Extensions of representations of analytic solvable groups. (English) Zbl 0695.22003

Let L denote a closed analytic subgroup of GL(n,\({\mathbb{R}})\) and G a closed analytic subgroup of L on which a finite dimensional representation \(\rho\) is defined with representation space V. The author addresses the question when a finite dimensional L-module W can be found containing a G-submodule isomorphic to V. He settles the issue for solvable L by presenting necessary and sufficient conditions. For a formulation of these results, let \(\rho '\) denote the semisimple representation on the direct sum \(V'\) of the factors of a Jordan-Hölder-series of V, called the associated semisimple representation, and let \(L'\) denote the commutator subgroup of L. The author establishes the following Theorem: If L is solvable, then the following conditions are necessary and sufficient for W to exist: (1) \(\rho '(G\cap L')=\{1\}\). (2) The representation \(\sigma\) of \(GL'\) defined on \(V'\) by \(\sigma (gu)=\rho '(g)\) (in view of (1)) is continuous for the subspace topology of \(GL'\subseteq L\). The proof, which is of considerable length, is based on methods first introduced by G. Hochschild and G. D. Mostow. The methods prepared for the proof yield further results on sufficient conditions for the extension of \(\rho\) to exist.
Reviewer: K.H.Hofmann


22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22E25 Nilpotent and solvable Lie groups
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