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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.

##### MSC:
 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods 2.2e+67 Analysis on and representations of infinite-dimensional Lie groups
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