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Globalizations of Harish-Chandra modules. (English) Zbl 0649.22010

Let G be a semisimple Lie group with finite center, K its maximal compact subgroup and \({\mathfrak g}\) its Lie algebra. In his paper [Proc. Cartan Symp. Lyon 1984, Astérisque 1985, 311-322 (1985; Zbl 0621.22014)] the first author introdupower series ring.
Es sei \(R\subset \bar R\) ein Paar von kommutativen Integritätsbereichen und \(\phi\) eine Bewertung von \(\bar R\) derart, daß R bei der \(\phi\) entsprechenden Topologie in \(\bar R\) dicht liegt. Man sagt, daß die Approximationseigenschaft für das Paar R, \(\bar R\) gilt, falls zu jeder Nullstelle \((\bar y_ 1,...,\bar y_ n)\in \bar R^ n\) einer endlichen Familie von Polynomen \(f_ i(Y_ 1,...,Y_ n)\in R[Y_ 1,...,Y_ n]\) Nullstellen \((y_ 1,...,y_ n)\in R^ n\) existieren, die beliebig nahe an \((\bar y_ 1,...,\bar y_ n)\) liegen. Diese Approximationseigenschaft ist von S. Lang [Ann. Math., II. Ser. 55, 373-390 (1952; Zbl 0046.262)] in etlichen wichtigen Fällen betrachtet worden.
\({\mathfrak O}\) sei ein vollständiger Bewertungsring vom Rang 1 und \(| |\) sei die entsprechende (multiplikative) Bewertung. Diese Bewertung sei auf den Potenzreihenring \({\mathfrak O}[[X_ 1,...,X_ n]]\) durch \(| \sum a_{\nu_ 1...\nu_ n}X_ 1^{\nu_ 1}...X_ n^{\nu_ n}| =\sup_{\nu_ 1,...,\nu_ n}| a_{\nu_ 1...\nu_ n}| \quad fortgesetzt\). \(A_ m\) bezeichne den algebraischen Abschluß von \({\mathfrak O}[X_ 1,...,X_ n]\) in \({\mathfrak O}[[X_ 1,...,X_ n]]\) bezüglich der durch \(| |\) definierten Topologie. Das Hauptergebnis der vorliegenden Arbeit besagt, daß die Approximationseigenschaft für das Paar \((A_ m,\bar A_ m)\) gilt.
Reviewer: C.U.Jensen

MSC:

22E46 Semisimple Lie groups and their representations
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
32M10 Homogeneous complex manifolds
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