#### 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.

Done