Thuses

A shortcut in Kapovich’s proof of Haupt’s theorem

The Teichmüller space $T_g$ of genus $g$ curves carries the Hodge bundle $\Omega T_g$ , the total space of which maps into the first cohomology space $V = H^1(S_g,\mathbb{C})$ via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image $per(\Omega T_g) \subset V$ in terms of the integral structure on $V = H^1(S, \mathbb{Z}) \otimes \mathbb{C}$ and …

The Steinberg Representation

In this post I want to describe a remarkable representation associated to finite groups of Lie type. For this, let $G$ be a connected reductive group over a finite field $k$ with $q = p^r$ elements, and let $U$ be the unipotent radical of some Borel $k$ -subgroup $B$ of $G$ . Steinberg constructed an irreducible representation $\mathrm{St}$ of $G(k)$ of …

Fun example: Empty colimit does not commute with empty limit

One important property of filtered colimits is that they commute with finite limits in the category of sets.

Theorem: Let $F \colon \mathcal{C}\times \mathcal{D} \to \mathbf{Sets}$ be a functor, where $\mathcal{C}$ is a filtered small category and $\mathcal{D}$ is a finite category. Then the natural mapping