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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 show that there is a unique order-reversing duality map No,c → LNo,c that has certain properties analogous to those of the original Lusztig–Spaltenstein duality map. This generalizes work of E. Sommers.

An order-reversing duality map for conjugacy classes in Lusztig's canonical quotient