Alvis–Curtis duality

Definition

The dual ζ* of a character ζ of a finite group G with a split BN-pair is defined to be :\zeta^*=\sum_{J\subseteq R}(-1)^{\vert J\vert}\zeta^G_{P_J} Here the sum is over all subsets J of the set R of simple roots of the Coxeter system of G. The character ζ is the truncation of ζ to the parabolic subgroup P**J of the subset J, given by restricting ζ to P**J and then taking the space of invariants of the unipotent radical of P**J, and ζ is the induced representation of G. (The operation of truncation is the adjoint functor of parabolic induction.)

Examples

• The dual of the trivial character 1 is the Steinberg character.
• showed that the dual of a Deligne–Lusztig character R is εGεT**R.
• The dual of a cuspidal character χ is (–1)|Δ|χ, where Δ is the set of simple roots.
• The dual of the Gelfand–Graev character is the character taking value |Z**F|q**l on the regular unipotent elements and vanishing elsewhere.

References