The regular-number character criterion for fixed Calogero–Moser spaces

Let $d$ be a regular number of $W$, let $\zeta_d$ be a primitive $d$-th root of unity, and choose a $\zeta_d$-regular element $w_d\in W$. Let ${\mathcal Z}_k^{{\boldsymbol\mu}_d}$ be the fixed-point variety for the group of $d$-th roots of unity, and let $({\mathcal Z}_k^{{\boldsymbol\mu}_d})_{\mathrm{max}}$ denote its maximal-dimensional irreducible component. For a ${\mathbb C}^\times$-fixed point $p$, let ${\mathfrak F}_p$ be its Calogero–Moser $k$-family.

Regular-number character criterion. Recall that $d$ is a regular number. For every $p\in{\mathcal Z}_k^{{\mathbb C}^\times}$, $p$ belongs to $({\mathcal Z}_k^{{\boldsymbol\mu}_d})_{\mathrm{max}}$ if and only if

$$\sum_{\chi\in{\mathfrak F}_p}|\chi(w_d)|^2\neq0.$$

This is the specialization of the fixed-point criterion to a regular element inducing an inner automorphism of $W$, for which $|\widetilde\chi(\tau_d)|^2=|\chi(w_d)|^2$. Its status is not separately established in the source and is therefore open.