Reflection positivity and conformal symmetry

A reflection positive Hilbert space is a triple (E,E+,θ)(E,E+,θ), where EE is a Hilbert space, θ   a unitary involution and E+E+ a closed subspace on which the hermitian form 〈v,w〉θ:=〈θv,w〉〈v,w〉θ:=〈θv,w〉 is positive semidefinite. For a triple (G,τ,S)(G,τ,S), where G is a Lie group, τ an involutive automorphism of G and S   a subsemigroup invariant under the involution s↦s♯=τ(s)−1s↦s♯=τ(s)−1, a unitary representation π of G   on (E,E+,θ)(E,E+,θ) is called reflection positive if θπ(g)θ=π(τ(g))θπ(g)θ=π(τ(g)) and π(S)E+⊆E+π(S)E+⊆E+. This is the first in a series of papers in which we develop a new and systematic approach to reflection positive representations based on reflection positive distributions and reflection positive distribution vectors. This approach is most natural to obtain classification results, in particular in the abelian case. Among the tools we develop is a generalization of the Bochner–Schwartz Theorem to positive definite distributions on open convex cones. We further illustrate our techniques with a non-abelian example by constructing reflection positive distribution vectors for complementary series representations of the conformal group O1,n+1+(R) of the sphere SnSn.