Room 217

Reducing submodules of Hilbert modules with an invariant kernel and analytic Chevalley-Shephard-Todd Theorem

Shibananda Biswas

IISER Kolkata

This talk is divided in to two parts. In the first part,  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 $\C[\boldsymbl
z]^{\mathfrak S_n}$, indexed by the partitions $\bl p$ of $n$. We then
discuss the problem of minimality and  inequivalence 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
$\lambda>0$. It seems that one way to deal with the equivalence problem is
to 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, more
generally, the analytic version do exist for finite pseudo-reflection
groups. In the process, we obtain a purely algebraic determinantal formula
that may also be of independent interest.