes de Lie nilpotents.) (French) Zbl 0567.22006
Homologie, groupes \(\mathrm{Ext}^ n\) représentations de longuer finie des groupes de Lie, Astérisque 124-125, 129-211 (1985).
The aim of this paper is to study for simply connected nilpotent Lie groups \(G\) the spaces \(\mathrm{Ext}^ n\) between unitary irreducible \(G\)-modules on the one hand and \(G\)-modules of finite length with unitary, simple subquotients on the other. Results of other authors in the semisimple and semidirect case suggest to consider the problem in the frame of the orbit method. Thus one expects that for orbits \(\mathcal O_ 1\) and \(\mathcal O_ 2\) with sufficiently big distance the associated \(\mathrm{Ext}\)-spaces are all zero. For modules, which are associated to the same orbit, one tries to determine \(\mathrm{Ext}^ n\) in terms of the stabilizer \(G(f)\) of a point \(f\) of the orbit.
First the author states that in order to obtain satisfactory results one has to replace the modules \((E,\pi)\) by the smoother ones \((E_{\infty},\pi)\). Then he formulates some conjectures (C\(_ 0\)), (C\(_ 1\)), (C\(_ 2\)), (C\(^{\prime}\)) and (C) which go back to Guichardet and Rosenberg and proves that (G\(^{\prime}\)) is equivalent to (G), that (G) and (G\(_ 2)\) are generally true and that (G) is true for flat orbits and nilpotent Lie groups with dimension \(\leq 6\). This is done with essentially algebraic methods.
[For the entire collection see Zbl 0547.00015.]
Reviewer: Joachim Boidol

MSC:
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods