Elyasheev Leibtag (Ghent University) : Weakly Almost Periodic G-flows and mixing properties for algebraic groups

Given a compact G-flow X, the Ellis enveloping semigroup E(X) is obtained by taking the closure of the image of G in the compact space X^X of all self-maps of X. In general, E(X) is a compact semigroup whose multiplication is continuous only on one side. When X is weakly almost periodic (WAP), E(X) becomes a semitopological compact semigroup. Thus WAP flows naturally give rise to semitopological semigroup compactifications of G, and there is a universal such compactification.

In this talk, I will discuss how locally compact groups sit inside their WAP compactifications. The main result is that algebraic groups over local fields of characteristic zero sit particularly rigidly inside these compactifications: Every point added at infinity has the same left and right G-orbit. I will explain this compactification-theoretic property and show how, it implies mixing results for unitary representations of these groups.