The map from nilpotent classes to conjugacy classes in the Weyl group introduced in this paper was computed
in type B,C,D in

N.Spaltenstein, Polynomials over local fields, nilpotent orbits and conjugacy classes in Weyl groups,
    Asterisque 168(1988),

N.Spaltenstein, A note on the Kazhdan-Lusztig map for even orthogonal Lie algebras, Arch.Math.(Basel)
55(1990),

and for most cases in exceptional types in

N.Spaltenstein, On the Kazhdan-Lusztig map for exceptional Lie algebras, Adv.Math. 83(1990).

The conjecture on the dimension of fixed point sets on an affine flag manifold in the Introduction 
was proved in

R.Bezrukavnikov, The dimension of the fixed point set on affine flag manifolds, Math.Res.Lett.3(1996).