We study the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group W. We determine some of (and conjecturally all) the C ×-fixed points of its unique irreducible component of maximal dimension in terms of the character table of W. This is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper [Bon3].