Hillel Furstenberg (Hebrew University of Jerusalem)
Aula 1E, Palazzina RM 004
Affine representations of topological groups
Abstract: An affine representation" of a group arises when the group acts on a compact convex set preserving the affine (convex) structure. As in the theory of linear representations, special interest is attached to Irreducible representations, namely, actions for which no proper closed convex subset is invariant. Unlike the linear case such an action may have a non-trivial, non-isomorphic "factor" action. Indeed ALL irreducible representations are derived as factors of a single universal representation. This universal affine representation can be made explicit for Lie groups. We discuss the case of PSL(2,R), or equivalently, the Mobius group of the unit disc. Remarkably this group has a unique non-trivial irreducible action; namely, the action on the space of probability measures on the boundary of the disc. From the irreducibility of this action we will rederive the boundary theory of bounded harmonic functions on the unit disc, with generalizations to more general symmetric spaces.