Let \(W=C_l \wr \mathfrak{S}_n\) be a wreath product of a cyclic group \(C_l\) and \(\overline{H}_c(W)\) the restricted rational Cherednik ablgebra at a generic parameter \(c\). Then the group algebra of \(W\) is naturally a subalgebra of \(\overline{H}_c(W)\), so that it is possible to restrict modules over \(\overline{H}_c(W)\) to representations of \(W\).
The irreducible representations \(\lambda\) of \(W\) are in bijection with the isomorphism classes of simple modules \(L(\lambda)\) of \(\overline{H}_c(W)\), and both are indexed by the \((l, n)\)-multipartitions \(\mathcal{P}(l, n)\). The article under review studies the restrictions of the simple modules \(L(\lambda)\) of \(\overline{H}_c(W)\) and their decompositions into irreducible representations of \(W\). More concretely, it establishes an explicit formula for their image \([L(\lambda)]\) in the Grothendieck group of \(\operatorname{mod} W\). This generalizes a result by I. Gordon [Bull. Lond. Math. Soc. 35, No. 3, 321–336 (2003; Zbl 1042.16017)], who has established such a formula in the case that \(W\) is a symmetric group. As in the case studied by Gordon, the question reduces to a question about the multiplicative structure on the Grothendieck group of \(\operatorname{mod} W\) which is induced by the tensor product. Hence the authors first collect the relevant results on the character theory of \(C_l \wr \mathfrak{S}_n\). Then they proceed by defining a generalization the \((t,t)\)-Kostka-Macdonald coefficients to a multipartition version \(K_{\mathbf{\lambda\mu}}(t,t)\), \(\mathbf{\lambda}, \mathbf{\mu}\in \mathcal{P}(l, n)\). They prove that with this definition \begin{align*} [L(\lambda)]=\sum_{\mu\in \mathcal{P}(l, n)}K _{\mathbf{\lambda\mu}}(t,t)[\mu], \end{align*} which generalizes the formula by Gordon for the symmetric group case. Finally, using this formula, they prove that the image of \([L(\lambda)]\) under the Frobenius character map is given by a specialization of the wreath Macdonald polynomial.