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We review the state-of-the-art concerning the freeness conjecture stated inthe 1990's by Brou\'e, Malle and Rouquier for generic Hecke algebras associatedto complex reflection groups, and in particular we expose in detail one of themain differences with the ordinary case, namely the lack of 0-Hecke algebras.We end the paper by proving a new case of this conjecture, the exceptionalgroup called G26 in Shephard-Todd classification, namely the largest lineargroup of automorphisms of the Hessian configuration.

The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group G26