  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).
