The Broué-Malle-Rouquier conjecture for the exceptional groups of rank 2

Proof of the Broué–Malle–Rouquier Conjecture in Characteristic Zero (After I. Losev and I. Marin—G. Pfeiffer)

“…This conjecture is currently known for all irreducible complex reflection groups except G 17 , ..., G 21 (according to the Shephard-Todd classification), and there is a hope that these cases can be proved as well using a sufficiently powerful computer (see [Cha1,Cha2] and [M3] for more details). Also, it is shown in [BMR] that to prove the conjecture, it suffices to show that H(W ) is spanned by |W | elements.…”

“…For instance, when W is a Coxeter group, then such a splitting is well known and is obtained by taking reduced expressions in the braid group. Such a version is currently available (over any base ring) for all irreducible complex reflection groups except G 17 , ..., G 21 , see [M3], [Cha1,Cha2].…”

“…We note that this argument works over an arbitrary base ring. A much more detailed description of this argument is given in [Cha2].…”