A counter-example to Martino's conjecture about generic Calogero-Moser families

Abstract

The Calogero-Moser families are partitions of the irreducible characters of a complex reflection group derived from the block structure of the corresponding restricted rational Cherednik algebra. It was conjectured by Martino in 2009 that the generic Calogero-Moser families coincide with the generic Rouquier families, which are derived from the corresponding Hecke algebra. This conjecture is already proven for the whole infinite series G(m,p,n) and for the exceptional group G4. A combination of theoretical facts with explicit computations enables us to determine the generic Calogero-Moser families for the nine exceptional groups G4, G5, G6, G8, G10, G23=H3, G24, G25, and G26. We show that the conjecture holds for all these groups - except surprisingly for the group G25, thus being the first and only-known counter-example so far.