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

Abstract

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 obtained from a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subset \Omega$ of codimension at least $2$. In this article, we prove that the compression of the multiplication operators onto ${\mathcal{M}}_q$ is homogeneous with respect to a suitable subgroup of the automorphism group Aut$(\Omega )$ of $\Omega$ depending upon a subgroup $G$ of Aut$(\Omega )$ whenever the tuple of multiplication operators on $\mathcal{M}$ is homogeneous with respect to $G$ and both $\mathcal{M}$ as well as ${\mathcal{M}}_q$ are in the Cowen-Douglas class. We show that these compression of multiplication operators might be reducible even if the tuple of multiplication operators on $\mathcal{M}$ is irreducible by exhibiting a concrete example. Moreover, the irreducible components of these reducible operators are identified as Generalized Wilkins' operators.

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