The Calogero-Moser partition for \(G(m,d,n)\). (Q2911016)

Let \(W\) be a finite complex reflection group. The rational Cherednik algebras associated to \(W\) depend on two parameters \(t\) and \(\mathbf c\). The restricted rational Cherednik algebra is a certain finite-dimensional quotient of the rational Cherednik algebra at \(t=0\). The blocks of the restricted rational Cherednik algebra induce a partitioning of the set \(\text{Irr}(W)\) of irreducible \(W\)-modules, called Calogero-Moser partition. Using the geometry of certain quiver varieties, \textit{I. G. Gordon} and \textit{M. Martino} gave an explicit combinatorial description of the Calogero-Moser partition [in Math. Res. Lett. 16, No. 2-3, 255-262 (2009; Zbl 1178.16030)] when \(W=G(m,1,n)\).NEWLINENEWLINE The present paper shows that Clifford theoretic arguments can be used to extend this result to the normal subgroups \(G(m,d,n)\) of \(G(m,1,n)\). Gordon and Martino conjectured [in loc. cit.] that the Calogero-Moser partition should be related in some precise way to the Rouquier blocks of a particular Hecke algebra associated to the same complex reflection group \(W\). This conjecture was refined by \textit{M. Martino} [in J. Algebra 323, No. 1, 193-205 (2010; Zbl 1219.20006)] and was verified in the case of \(W=G(m,1,n)\). A consequence of the main result of the present paper is that the conjecture as stated by \textit{M. Martino} [loc. cit., Conjecture 2.7 (i)] is true for all \(G(m,d,n)\). However, it should be noted that, when \(n=2\) and \(d\) is even, there are certain unequal parameter cases where the method used in the present paper fails. In these cases, the Calogero-Moser partition is not known.