  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.
