Integrability of unitary representations on reproducing kernel spaces

Let g be a Banach-Lie algebra and τ: g → g an involution. Write g = h⊕iq for the eigenspace decomposition of g with respect to τ and gc := h⊕iq for the dual Lie algebra. In this article we show the integrability of two types ofinfinitesimally unitary representations of gc. The first class of representation isdetermined by a smooth positive definite kernel K on a locally convex manifoldM. The kernel is assumed to satisfy a natural invariance condition with respectto an infinitesimal action ß: g → ν(M) by locally integrable vector fields thatis compatible with a smooth action of a connected Lie group H with Lie algebrah. The second class is constructed from a positive definite kernel correspondingto a positive definite distribution K ∈ C-∞(M ×M) on a finite dimensionalsmooth manifold M which satisfies a similar invariance condition with respectto a homomorphism ß: g → ν(M). As a consequence, we get a generalizationof the Lüscher-Mack Theorem which applies to a class of semigroups that neednot have a polar decomposition. Our integrability results also apply naturallyto local representations and representations arising in the context of reflectionpositivity