We provide an alternative description of the restriction to spherical
unipotent conjugacy classes, of Lusztig's map Psi from the set of unipotent
conjugacy classes in a connected reductive algebraic group to the set of
conjugacy classes of its Weyl group. For irreducible root systems, we analyze
the image of this restricted map and we prove that a conjugacy class in a
finite Weyl group has a unique maximal length element if and only if it has a
maximum.Comment: Refereed version, one reference added. The final version will appear
in the Bulletin of the London Mathematical Societ
