In Liu and Zhu’s article, remark 1.10(i), the following result from the article of Simpson, Corollary 4.2 is cited:
If $L$ is a Betti local system on a complex smooth projective variety that underlies a variation of Hodge structure, then the corresponding Higgs bundle $(M, \theta )$ under the (classical) Simpson correspondence is nilpotent.
However the original statement from Simpson’s article is
The representations of $\pi_{1}(X)$ which come from complex variations of Hodge structure are exactly the semisimple ones which are fixed by the action of $\mathbb{C}^*$.
It confused me how we get Liu&Zhu’s statement from Simpson’s original form. Also, I’m not cleared about the definition of a nilpotent Higgs field, mainly confused in how they define the iteration of it. I guess it should be $$\theta^n: E\otimes \Omega^{n-1} \rightarrow E\otimes \Omega^{1}\otimes \Omega^{n-1}, \theta\otimes Id $$ for Higgs field $\theta $ over holomorphic vector bundle $E$.