  In this article, we explore Higgs bundles on a projective manifold $X$,
focusing on their spectral bases, a concept introduced by T.Chen and B.Ng\^{o}.
The spectral base is a specific closed subscheme within the space of symmetric
differentials. We observe that if the spectral base vanishes, then any
reductive representation $\rho: \pi_1(X) \to \text{GL}_r(\mathbb{C})$ is both
rigid and integral. Additionally, we prove that for $X=\Omega/\Gamma$, a
quotient of a bounded symmetric domain $\Omega$ of rank at least $2$ by a
torsion-free cocompact irreducible lattice $\Gamma$, the spectral base indeed
vanishes, which generalizes a result of B.Klingler.
