Infinitesimal random dynamics of certain Veech groups on SU(2)-character varieties

Almost 20 years ago, the first and fourth authors found examples of SL(2,R)-invariant subbundles of Hodge bundles over Teichmüller curves having maximally degenerate Lyapunov spectrum. For these same surfaces, we show that a natural non-Abelian analogue has no zero Lyapunov exponents.