Alvis–Curtis duality
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
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 PJ of the subset J, given by restricting ζ to PJ and then taking the space of invariants of the unipotent radical of PJ, and ζ is the induced representation of G.Examples
