Curtis duality

In mathematics, Alvis–Curtis duality is a duality operation on the characters of a reductive group over a finite field, introduced by Charles W. Curtis (1980) and studied by his student Dean Alvis (1979). Kawanaka (1981, 1982) introduced a similar duality operation for Lie algebras.

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

Carter (1985, 8.2) discusses Alvis–Curtis duality in detail.

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
_PJ 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^G
_PJ is the induced representation of G. (The operation of truncation is the adjoint functor of parabolic induction.)

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