Espaces de repr\'esentations compl\`etement r\'eductibles

TL;DR

This paper investigates geometric properties of group actions on nonpositively curved spaces, focusing on complete reducibility and semisimplicity, and characterizes the space of completely reducible classes as a maximal Hausdorff quotient.

Contribution

It establishes a link between complete reducibility in representations and the topological structure of the representation space, identifying the maximal Hausdorff quotient.

Findings

The space of completely reducible classes forms a maximal Hausdorff quotient.

Provides geometric criteria for complete reducibility in nonpositively curved spaces.

Connects algebraic properties of representations with topological features of the representation space.

Abstract

We study some geometric properties of actions on nonpositively curved spaces related to complete reducibility and semisimplicity, focusing on representations of a finitely generated group in the group G of rational points of a reductive group over a local field, acting on the associated space (symmetric space or affine building). We prove that the space of completely reducible classes is the maximal Hausdorff quotient space for the conjugacy action of G on the representation space.