Yvonne–Rouquier conjecture on multiplicities for wreath products

Let $l\geq 1$, let $\mathcal{P}^l_n$ denote the set of $l$-multipartitions of $n$, and let $\Delta(\boldsymbol{\lambda})$ and $L(\boldsymbol{\mu})$ denote respectively the standard and simple modules in category $\mathcal{O}$ for $\mathcal{H}_{\mathbf{c}}(\mathfrak{s}_n\wr\mathbb{Z}_l)$. If $d_{\boldsymbol{\lambda},\boldsymbol{\mu}}(q)$ are the coefficients defined by the Uglov canonical basis expansion and $\boldsymbol{\lambda}^{\dagger},\boldsymbol{\mu}^{\dagger}$ denote the corresponding dual multipartitions, then Yvonne–Rouquier conjecture. For all $\boldsymbol{\lambda},\boldsymbol{\mu}\in\mathcal{P}^l_n$,

$$[\Delta(\boldsymbol{\lambda}):L(\boldsymbol{\mu})]=d_{\boldsymbol{\lambda}^{\dagger},\boldsymbol{\mu}^{\dagger}}(1).$$

This conjecture relates composition multiplicities of standard modules for cyclotomic rational Cherednik algebras to canonical-basis coefficients in Uglov's level-$l$ Fock spaces. It was originally due to Yvonne, with the stated generality attributed to Rouquier; the supplied material gives no resolution status.