Affine representations, harmonic functions, and group boundaries

If  Q is a convex set, a transformation T: Q ->Q is affine if it preserves the convex structure of Q.  An affine representation of a group is a homomorphism of  G  to the group of invertible affine transformations of a compact convex  Q.  It is irreducible if no proper closed, convex subset of Q is left invariant.  Abelian and compact groups have no non-trivial irreducible affine representations.   From the classical theory of harmonic functions we show that any bounded harmonic function on the upper half plane leads to an irreducible affine representation of SL(2,R).

We discuss generalizations leading to the notion of the Poisson boundary of a group.