Higgs bundles and representation spaces associated to morphisms

Abstract: 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 $\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_* : \pi_1(X) \rightarrow\pi_1(Y)$ is surjective. Define $$ {\mathcal R }^f(\pi_1(X),\, G)\,=\, {\rho\, \in\, \text{Hom}(\pi_1(X),\, G)\, \mid\, A\circ\rho \ \text{ factors through }~ f_}\, , $$ $$ {\mathcal R }^f(\pi_1(X),\, K)\,=\, {\rho\, \in\, \text{Hom}(\pi_1(X),\, K)\, \mid\, A\circ\rho \ \text{ factors through }~ f_}\, , $$ where $A: G \rightarrow \text{GL}(\text{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal R }^f(\pi_1(X, x_0), G)/!!/G$ admits a deformation retraction to ${\mathcal R }^f(\pi_1(X, x_0),\, K)/K$. 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 $K$.