Higgs bundles and representation spaces associated to morphisms

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 $f: X \rightarrow Y$ an algebraic morphism, such that $π_1(Y)$ is virtually nilpotent and the homomorphism $f_* : π_1(X) \rightarrowπ_1(Y)$ is surjective. Define ${\mathcal R }^f(π_1(X),\, G)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, G)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, ,$ ${\mathcal R }^f(π_1(X),\, K)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, K)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, ,$ where $A: G \rightarrow \text{GL}(\text{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal R }^f(π_1(X, x_0), G)/\!\!/G$ admits a deformation retraction to ${\mathcal R }^f(π_1(X, x_0),\, K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements in $G$ admits a deformation retraction to the space of conjugacy classes of $n$ almost commuting elements in $K$.

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