Calogero-Moser versus Kazhdan-Lusztig cells. (Q1951181)

The paper under review proposes a definition of left cells and two-sided cells for a complex reflection group \(G\) in virtue of the ramification theory for Calogero-Moser spaces. It is known that any Coxeter group \(W\) can be canonically regarded as a complex reflection group; that the left cells and two-sided cells of \(W\) were defined combinatorially by \textit{D. Kazhdan} and \textit{G. Lusztig} in 1979 [see Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] and that \textit{G. Lusztig} introduced the concept of families of irreducible characters of \(W\) [see Characters of reductive groups over a finite field. Ann. Math. Stud. 107. Princeton: Princeton University Press (1984; Zbl 0556.20033)]. The authors of the paper conjecture that their definition of cells should coincide with that of Kazhdan-Lusztig and that the Calogero-Moser families of irreducible characters of \(G\) coincide with the Lusztig families when \(G\) is a Coxeter group. The authors announce that their forthcoming paper will provide certain evidences to support the above conjectures [see \textit{C. Bonnafé} and \textit{R. Rouquier}, ``Cellules de Calogero-Moser'', in preparation].