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Analytic continuation of local representations of symmetric spaces. (English) Zbl 0608.22010

Let \(G\) be a Lie group, \(K\) a closed subgroup, and \(\sigma\) an involutive automorphism with \(K\) as fixed-point subgroup. If \(\mathfrak g=\mathfrak k+\mathfrak m\) is the corresponding symmetric Lie algebra, we form \(\mathfrak g^*=\mathfrak k+i\mathfrak m\), and let \(G^*\) denote the simply connected Lie group with \(\mathfrak g^*\) as Lie algebra. We consider virtual representations \(\pi\) of \(G\) on a fixed complex Hilbert space \(\mathcal H\), adopting the definitions due to J. Fröhlich, K. Osterwalder, and E. Seiler [Ann. Math. (2) 118, 461–489 (1983; Zbl 0537.22017)]; in particular, \(\pi (g^{-1})\subset \pi (\sigma (g))^*\) (possibly unbounded operators) for \(g\) in a neighborhood of \(e\) in \(G\). We prove that every such \(\pi\) continues analytically to a strongly continuous unitary representation of \(G^*\) on \(\mathcal H\). Our theorem extends results due to Klein-Landau, Fröhlich et al., and others, earlier, for special cases. Previous results were known only for special \((G,K,\sigma)\), and then only for certain \(\pi\).

MSC:

22E46 Semisimple Lie groups and their representations
47L60 Algebras of unbounded operators; partial algebras of operators
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22E70 Applications of Lie groups to the sciences; explicit representations
53C35 Differential geometry of symmetric spaces

Citations:

Zbl 0537.22017
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References:

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