Integrability of unitary representations on reproducing kernel spaces

Abstract

Let g be a Banach Lie algebra and \tau : g ---> g an involution. Write g=h+q for the eigenspace decomposition of g with respect to \tau and g^c := h+iq for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of g^c. The first class of representation is determined by a smooth positive definite kernel K on a locally convex manifold M. The kernel is assumed to satisfying a natural invariance condition with respect to an infinitesimal action \beta : g \to V(M) by locally integrable vector fields that is compatible with a smooth action of a connected Lie group $H$ with Lie algebra h. The second class is constructed from a positive definite kernel corresponding to a positive definite distribution K \in C^{-\infty}(M \times M) on a finite dimensional smooth manifold M which satisfies a similar invariance condition with respect to a homomorphism \beta : g \to V(M). As a consequence, we get a generalization of the Luscher--Mack Theorem which applies to a class of semigroups that need not have a polar decomposition. Our integrability results also apply naturally to local representations and representations arising in the context of reflection positivity.