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Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\subset G$ be a maximal compact subgroup. Let $X$, $Y$ be irreducible smooth complex projective varieties and $f:X\to Y$ an algebraic morphism, such that ${\pi }_1\left(Y\right)$ is virtually nilpotent and the homomorphism $f_{*}:{\pi }_1\left(X\right)\to {\pi }_1\left(Y\right)$ is surjective. Define $$\begin{array}{cc}\hfill {ℛ}^f({\pi }_1\left(X\right),G)& =\{\rho \in Hom({\pi }_1\left(X\right),G)\mid A\circ \rho \text{factors}\text{through}{f}_{*}\},\hfill \\ \hfill {ℛ}^f({\pi }_1\left(X\right),K)& =\{\rho \in Hom({\pi }_1\left(X\right),K)\mid A\circ \rho \text{factors}\text{through}{f}_{*}\},\hfill \end{array}$$ where $A:G\to GL(Lie(G\left)\right)$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${ℛ}^f({\pi }_1(X,x_0),G)//G$ admits a deformation retraction to ${ℛ}^f({\pi }_1(X,x_0),K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements...