Integrability of unitary representations on reproducing kernel spaces

Abstract:

Let $\mathfrak {g}$ be a Banach–Lie algebra and $\tau : \mathfrak {g} \to \mathfrak {g}$ an involution. Write $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {q}$ for the eigenspace decomposition of $\mathfrak {g}$ with respect to $\tau$ and $\mathfrak {g}^c := \mathfrak {h}\oplus i\mathfrak {q}$ for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of $\mathfrak {g}^c$. 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 satisfy a natural invariance condition with respect to an infinitesimal action $\beta \colon \mathfrak {g} \to \mathcal {V}(M)$ by locally integrable vector fields that is compatible with a smooth action of a connected Lie group $H$ with Lie algebra $\mathfrak {h}$. 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 $\beta \colon \mathfrak {g} \to \mathcal {V}(M)$. 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.