# On some modules of covariants for a reflection group

## TL;DR

This paper constructs a graded map linking modules of covariants associated with a reflection group and proves an enhanced form of Reeder's conjecture for the adjoint representation of a simple Lie algebra.

## Contribution

It introduces a new graded map between modules of covariants and proves an improved version of Reeder's conjecture for the adjoint representation.

## Key findings

- Established a graded map between covariant modules.
- Proved an enhanced form of Reeder's conjecture.
- Provided new insights into modules of covariants for reflection groups.

## Abstract

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak g)^\mathfrak g$ of $(\bigwedge \mathfrak g)^\mathfrak g\cong S(\mathfrak h)^W$-modules, where $\mathcal H$ is the space of $W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation. New version with different title. Various improvements. New section 7.