From Weyl's Unitarian Trick to Berkovich's compactifications of Bruhat-Tits buildings. A tale about trees, norms and compact subgroups

In this mini-course we will be interested in the relationship between the representations of a linear algebraic group and its topology over a complete field -- more precisely the structure of its compact subgroups.

In the complex case, this relationship stems back to Weyl (relying on previous work of Hurwitz) and it is based on integration on Zariski-dense compact subgroups (a technique that he called "Unitarian Trick").

In order to investigate an analogous correspondence in the p-adic case, Goldman and Iwahori introduced the space of p-adic norms. This marked the beginning of the theory of buildings, whose development in the following twenty years enlisted mathematicians of the calibre of Bruhat, Tits, Rousseau, Satake...

In the first lecture we will try to sum up these "classical" facts trying to focus on the analogies between complex and p-adic case. In the second lecture we will adopt a new point of view introduced by Berkovich and (further developed by Rémy, Thuillier and Werner) that permits to understand and sometimes circumvent some pathologies that arise in the classical point of view. Finally, we will prove an analogue of Weyl's Unitarian Trick in this new framework.