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This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let 
$G$ be a compact group and 
$H$ a closed subgroup of $G$. Let 
$\mu $ be the normalized $G$-invariant measure over the compact homogeneous space 
$G/H$ associated with Weil's formula and 
$1\,\le \,p\,<\,\infty $. We then present a structured class of abstract linear representations of the Banach convolution function algebras 
${{L}^{p}}\left( G/H,\,\mu \right)$.
