Lusztig constants and endoscopy

In the 80’s, Lusztig described the character table of G(F_q) in terms of generalized Green functions, which arise from equivariant perverse sheaves on the unipotent variety. For large p we can pass to the Lie algebra, and Lusztig showed that the building blocks - cuspidal equivariant perverse sheaves on the nilpotent cone - are invariant under Deligne-Fourier transforms. Consequently, a corresponding function is equal to its Fourier transform up to a 4th root of unity, sometimes called a Lusztig constant.