  The concept of reflection positivity has its origins in the work of
Osterwalder--Schrader on constructive quantum field theory and duality between
unitary representations of the euclidean motion group and the Poincare group.
  On the mathematical side this duality can be made precise as follows. If $\g$
is a Lie algebra with an involutive automorphism $\tau$. Decompose $\g = \fh
\oplus \fq = \ker(\tau - \1) \oplus \ker(\tau + \1)$ into $\tau$-eigenspaces
and let $\g^c := \fh \oplus i \fq$. At the core of the notion of reflection
positivity is the idea that this duality can sometimes be implemented on the
level of unitary representations. The idea is simple on the Lie algebra level:
Let $(\pi,\cH^0)$ be a representation of $\g$ where $\pi$ acts by
skew-symmetric operators. Assume that there exists a unitary operator $J$ of
order two such that $J \pi J = \pi\circ \tau$ and a $\g$-invariant subspace
$\cK^0$ which is {\it $J$-positive}. Then complex linear extension leads to a
representation of $\g^c$ on $\cK^0$ by operators which are skew-symmetric with
respect to $h_J$, so that we obtain a "unitary" representation of $\g^c$ on the
pre-Hilbert space $\cK_J^0 := \cK_J/{v \: \cK^0 \: h_J(v,v)=0}$.
  The aim of this article is twofold. First we discuss reflection positivity in
an abstract setting using {\it reflection positive distributions} on the Lie
group $G_\tau =G\rtimes {1,\tau}$ and {\it reflection positive distribution
vectors} of a unitary representation of $G_\tau$. Then we apply these ideas to
the conformal group $\OO_{1,n+1}^+(\R)$ of the sphere $\bS^n$ as well as the
the half-space picture mostly used in physics.
