Refection Positivity is a central theme at the crossroads of Lie group representations
euclidean and abstract harmonic analysis constructive quantum field theory and stochastic
processes. This book provides the first presentation of the representation theoretic aspects of
Refection Positivity and discusses its connections to those different fields on a level
suitable for doctoral students and researchers in related fields. It starts with a general
introduction to the ideas and methods involving refection positive Hilbert spaces and the
Osterwalder--Schrader transform. It then turns to Reflection Positivity in Lie group
representations. Already the case of one-dimensional groups is extremely rich. For the real
line it connects naturally with Lax--Phillips scattering theory and for the circle group it
provides a new perspective on the Kubo--Martin--Schwinger (KMS) condition for states of
operator algebras. For Lie groups Reflection Positivity connects unitary representations of a
symmetric Lie group with unitary representations of its Cartan dual Lie group. A typical
example is the duality between the Euclidean group E(n) and the Poincare group P(n) of special
relativity. It discusses in particular the curved context of the duality between spheres and
hyperbolic spaces. Further it presents some new integration techniques for representations of
Lie algebras by unbounded operators which are needed for the passage to the dual group.
Positive definite functions kernels and distributions and used throughout as a central tool.
