Finite-dimensional representation conjecture for rational Cherednik algebras of type A

Let $c=r/n>0$. The algebra ${\mathsf{H}}_c$ is the rational Cherednik algebra associated with $S_n$, and $V$ denotes a finite-dimensional ${\mathsf{H}}_c$-module.

Finite-dimensional representation conjecture. The algebra ${\mathsf{H}}_c$ has a nonzero finite-dimensional representation $V$ if and only if $r$ is coprime to $n$; moreover, in this case the integer $\ell$ in Theorem (ii) satisfies $\ell=1$.

This conjecture characterizes the positive rational parameters for which finite-dimensional representations exist and specifies their restriction to $S_n$. It was proved in the cited work, so the conjecture is solved.