Pressure forms on pure imaginary directions

Anosov groups are a class of discrete subgroups of semi-simple algebraic groups
analogue to what is known as \emph{convex-co-compact groups} in negative curvature.
Thermodynamical constructions equip the (regular points of the) moduli space of
Anosov representations from $\Gamma$ to $G$ with natural positive semi-definite
bi-linear forms, known as pressure forms. Determining whether such a pressure form
is Riemannian requires non-trivial work.

The purpose of the lecture is to explain some geometrical meaning of these forms,
via a higher rank version of a celebrated result for quasi-Fuchsian space by
Bridgeman-Taylor and McMullen on the Hessian of Hausdorff dimension on pure bending
directions. This is work in collaboration with M. Bridgeman, B. Pozzetti and A.
Wienhard.