Alvis–Curtis duality explained

In mathematics, the Alvis–Curtis duality is a duality operation on the characters of a reductive group over a finite field, introduced by and studied by his student . introduced a similar duality operation for Lie algebras.

Alvis–Curtis duality has order 2 and is an isometry on generalized characters.

discusses Alvis–Curtis duality in detail.

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} \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 ε_Gε_TR.
• 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.