Stratified bundles and Representation spaces

For a given stratified bundle $E$ on $X$, we construct an irreducible closed subvariety $\sN(E)_S$ of the so called representation space $R(\sO_{X_S},\xi_S,P)\to S$ such that $\sN(E)_S(\overline{\mathbb{F}}_q)$ contains a dense set of $(V,\beta)$ where $V$ is induced by a representation of $\pi_1^{{\rm \acute{e}t}}(X)$ (Theorem \ref{thm3.7}). As an application, we give a simply proof of the main theorem of \cite{EM} and its relative version (Theorem \ref{thm4.2}).