Analytic continuation of local representations of Lie groups

Abstract

We consider a symmetric space (G, K, ) where G is a Lie group, K a closed subgroup, and the involutive automorphism defining the space. A local representation is defined for g in a neighborhood of e in G\ and the operator (g) is unbounded and defined on a dense subspace in a Hubert space where the identity holds. We study analytic continuations of to unitary representations of a group G* which is dual to G. 1. Introduction. Recent papers by Klein and Landau [12], and Frhlich-Osterwalder-Seiler [4] have studied local representations of Lie groups in connection with Euclidean quantum field theory. See also [16, 19, 21]. Important physical ideas seem to have originated in part with the earlier Lscher-Mack paper [13]. Earlier mathematical papers where local representations and analytic continuation played an important role, are [6] and [18]. The representations in question have become known as "virtual representations" of symmetric spaces [4].