 
Summary: AN ORDERREVERSING DUALITY MAP FOR CONJUGACY CLASSES IN
LUSZTIG'S CANONICAL QUOTIENT
PRAMOD N. ACHAR
Abstract. We define a partial order on the set No,¯c of pairs (O, C), where O is a nilpotent orbit and C
is a conjugacy class in ¯A(O), Lusztig's canonical quotient of A(O). We then construct an orderreversing
duality map No,¯c L
No,¯c that satisfies many of the properties of the original Spaltenstein duality map.
This generalizes work of Sommers [16].
1. Introduction
Let G be a connected complex reductive algebraic group, and g its Lie algebra. Let N be the nilpotent
cone in g; let No be the set of Gorbits in N, and let Nsp
o No be the set of special orbits. Given a nilpotent
orbit O No, let A(O) be the component group of the isotropy group of O in the adjoint group for g, and
let ¯A(O) denote Lusztig's canonical quotient of A(O) (see [10] and [16]). Let No,c denote the set of pairs
(O, C), where O is a nilpotent orbit, and C is a conjugacy class in A(O). We define No,¯c the same way,
except that we take conjugacy classes in ¯A(O) rather than A(O). There is a projection No,c No,¯c arising
from the projection A(O) ¯A(O). We also have a projection pr1 : No,c No, as well as an inclusion
i : No No,c which pairs O with the trivial conjugacy class in A(O). Finally, let L
G be the Langlands dual
group of G, and define L
