The character-sum refinement for fixed Calogero–Moser components

Let $V$, $W$, $\tau$, $\aleph$, $\aleph_\tau$, $\lambda$, and the Calogero–Moser spaces be as above. Assume that the fixed-component Calogero–Moser isomorphism conjecture holds. For a ${\mathbb C}^\times$-fixed point $p$ on the maximal component, let ${\mathfrak F}_{\lambda(k)}^{(\tau)}$ be the Calogero–Moser $\lambda(k)$-family of $W^\tau$ corresponding to $p$ under $\iota_k$.

Character-sum refinement. Assume that $\tau$ is regular and that the fixed-component Calogero–Moser isomorphism conjecture holds. If $p\in({\mathcal Z}_k^\tau)_{\mathrm{max}}^{{\mathbb C}^\times}$, then

$$\sum_{\chi\in{\mathfrak F}_p^\tau}|\widetilde\chi(\tau)|^2=\sum_{\psi\in{\mathfrak F}_{\lambda(k)}^{(\tau)}}\psi(1)^2.$$

This conjecture makes the preceding fixed-point criterion more precise by predicting equality with the corresponding character-degree sum for the fixed reflection group. It is conditional on the fixed-component isomorphism conjecture and is not resolved in the source.