Alvis–Curtis duality

The dual ζ* of a character ζ of a finite group G with a split BN-pair is defined to be

:$\backslash zeta^*=\backslash sum\_\{J\backslash subseteq\; R\}(-1)^\{\backslash vert\; J\backslash vert\}\backslash zeta^G\_\{P\_J\}$

Here the sum is over all subsets J of the set R of simple roots of the Coxeter system of G. The character ζ{{su|p=|b=P_J}} 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{{su|p=G|b=P_J}} is the induced representation of G. (The operation of truncation is the adjoint functor of parabolic induction.)

• The dual of the trivial character 1 is the Steinberg character.
• {{harvtxt|Deligne|Lusztig|1983}} showed that the dual of a Deligne–Lusztig character R{{su|b=T|p=θ}} is ε_Gε_TR{{su|b=T|p=θ}}.
• 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.

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Category:Representation theory

Category:Duality theories