From conjugacy classes in the Weyl group to unipotent classes, III
  Let G be an affine algebraic group over an algebraically closed field such that the identity component G^0 of G is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G^0 is a unipotent element.   In this paper we define a map from the set of "twisted conjugay classes" in W to the set of unipotent G^0-conjugacy classes in D, generalizing an earlier construction which applied when G is connected. 
G. Lusztig
math.RT