  We define a partial order on the set of pairs {(O,C)}, where O is a nilpotent
orbit and C is a conjugacy class in Lusztig's canonical quotient of A(O). We
then show that there is a unique order-reversing duality map on this set that
has certain properties analagous to those of the original Lusztig-Spaltenstein
duality map. This generalizes work of E. Sommers.
  In this revised version, a new approach leads to a strengthened version of
Theorem 2, and applications of the results are discussed in greater detail.
