For a closed surface $S$, the Hitchin component $\mathrm{Hit}_n (S)$ is a preferred component of the character variety consisting of group homomorphisms from the fundamental group $\pi_1(S)$ to the Lie group $\mathrm{PSL}_n (\mathbb{R})$. We construct a parametrization of the Hitchin component that is well-adapted to a geodesic lamination $\lambda$ on the surface. This is a natural extension of Thurston’s parametrization of the Teichmüller space $\mathcal{T}(S)$ by shearing coordinates associated with $\lambda$, corresponding to the case $n=2$. However, significantly new ideas are needed in this higher-dimensional case. The article concludes with a few applications.

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $\lvert E \rvert$ lines, unless all the points are contained in a line. The result was extended to higher dimensions by Motzkin and others, who showed that every set of points $E$ in a projective space determines at least $\lvert E \rvert$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of an $r$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E$, there are at least as many $(r-p)$-dimensional subspaces as there are $p$-dimensional subspaces, for every $p$ at most $\frac{1}{2} r$. This confirms the “top-heavy” conjecture by Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for $\ell$-adic intersection complexes.

Let $G/H$ be a unimodular real spherical space which is either absolutely spherical, i.e. the real form of a complex spherical space, or of wave-front type. It is shown that every tempered representation for $G/H$ embeds into a twisted discrete series for a boundary degeneration of $G/H$. If $G/H$ is of wave-front type it follows that the tempered representation is parabolically induced by a twisted discrete series representation for a real spherical space formed by a Levi subgroup.