Generalised Trinh–Xue conjecture for spetsial reflection groups

Let $W$ be a finite spetsial complex reflection group on $V=\mathbb C^n$, let $\vhi\in\operatorname{GL}(V)$ normalise $W$, and write ${\mathbb G}=(V,W\vhi)$. For a root of unity $\ze$ set $\Phi=x-\ze$, and for a $\Phi$-cuspidal pair $({\mathbb L},\lambda)$ let $\Psi_{{\mathbb L},\lambda}$ be the bijection from the irreducible characters of the associated Hecke algebra to its $\Phi$-Harish-Chandra series. Let $({\mathbb M},\mu)$ be a $\Phi'=x-\ze'$-cuspidal pair.

Generalised Trinh–Xue conjecture. The intersection of the $\Phi$- and $\Phi'$-Harish-Chandra series

$$\Psi_{{\mathbb L},\lambda}(\operatorname{Irr}({\mathcal H}(W_{\mathbb G}({\mathbb L},\lambda),x)))\cap\Psi_{{\mathbb M},\mu}(\operatorname{Irr}({\mathcal H}(W_{\mathbb G}({\mathbb M},\mu),x)))$$

admits a partition $\bigsqcup B_i$ such that, for every $i$, $\Psi_{{\mathbb L},\lambda}^{-1}(B_i)$ is a block of ${\mathcal H}(W_{\mathbb G}({\mathbb L},\lambda),\ze')$ and $\Psi_{{\mathbb M},\mu}^{-1}(B_i)$ is a block of ${\mathcal H}(W_{\mathbb G}({\mathbb M},\mu),\ze)$. Thus the partition defines a bijection between the blocks of the two algebras appearing in the intersection.

This extends the proposed block correspondence from finite reductive groups to spetsial complex reflection groups and their unipotent characters. It is presented as a proposal in the source, with no resolution supplied.