Twisted Reflection Positivity

Reflection positivity has several applications in both mathematics and physics. For example, reflection positivity induces a duality between group representations. In this thesis, we coin a new definition for a new kind of reflection positivity, namely, twisted reflection positive representation on a vector space. We show that all of the non-compactly causal symmetric spaces give rise to twisted reflection positive representations. We discover examples of twisted reflection positive representations on the sphere and on the Grassmannian manifold which are not unitary, namely, the generalized principle series with the Cosine transform as an intertwining operator. We give a direct proof for the reflection positivity of the Cosine transform on SO(n). On the other hand, we generalize an integrability theorem to the case of non-positive definite distribution. As a result, we give a relation between the non-compactly causal symmetric spaces and the reflection positive distributions. Cocycle conditions are also treated. We construct a general method to generate twisted reflection positive representations and then we apply it to get twisted reflection positive representations on the Euclidean space. Finally, we introduce a reflection positive cyclic distribution vector for the circle case. Then we prove that this distribution vector generates a well known reflection positive function