Alvis–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 () and studied by his student Dean Alvis (). Kawanaka (, ) introduced a similar duality operation for Lie algebras.
Alvis–Curtis duality has order 2 and is an isometry on generalized characters.
, 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 ζG
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. (The operation of truncation is the adjoint functor of parabolic induction.)
Examples
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, MR546315, http://dx.doi.org/10.1090/S0273-0979-1979-14690-1
- 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, MR794307, http://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, MR563231, http://dx.doi.org/10.1016/0021-8693(80)90185-4
- 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, ISSN 0386-2194, MR637555, 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, MR679766, http://dx.doi.org/10.1007/BF01389363
Retrieved from : http://en.wikipedia.org/wiki/Alvis%E2%80%93Curtis_duality
