Positivity and higher Teichmüller theory

Abstract

We introduce $\Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $\Theta$-positivity generalizes at the same time Lusztig’s total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, $\mathrm{SO}(p,q)$ for $p\ne q$ and a family of exceptional Lie groups, which admit a $\Theta$-positive structure. We describe key aspects of $\Theta$-positivity and make a connection with representations of surface groups and higher Teichmüller theory.