The fixed-component Calogero–Moser isomorphism conjecture

Let $V$ be the reflection representation of a finite complex reflection group $W$, let $\tau$ be a regular element normalizing $W$, and let $\aleph$ and $\aleph_\tau$ be the parameter-indexing sets for $(V,W)$ and $(V^\tau,W^\tau)$, respectively. Let ${\mathcal Z}_k$ and ${\mathcal Z}_l(V^\tau,W^\tau)$ denote the corresponding Calogero–Moser spaces, and let $({\mathcal Z}_k^\tau)_{\mathrm{max}}$ be the maximal-dimensional irreducible component of the fixed-point variety.

Fixed-component Calogero–Moser isomorphism conjecture. Assume that $\tau$ is regular. There exists a linear map

$$\lambda:{\mathbb C}^{\aleph}\longrightarrow{\mathbb C}^{\aleph_\tau}$$

and, for each $k\in{\mathbb C}^{\aleph}$, a ${\mathbb C}^\times$-equivariant isomorphism of Poisson varieties

$$\iota_k:({\mathcal Z}_k^\tau)_{\mathrm{max}}\stackrel{\sim}{\longrightarrow}{\mathcal Z}_{\lambda(k)}(V^\tau,W^\tau).$$

This is a special case of a conjecture attributed in the source to Bellamy and Schedler. The expected isomorphism would identify the maximal fixed component with a Calogero–Moser space for the fixed reflection group; the source notes that the conjecture is known in some cases, but not in general.