On $m$-quasi-invariants of a Coxeter group

Let $W$ be a finite Coxeter group in a Euclidean vector space $V$, and let m be a $W$-invariant $\mathbb Z_+$-valued function on the set of reflections in $W$. Chalykh and Veselov introduced an interesting algebra $Q_m$, called the algebra of $m$-quasi-invariants for $W$, such that $\mathbb C[V]_W\subseteq Q_m\subseteq\mathbb C[V]$, $Q_0=\mathbb C[V]$ and $Q_m\supseteq Q_{m'}$ whenever $m\leq m'$. Namely, $Q_m$ is the algebra of quantum integrals of the rational Calogero–Moser system with coupling constant $m$. Feigin and Veselov proposed a number of interesting conjectures concerning the structure of $Q_m$ and verified them for dihedral groups and constant functions $m$. Our objective is to prove some of these conjectures in the general case.