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
\( {\displaystyle \zeta ^{*}=\sum _{J\subseteq R}(-1)^{J}\zeta _{P_{J}}^{G}} \)
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 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 ζG
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 Rθ
T is εGεTRθ
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 |ZF|ql 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
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