Tuesday, March 19 2019
11:30 - 12:45

This talk is divided in to two parts. In the first part,  we showthat $\mathcal H$, 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 $\C[\boldsymblz]^{\mathfrak S_n}$, indexed by the partitions $\bl p$ of $n$. We thendiscuss the problem of minimality and  inequivalence of thesubmodules $\mathbb P_{\bl p} \m H$, particularly in the case when$\mathcal H$ is the weighted Bergman space $\mb A^{(\lambda)}(\mb D^n)$, for$\lambda>0$. It seems that one way to deal with the equivalence problem isto have an analogue of Chevalley-Shephard-Todd Theorem for $\mathfrak S_n$in the analytic setup. In the second part of this talk, we show that, moregenerally, the analytic version do exist for finite pseudo-reflectiongroups. In the process, we obtain a purely algebraic determinantal formulathat may also be of independent interest.

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