Reflection Positive Distribution Vectors

Abstract

In this chapter we first introduce the concept of a distribution vector of a unitary representation (Sect. 7.1). It turns out that certain distribution vectors semi-invariant under a subgroup H correspond naturally to realizations of the representation in a Hilbert space of distributions on the homogeneous space G / H. In this context reflection positive representations can be constructed from reflection positive distributions on G / H (Sect. 7.2). Such distributions can often be found and even classified in terms of the geometry of the homogeneous space.