I will present arguably the most basic one among the set of conjectures stated in 1998 by Broue, Malle and Rouquier (following early work by Broue and Malle) about the generalized Iwahori-Hecke algebras associated to complex reflection groups. By a combination of several kind of arguments and lots of hand-writen as well as computer-assisted calculations, it seems that a complete proof is now within reach. I will report on recent progress by my PhD student E. Chavli, as well as on a recent work by G. Pfeiffer and myself on this topic.[-]

A well-known result of Davis-Januszkiewicz is that every right-angled Artin group (RAAG) is commensurable to some rightangled Coxeter group (RACG). In this talk we consider the converse question: which RACGs are commensurable to some RAAG? To do so, we investigate some natural candidate RAAG subgroups of RACGs and characterize when such subgroups are indeed RAAGs. As an application, we show that a 2-dimensional, one-ended RACG with planar defining graph is quasiisometric to a RAAG if and only if it is commensurable to a RAAG. This talk is based on work joint with Pallavi Dani.[-]

Recently several papers appears on ArXiv, on various topics apparently unrelated such as: spin system observable (T. Helmuth, A. Shapira), Fibonacci polynomials (A. Garsia, G. Ganzberger), fully commutative elements in Coxeter groups (E. Bagno, R. Biagioli, F. Jouhet, Y. Roichman), reciprocity theorem for bounded Dyck paths (J. Cigler, C. Krattenthaler), uniform random spanning tree in graphs (L. Fredes, J.-F. Marckert). In each of these papers the theory of heaps of pieces plays a central role. We propose a walk relating these topics, starting from the well-known loop erased random walk model (LERW), going around the classical bijection between lattice paths and heaps of cycles, and a second less known bijection due to T. Helmuth between lattice paths and heaps of oriented loops, in relation with the Ising model in physics, totally non-backtracking paths and zeta function in graphs. Dyck paths, these two bijections involve heaps of dimers and heaps of segments. A duality between these two kinds of heaps appears in some of the above papers, in relation with orthogonal polynomials and fully commutative elements. If time allows we will finish this excursion with the correspondence between heaps of segments, staircase polygons and q-Bessel functions.[-]

Using the representation theory of Cherednik algebra at t= 0, we define a family of "Calogero-Moser cellular characters" for any complex reflection group $W$. Whenever $W$ is a Coxeter group, we conjecture that they coincide with the "Kazhdan-Lusztig cellular characters". We shall give some evidences for this conjecture. Our main result is that, whenever the associated Calogero-Moser space is smooth, then all the Calogero-Moser cellular characters are irreducible. This implies in particular that our conjecture holds in type $A$ and for some particular choices of the parameters in type $B$.[-]