Let k be a non-archimedean complete valued field and X be a k-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: 1) for every complete valued extension k ′ of k, every coherent sheaf on X × k k ′ is acyclic; 2) X is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex) and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, X is compact); 3) X admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity.When X has no boundary the characterization is simpler: in 2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in 3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl's definition of Stein space.

Let k be a complete non-archimedean field (non trivially valued). Given a reductive k-group G, we prove that hyperspecial subgroups of G(k) (i.e. those arising from reductive models of G) are maximal among bounded subgroups. The originality resides in the argument: it is inspired by the case of GLn and avoids all considerations on the Bruhat-Tits building of G.

Let k k be a non-Archimedean complete valued field and let X X be a k k -analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: (1) for every complete valued extension  k ′ k’ of  k k , every coherent sheaf on  X × k k ′ X \times _{k} k’ is acyclic; (2) X X is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex), and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, X X is compact); (3) X X admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. When  X X has no boundary the characterization is simpler: in (2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in (3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl’s definition of Stein space.

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In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of PGL(4, C).