  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 ${\mathbb{C}}^\times$-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 will be the theme of a forthcoming paper.
