Antonin Guilloux

(ENS Lion)

Representations of linear groups : A dynamical point of view

It is possible to study the theory of unitary representations of linear groups - for example SL(2,R) - from a dynamical point of view. And  stating some theorems with the language of dynamical systems yields a good understanding of certain groups actions, namely actions of lattices in linear groups - for example the subgroup of matrices with integral entries SL(2,Z).

Here we will present a simple case, due to Margulis. We will introduce the hyperbolic plane and geometry together with the action of the groups SL(2,R) and SL(2,Z). Then we will state some results about representations of SL(2) and see their dynamical interpretations to  eventually count the cardinality of certain subsets of the hyperbolic plane.

First lecture : The hyperbolic plane, the action of SL(2,R), SL(2,Z). We will state the main result of the course.

Second lecture : Unitary representations of linear groups. We will focus on the example of SL(2,R) and then on the dynamical aspects.

Third lecture :  Equirepartition and counting. We will see how the tools developed during the second lecture apply to the problem stated during the first.