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This work investigates analytic Hilbert modules $\mathcal{H}$, over the polynomial ring, consisting of holomorphic functions on a $G$-space $\Omega \subset \mathbb{C}^m$ that are homogeneous under the natural action of the group $G$. In a departure from the past studies of such questions, here we don't assume transitivity of the group action. The primary finding reveals that unitary invariants such as curvature and the reproducing kernel of a homogeneous analytic Hilbert module can be deduced from their values on a fundamental set $\Lambda$ of the group action. Next, utilizing these techniques, we examine the analytic Hilbert modules associated with the symmetrized bi-disc $\mathbb{G}_2$ and its homogeneity under the automorphism group of $\mathbb{G}_2$. It follows from one of our main theorems that none of the weighted Bergman metrics on the symmetrized bi-disc is K\"{a}hler-Einstein., Comment: 38 pages

Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $\mathbb D^n$ in $\mathbb C^n.$ A $\boldsymbol\Theta_n$-contraction is a commuting tuple of operators on a Hilbert space having $$\overline{\boldsymbol\Theta}_n:=\{\boldsymbol\theta(z)=(\theta_1(z),\ldots,\theta_n(z)):z\in\overline{\mathbb D}^n\}$$ as a spectral set, where $\{\theta_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G(m,p,n).$ A plethora of examples of $\boldsymbol\Theta_n$-contractions is exhibited. Under a mild hypothesis, it is shown that these $\boldsymbol\Theta_n$-contractions are mutually unitarily inequivalent. These inequivalence results are obtained concretely for the weighted Bergman modules under the action of the permutation groups and the dihedral groups. The division problem is shown to have negative answers for the Hardy module and the Bergman module on the bidisc. A Beurling-Lax-Halmos type representation for the invariant subspaces of $\boldsymbol\Theta_n$-isometries is obtained., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1301.2837

Let $\Omega \subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(\Omega)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring $\mathbb C[\boldsymbol{z}]\subseteq \mathcal H$ is dense and the point-wise multiplication induced by $p\in \mathbb C[\boldsymbol{z}]$ is bounded on $\mathcal H$. We fix an ideal $\mathcal I \subseteq \mathbb C[\boldsymbol{z}]$ generated by $p_1,\ldots,p_t$ and let $[\mathcal I]$ denote the completion of $\mathcal I$ in $\mathcal H$. The sheaf $\mathcal S^\mathcal H$ associated to analytic Hilbert module $\mathcal H$ is the sheaf $\mathcal O(\Omega)$ of holomorphic functions on $\Omega$ and hence is free. However, the subsheaf $\mathcal S^{\mathcal [\mathcal I]}$ associated to $[\mathcal I]$ is coherent and not necessarily locally free. Building on the earlier work of \cite{BMP}, we prescribe a hermitian structure for a coherent sheaf and use it to find tractable invariants. Moreover, we prove that if the zero set $V_{[\mathcal I]}$ is a submanifold of codimension $t$, then there is a unique local decomposition for the kernel $K_{[\mathcal I]}$ along the zero set that serves as a holomorphic frame for a vector bundle on $V_{[\mathcal I]}$. The complex geometric invariants of this vector bundle are also unitary invariants for the submodule $[\mathcal I] \subseteq \mathcal H$., Comment: To appear in Complex Analysis and Operator Theory

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., Comment: Primary version

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue $L^2$-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan are obtained on rational curves., Comment: 8 pages

Let $G$ be a finite pseudoreflection group, $\Omega\subseteq \mathbb C^n$ be a bounded domain which is a $G$-space and $\mathcal H\subseteq\mathcal O(\Omega)$ be an analytic Hilbert module possessing a $G$-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator $\mathbf M_{\boldsymbol\theta}$ on $\mathcal H,$ where $\{\theta_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\boldsymbol\theta = (\theta_1, \ldots, \theta_n)$ is a polynomial map of $\mathbb C^n$. We show that it admits a family $\{\mathbb P_\varrho\mathcal H:\varrho\in\widehat G\}$ of non-trivial joint reducing subspaces, where $\widehat G$ is the set of all equivalence classes of irreducible representations of $G.$ We prove a generalization of Chevalley-Shephard-Todd theorem for the algebra $\mathcal O(\Omega)$ of holomorphic functions on $\Omega$. As a consequence, we show that for each $\varrho\in \widehat G,$ the multiplication operator $\mathbf M_{\boldsymbol\theta}$ on the reducing subspace $\mathbb P_\varrho \mathcal H$ can be realized as multiplication by the coordinate functions on a reproducing kernel Hilbert space of $\mathbb C^{(\mathrm{deg}\,\varrho)^2}$-valued holomorphic functions on $\boldsymbol\theta(\Omega)$. This, in turn, provides a description of the structure of joint reducing subspaces of the multiplication operator induced by a representative of a proper holomorphic map from a domain $\Omega$ in $\mathbb C^n$ which is factored by automorphisms $G\subseteq {\rm Aut}(\Omega).$, Comment: Extensively revised with new results and applications that includes version 1 as one of the sections

The weighted Bergman spaces on the polydisc, $\mathbb A^{(\lambda)}(\mathbb D^n)$, $\lambda>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ indexed by the partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $\chi_{\boldsymbol p}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\boldsymbol p$ and $\boldsymbol q$, if $\chi_{\boldsymbol p}(1) \ne \chi_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbol p}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbol q}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$, respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent., Comment: 22 pages

Let $\mathcal{H}_d^{(t)}$ ($t \geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $\mathbb{B}_d$ with kernel \[ k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} . \] We prove that if an ideal $I \triangleleft \mathbb{C}[z_1, \ldots, z_d]$ (not necessarily homogeneous) has what we call the "approximate stable division property", then the closure of $I$ in $\mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $\mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$., Comment: Some mistakes fixed and details added to the proof of the main result. 17 pages

In this note we characterize the distinguished boundary of the symmetrized polydisc and thereby develop a model theory for $\Gamma_n$-isometries along the lines of \cite{AY}. We further prove that for invariant subspaces of $\Gamma_n$-isometries, similar to the case $n=2$ \cite{S}, Beurling-Lax-Halmos type representation holds., Comment: Submitted

The curvature $\mathcal K_T(w)$ of a contraction $T$ in the Cowen-Douglas class $B_1(\mathbb D)$ is bounded above by the curvature $\mathcal K_{S^*}(w)$ of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this note, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle $E_T$ corresponding to the operator $T$ in the Cowen-Douglas class $B_1(\mathbb D)$ which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the class $B_1(\Omega)$, for a bounded domain $\Omega$ in $\mathbb C^m$., Comment: 14 pages

A short proof of the "Rigidity theorem" using the sheaf theoretic model for Hilbert modules over polynomial rings is given. The joint kernel for a large class of submodules is described. The completion $[\mathcal I]$ of a homogeneous (polynomial) ideal $\mathcal I$ in a Hilbert module is a submodule for which the joint kernel is shown to be of the form $$ \{p_i(\tfrac{\partial}{\partial \bar{w}_1}, ..., \tfrac{\partial}{\partial \bar{w}_m}) K_{[\mathcal I]}(\cdot,w)|_{w=0}, 1 \leq i \leq n\}, $$ where $K_{[\mathcal I]}$ is the reproducing kernel for the submodule $[\mathcal I]$ and $p_1, ..., p_n$ is some minimal "canonical set of generators" for the ideal $\mathcal I$. The proof includes an algorithm for constructing this canonical set of generators, which is determined uniquely modulo linear relations, for homogeneous ideals. A set of easily computable invariants for these submodules, using the monoidal transformation, are provided. Several examples are given to illustrate the explicit computation of these invariants.

A refined notion of curvature for a linear system of Hermitian vector spaces, in the sense of Grothendieck, leads to the unitary classification of a large class of analytic Hilbert modules. Specifically, we study Hilbert sub-modules, for which the localizations are of finite (but not constant) dimension, of an analytic function space with a reproducing kernel. The correspondence between analytic Hilbert modules of constant rank and holomorphic Hermitian bundles on domains of $\mathbb C^n$ due to Cowen and Douglas, as well as a natural analytic localization technique derived from the Hochschild cohomology of topological algebras play a major role in the proofs. A series of concrete computations, inspired by representation theory of linear groups, illustrate the abstract concepts of the paper.