Stratified bundles and Representation spaces

For a given stratified bundle $E$ on $X$, we construct an irreducible closed subvariety $\text{sN}(E{)}_S$ of the so called representation space $R({\text{sO}}_{X_S},{\xi }_S,P)\to S$ such that $\text{sN}(E{)}_S({\bar{\mathbb{F}}}_q)$ contains a dense set of $(V,\beta )$ where $V$ is induced by a representation of ${\pi }_1^{\acute{\mathrm{e}}\mathrm{t}}(X)$ (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}).