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Monday, November 18, 2019 - 15:35 to 16:35
Seminar Room, School of Mathematical Sciences
Dr. Shibananda Biswas
Indian Institute of Science Education and Research Kolkata
Reducing submodules of Hilbert modules with an invariant kernel

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. 

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