# 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 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.

## 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^{}_{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.)

## Examples

- The dual of the trivial character 1 is the Steinberg character.
- Deligne & Lusztig (1983) showed that the dual of a Deligne–Lusztig character
*R*^{θ}_{T}is ε_{G}ε_{T}*R*^{θ}_{T}. - 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

- Alvis, Dean (1979), "The duality operation in the character ring of a finite Chevalley group",
*American Mathematical Society. Bulletin. New Series*,**1**(6): 907–911, doi:10.1090/S0273-0979-1979-14690-1, ISSN 0002-9904, MR 0546315 - Carter, Roger W. (1985),
*Finite groups of Lie type. Conjugacy classes and complex characters.*, Pure and Applied Mathematics (New York), New York: John Wiley & Sons, ISBN 978-0-471-90554-7, MR 0794307 - Curtis, Charles W. (1980), "Truncation and duality in the character ring of a finite group of Lie type",
*Journal of Algebra*,**62**(2): 320–332, doi:10.1016/0021-8693(80)90185-4, ISSN 0021-8693, MR 0563231 - Deligne, Pierre; Lusztig, George (1982), "Duality for representations of a reductive group over a finite field",
*Journal of Algebra*,**74**(1): 284–291, doi:10.1016/0021-8693(82)90023-0, ISSN 0021-8693, MR 0644236 - Deligne, Pierre; Lusztig, George (1983), "Duality for representations of a reductive group over a finite field. II",
*Journal of Algebra*,**81**(2): 540–545, doi:10.1016/0021-8693(83)90202-8, ISSN 0021-8693, MR 0700298 - Kawanaka, Noriaki (1981), "Fourier transforms of nilpotently supported invariant functions on a finite simple Lie algebra",
*Japan Academy. Proceedings. Series A. Mathematical Sciences*,**57**(9): 461–464, doi:10.3792/pjaa.57.461, ISSN 0386-2194, MR 0637555 - Kawanaka, N. (1982), "Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field",
*Inventiones Mathematicae*,**69**(3): 411–435, doi:10.1007/BF01389363, ISSN 0020-9910, MR 0679766