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Let $ \Omega \subset \mathbb{C}^m $ be an open, connected and bounded set and$\mathcal{A}(\Omega)$ be a function algebra of holomorphic functions on$\Omega$. Suppose that $\mathcal{M}_q$ is the quotient Hilbert module obtainedfrom a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing toorder $k$ along a smooth irreducible complex analytic set$\mathcal{Z}\subset\Omega$ of codimension at least $2$. In this article, weprove that the compression of the multiplication operators onto $\mathcal{M}_q$is homogeneous with respect to a suitable subgroup of the automorphism groupAut$(\Omega)$ of $\Omega$ depending upon a subgroup $G$ of Aut$(\Omega)$whenever the tuple of multiplication operators on $\mathcal{M}$ is homogeneouswith respect to $G$ and both $\mathcal{M}$ as well as $\mathcal{M}_q$ are inthe Cowen-Douglas class. We show that these compression of multiplicationoperators might be reducible even if the tuple of multiplication operators on$\mathcal{M}$ is irreducible by exhibiting a concrete example. Moreover, theirreducible components of these reducible operators are identified asGeneralized Wilkins' operators.

On irreducibility of a certain class of homogeneous operators obtained from quotient modules