Applications des immeubles en th\'eorie des repr\'esentations

TL;DR

This survey explores the relationship between semisimple group representation theory and affine building geometry, focusing on simplifying the proof of the saturation theorem related to folded triangles.

Contribution

It provides a simplified proof of the saturation theorem of Kapovich and Millson, connecting representation theory with affine building geometry.

Findings

Simplified proof of the saturation theorem

Enhanced understanding of folded triangles

Clarified link between representation theory and affine buildings

Abstract

This is a survey about the connection between the representation theory of a semisimple group and the geometry of an affine building. The latter is, actually, associated to the Langlands'dual of the semisimple group. We deal, mainly, with the proof of the saturation theorem of Kapovich and Millson. We obtain a simplification of their proof regarding the characterization of folded triangles. The article is written in french for it is the outcome of a seminar that took place in Nancy last year.