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) 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

[math]\displaystyle{ \zeta^*=\sum_{J\subseteq R}(-1)^{\vert J\vert}\zeta^G_{P_J} }[/math]

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 ζGP_J 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) showed that the dual of a Deligne–Lusztig character RθT is ε_Gε_TRθT.
• 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", Bulletin of the American Mathematical Society, New Series 1 (6): 907–911, doi:10.1090/S0273-0979-1979-14690-1, ISSN 0002-9904 
• 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, https://books.google.com/books?id=LvvuAAAAMAAJ 
• 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 
• 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 
• 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 
• 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, http://projecteuclid.org/getRecord?id=euclid.pja/1195516260 
• 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