  In this note we are interested in the dynamics of the linear flow on infinite
periodic $\mathbb{Z}^d$-covers of Veech surfaces. An elementary remark allows
us to show that the kernel of some natural representations of the Veech group
acting on homology is "big". In particular, the same is true for the Veech
group of the infinite surface, answering a question of Pascal Hubert. We give
some applications to the dynamics on wind-tree models where the underlying
compact translation surface is a Veech surface.
