We present a new characterization of Lusztig's canonical quotient group. We also define a duality map: to a pair consisting of a nilpotent orbit and a conjugacy class in its fundamental group, the map assigns a nilpotent orbit in the Langlands dual Lie algebra.
Tempered Representations and Nilpotent Orbits
Quantizations of nilpotent orbits vs 1-dimensional representations of W-algebras
On Harish-Chandra modules over quantizations of nilpotent orbits
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