We show that $\mathcal H$, with an $\mathfrak S_n$ invariant reproducing kernel $K$on an $\mathfrak S_n$ domain in $\C^n$, splits into reducing submodules $\mathbb P_{\bl p} \m H$, over the invariant ring $\C[\boldsymbol z]^{\mathfrak S_n}$, indexed by the partitions $\bl p$ of $n$. We then discuss the problem of minimality, inequivalence and realization of the submodules $\mathbb P_{\bl p} \m H$, particularly in the case when $\mathcal H$ is the weighted Bergman space $\mb A^{(\lambda)}(\mb D^n)$, for $\lambda>0$. One way to deal with the equivalence problem is through the realization and for which an analogue of Chevalley-Shephard-Todd Theorem for $\mathfrak S_n$ in the analytic setup seems quintessential. In fact, we show that the analytic version do exist for the most general version, that is, for finite pseudo-reflection groups. These results are from the joint works with Swarnendu Datta, Gargi Ghosh, Gadadhar Misra and Subrata Shyam Roy.
Venue
Seminar Room, School of Mathematical Sciences
Speaker
Dr. Shibananda Biswas
Affiliation
Indian Institute of Science Education and Research Kolkata
Title
Reducing submodules of Hilbert modules with an invariant kernel