Integrability of unitary representations on reproducing kernel spaces

Abstract

Let g \mathfrak {g} be a Banach–Lie algebra and τ : g → g \tau : \mathfrak {g} \to \mathfrak {g} an involution. Write g = h ⊕ q \mathfrak {g}=\mathfrak {h}\oplus \mathfrak {q} for the eigenspace decomposition of g \mathfrak {g} with respect to τ \tau and g c := h ⊕ i q \mathfrak {g}^c := \mathfrak {h}\oplus i\mathfrak {q} for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of g c \mathfrak {g}^c . The first class of representation is determined by a smooth positive definite kernel K K on a locally convex manifold M M . The kernel is assumed to satisfy a natural invariance condition with respect to an infinitesimal action β : g → V ( M ) \beta \colon \mathfrak {g} \to \mathcal {V}(M) by locally integrable vector fields that is compatible with a smooth action of a connected Lie group H H with Lie algebra h \mathfrak {h} . The second class is constructed from a positive definite kernel corresponding to a positive definite distribution K ∈ C − ∞ ( M × M ) K \in C^{-\infty }(M \times M) on a finite dimensional smooth manifold M M which satisfies a similar invariance condition with respect to a homomorphism β : g → V ( M ) \beta \colon \mathfrak {g} \to \mathcal {V}(M) . As a consequence, we get a generalization of the Lüscher–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.