Martino's conjecture on Calogero–Moser and Hecke families

Assume ${\mathbf k}={\mathbb C}$, let $c$ be a conjugation-invariant parameter, let $k$ be the associated real Hecke parameter, and set $k^\sharp_{\Omega,j}=k_{\Omega,-j}$. For a Calogero–Moser block idempotent $b\in {\mathrm{Idem_{pr}}}({\bar Z}_c)$, write $\operatorname{Irr}_{\mathbf H}(W,b)$ for the corresponding Calogero–Moser family and let $b^{\mathcal H}$ be a central idempotent of the associated Hecke algebra.

Martino's conjecture. For every $b\in {\mathrm{Idem_{pr}}}({\bar Z}_c)$, there exists a central idempotent $b^{\mathcal H}$ of ${\mathcal O}^{\mathrm{cyc}}[\mathbf q^{\mathbb R}]{\mathcal H}_W^{\mathrm{cyc}}(k^\sharp)$ such that

$$\operatorname{Irr}_{\mathbf H}(W,b)=\operatorname{Irr}_{\mathcal H}(W,b^{\mathcal H})$$

and

$$\dim_{\mathbb C}({\bar Z}b)=\dim_{F(\mathbf q^{\mathbb R})}\bigl(F(\mathbf q^{\mathbb R}){\mathcal H}_W^{\mathrm{cyc}}(k^\sharp)b^{\mathcal H}\bigr).$$

In particular, every Calogero–Moser $c$-family is a union of $k^\sharp$-families of the Hecke algebra.

This conjecture relates Calogero–Moser families to Hecke-algebra families. The paper recalls it as Martino's conjecture and discusses the state of knowledge toward its proof; no complete resolution is given in the supplied text.