×

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

MSC:

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Bialynicki-Birula, A., Hochschild, G., Mostow, G.D.: Extensions of representations of algebraic linear groups. Am. J. Math.85, 131-144 (1963) · Zbl 0116.02302
[2] Chen, P.B., Wu, T.S.: On solvable groups. Math. Ann.276, 43-51 (1986) · Zbl 0589.22008
[3] Chevalley, C.: Theorie des groupes de Lie, Tome II, Tome III. Paris: Herman 1951, 1955 · Zbl 0054.01303
[4] Hochschild, G.: The structure of Lie groups. San Francisco: Holden-Day 1965 · Zbl 0131.02702
[5] Hochschild, G.: Basic theory of algebraic groups and Lie algebras. New York: Springer 1981 · Zbl 0589.20025
[6] Hochschild, G., Mostow, G.D.: Extension of representations of Lie groups and Lie algebras I. Am. J. Math.79, 924-942 (1957) · Zbl 0080.25201
[7] Hochschild, G., Mostow, G.D.: Representations and representative functions of Lie groups. Ann. Math.66, 495-542 (1957) · Zbl 0080.25101
[8] Hochschild, G., Mostow, G.D.: Representations and representative functions of Lie groups II. Ann. Math.,68, 295-313 (1958) · Zbl 0085.01803
[9] Hochschild, G., Mostow, G.D.: On the algebra of representative functions. Am. J. Math.83, 111-136 (1961) · Zbl 0116.02203
[10] Mostow, G.D.: Extensions of representations of Lie groups II. Am. J. Math.80, 331-347 (1958) · Zbl 0085.01802
[11] Mostow, G.D.: Fully reducible subgroups of algebraic groups. Am. J. Math.78, 200-221 (1956) · Zbl 0073.01603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.