Suppose that $X$ is a connected smooth manifold and $Gamma$ is a group acting smoothly, freely, properly and discretely on $X$, so that $Y=X/Gamma$ is another smooth manifold endowed with a covering map $pi:Xrightarrow Y$.

Suppose that $G$ is a Lie group and that $rho:Gamma rightarrow G$ is a group homomorphism. Then we can consider the quotient $E_rho =(Xtimes G)/Gamma$, where the action of $Gamma$ on $G$ is given by composing $rho$ with the adjoint action of $G$ on itself. $E_rho$ is naturally a principal $G$-bundle over $Y$.

My question is if there exists a condition for a principal $G$-bundle $E$ on $Y$ to be isomorphic to an $E_rho$, for some homomorphism $rho:Gamma rightarrow G$.

This can be interpreted as a “relative” notion of a flat connection since, if $X$ is the universal covering space of $Y$ and $Gamma=pi_1(Y)$, then the condition for $E$ to be of the form $E_rho$ is that $E$ admits a flat connection.

Moreover, the same question can be extended to the holomorphic category. For example, we can take $X$ a Riemann surface and $G=U(n)$. In that situation, if $X$ is the hyperbolic plane and $Y$ is a compact Riemann surface of genus $geq 2$, the condition for a holomorphic Hermitian vector bundle $E$ to be of the form $E_rho$ is that it is stable of degree $0$ (this is the Theorem of Narasimhan-Seshadri).