Ergodic properties of infinite extension of symmetric interval exchange transformations

We prove that skew products with the cocycle given by the function $f(x)=a(x-1/2)$ with $a\neq 0$ are ergodic for every ergodic symmetric IET in the base, thus giving the full characterization of ergodic extensions in this family. Moreover, we prove that under an additional natural assumption of unique ergodicity on the IET, we can replace $f$ with any differentiable function with a non-zero sum of jumps. Finally, by considering weakly mixing IETs instead of just ergodic, we show that the skew products with cocycle given by $f$ have infinite ergodic index.