Relations between the Veech group and the automorphism group

Question

Let $(X,\omega)$ be a compact (half-)translation surface: $X$ is a compact riemann surface and $\omega$ a (quadratic) differential on it.

One can define a distinguished local coordinate $z$ on $X$: $$z(p)=\int^p \omega\qquad \mbox{(}z(p)=\int^p \sqrt{\omega} \quad \mbox{in the hal-translation case)}.$$ It is defined up to (half-)translations: $z\leadsto (\pm) z +c$ where $c$ is a complex constant.

Veech has introduced the group $Aff_+(X,\omega)$ of orientation-preserving diffeomorphisms $h:X\rightarrow X$ that are affine relatively to the distinguished coordinate $z$.

Question: how is $Aff_+(X,\omega)$ related to the group ${\rm Aut}(X)$ of holomorphic isomorphisms of $X$?

Answer 1

They're distinct objects. There is some overlap; those elements of $\mathrm{Aff}_+(X,\omega)$ whose derivatives lie in $SO(2)$ are holomorphic automorphisms of $X$.