Analytic continuation of local representations of symmetric spaces

AbstractWe prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space (G, K, σ). The problem of passing from the Euclidean group to the Poincaré group appears first to have been addressed and solved this way by Klein and Landau. Let G be a Lie group, K a closed subgroup, and σ an involutive automorphism with K as fixed-point subgroup. If g = k + m is the corresponding symmetric Lie algebra, we form g∗ = k + im, and let G∗ denote the simply connected Lie group with g∗ as Lie algebra. We consider virtual representations π of G on a fixed complex Hilbert space H, adopting the definitions due to J. Fröhlich, K. Osterwalder, and E. Seiler; in particular, π(g−1) ⊂ π(σ(g))∗ (possibly unbounded operators) for g in a neighborhood of e in G. We prove that every such π continues analytically to a strongly continuous unitary representation of G∗ on 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, σ), and then only for certain π