Itamar Vigdorovich: Characters and their dynamics

A character on a locally compact group G is a normalized, continuous, positive-definite function on G which is conjugation-invariant and moreover extremal for these properties. The space of characters was suggested by Thoma to serve as a dual object for harmonic analysis on arbitrary groups. It generalizes Pontryagin duality on abelian groups, and Peter-Weyl on compact groups.

In this talk I'll introduce the basics of this theory and describe the space of characters of certain groups. The main example will be the discrete Heisenberg group whose space of characters turns out to be quite remarkable. I will state some results regarding dynamics on such spaces, that generalize Furstenberg's classical stiffness result for dynamics on the torus. Applications to rigidity of arithmetic groups will be mentioned as well. The talk is based on a joint work with Uri Bader.