# Decomposition of the polynomials over the spherical subalgebra

## TL;DR

This paper explores the structure of modules over the spherical subalgebra of the rational Cherednik algebra, establishing a lowest weight category description and drawing analogies with invariant differential operator decompositions.

## Contribution

It introduces a lowest weight category framework for modules over the spherical Cherednik algebra, generalizing known decompositions for invariant differential operators across all finite reflection groups.

## Key findings

- Established a lowest weight description for the category of D_c-modules over the polynomial ring.
- Identified the algebra Rc_c as an analogue of the Cartan algebra in this setting.
- Extended the analogy between Cherednik algebra modules and invariant differential operator modules.

## Abstract

Given a finite subgroup $W \subset \GL(\fh)$ of the linear group of a finite-dimensional complex vector field $\fh$, it is a well-studied problem to describe the structure of the symmetric algebra $B= \sym(\fh^*)$ as a representation of $G$, and also as a module over the ring of invariant differential operators under $W$ in the ring $\D(\fh)$ of differential operators on $\fh$. Since the rational Cherednik algebra $H_c(W,\fh)$ and the spherical algebra $eH_ce$ are respectively universal deformations of the ring $\D(\fh)$ and the ring $\D(\fh)^W$of $W$-invariant differential operators, we would like to build an analogy between the decomposition of modules over the invariant differential operators in \cite{ Nonk1, Nonk2, Nonk3} and the decomposition of modules over the sperical subalgebra of the rational Cherednik algebra. The ring $\D_c= eH_ce$ inherits the natural grading of $B$, and we let $\D_c^0 \subset \D_c$ and $\D_c^{-} \subset \D_c$ be subset of elements of degree 0 end strictly negative degree, respectively. Our main result is that there is for all finite reflection groups a lowest weight description of the category of $\D_c$-modules of $B$ where the ring $\Rc_c= \D_c^0 / \Dc_c^0 \cap \D_c \D_c^{-}$ plays the very important role of Cartan algebra.