  Inspired by classical results by Benoist, we introduce and study natural
objects associated to an integrable tangent vector to the character variety
$\mathfrak X(\Gamma,\mathsf G)$ of a semi-group $\Gamma$, with values on a
semi-simple real algebraic group $\mathsf G$ of the non-compact type. We obtain
non-empty interior of the cone of Jordan variations and, when $\mathsf G$ is
split and Benoist's limit cone is sparse, we obtain double-density results
associated to these variations, giving in particular non-empty interior of the
set of length-normalized variations. We then apply the developed techniques to
study pressure forms on the space of Anosov representations, in particular to
higher-rank Teichm\"uller spaces. Among other things we exhibit an explicit
functional $\varphi\in\mathfrak a^*$ whose pressure form is compatible with
Goldman's symplectic form at the Fuchsian points of the Hitchin component. We
finally exhibit a Diophantine equation that governs the degeneration of the
Hausdorff dimension of higher-quasi-circles.
