Title

Integrability of unitary representations on reproducing kernel spaces

Document Type

Article

Publication Date

1-1-2015

Abstract

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.

Publication Source (Journal or Book title)

Representation Theory

First Page

24

Last Page

55

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