Cells in Coxeter groups, I

The purpose of this article is to shed new light on the combinatorial structure of Kazhdan–Lusztig cells in infinite Coxeter groups W. Our main focus is the set D of distinguished involutions in W, which was introduced by Lusztig in one of his first papers on cells in affine Weyl groups. We conjecture that the set D has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of D we assign an explicitly defined set of equivalence relations on W that altogether conjecturally determine the partition of W into left (right) cells. We are able to prove these conjectures only in a special case, but even from these partial results we can deduce some interesting corollaries.