**Speaker:** Maria Chlouveraki (Versailles)

**Title:** *Two Fundamental Conjectures on the Structure of Hecke Algebras. Part II: The BMM*

**Abstract:**

The work of BrouĂ©, Malle and Rouquier described in Part I was motivated by the theory, developed together with Michel, of "Spetses", which are objects that generalise finite reductive groups. In their article "Towards Spetses I", BrouĂ©, Malle and Michel stated the BMM symmetrising trace conjecture, according to which there exists a symmetrising trace function on the Hecke algebra that satisfies certain canonicality conditions. The existence of such a trace gives us a lot of insight into the modular representation theory of the Hecke algebra, that is, when its parameters specialise to complex numbers. Until recently, this conjecture was known to hold for very few non-real reflection groups. In this talk we explain the proof of this conjecture for 5 new cases, using most of the times the bases described in Part I.