  Using the geometry of the associated Calogero-Moser space, R. Rouquier and
the author have attached to any finite complex reflection group $W$ several
notions (Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular
characters), completing the notion of Calogero-Moser families defined by
Gordon. If moreover $W$ is a Coxeter group, they conjectured that these notions
coincide with the analogous notions defined using the Hecke algebra by Kazhdan
and Lusztig (or Lusztig in the unequal parameters case). In the present paper,
we aim to investigate these conjectures whenever $W$ is a dihedral group.
