Connectedness conjecture for the stabilizers of Calogero–Moser spaces

For each $n\geq 0$, let $G_n$ be the stabilizer in $G$ of a point of the Calogero–Moser space $\mathcal{C}_n$. Since the action is algebraic, $G_n$ is a closed subgroup of the ind-group $G$.

Connectedness conjecture for the stabilizers. The groups $G_n$ are connected and hence irreducible for all $n\geq 0$.

The conjecture concerns the ind-algebraic structure of the stabilizers $G_n$. The paper establishes that these stabilizers are closed algebraic subgroups, but leaves their connectedness and irreducibility as an open question.