Reflection negative kernels and fractional Brownian motion

In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space E and show in particular that fractional Brownian motion for Hurst index 0<H\le 1/2 is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if 0<H <1/2. We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of GL(2,R). We relate this to a measure preserving action on a Gaussian L^2-Hilbert space L^2(E).