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Reflection positivity and conformal symmetry. (English) Zbl 1290.22006
The paper under review is the first in a series of papers in which the authors plan to describe representation theoretic aspects of reflection positive representations of Lie groups.
A reflection positive Hilbert space is a triple \((\mathcal{E},\mathcal{E}_+,\theta)\) where \(\mathcal{E}\) is a Hilbert space, \(\theta\) is a unitary involution on \(\mathcal{E}\), and \(\mathcal{E}_+\) is a closed \(\theta\)-positive subspace. Now, given a triple \((G,\tau,S)\) consisting of a Lie group \(G\), an involution \(\tau\) on \(G\) and an open \(\tau(\cdot)^{-1}\)-invariant subsemigroup \(S\) of \(G\), a unitary representation \(\pi\) of \(G\) on \(\mathcal{E}\) is reflection positive provided that \(\theta\pi(g)\theta=\pi(\tau(g))\) for all \(g\in G\) and, moreover, \(\pi(S)\mathcal{E}_+\subset\mathcal{E}_+\).
The authors propose an approach for the description of reflection positive representations in terms of cyclic distribution vectors satisfying a reflection positivity condition. This leads to some classification results, notably, in the abelian case. The same approach is applicable to some examples in the non-abelian case, for instance, the authors consider in detail the complementary series representations of the conformal group of an \(n\)-dimension sphere. Finally, the authors also generalize the Bochner-Schwartz Theorem to positive distributions on open convex cones.

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E66 Analysis on and representations of infinite-dimensional Lie groups
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