A coding-free simplicity criterion for the Lyapunov exponents of Teichmüller curves

In this note we show that the results of Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich–Zorich cocycle of Teichmüller curves in moduli spaces of Abelian differentials without the usage of codings of the Teichmüller flow. As an application, we show the simplicity of some Lyapunov exponents in the setting of (some) Prym Teichmüller curves of genus 4 where a coding-based approach seems hard to implement because of the poor knowledge of the Veech groups of these Teichmüller curves. Finally, we extend the discussion in this note to show the simplicity of Lyapunov exponents coming from (high weight) variations of Hodge structures associated to mirror quintic Calabi–Yau threefolds.