mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Alvis–Curtis duality is a duality operation on the

characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...

of a

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

, introduced by and studied by his student . introduced a similar duality operation for Lie algebras. Alvis–Curtis duality has order 2 and is an isometry on generalized characters. discusses Alvis–Curtis duality in detail.

Definition

The dual ζ* of a character ζ of a finite group ''G'' with a split

BN-pair In mathematics, a (''B'', ''N'') pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar ...

is defined to be :

\zeta^*=\sum_(-1)^\zeta^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 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 is the induced representation of ''G''. (The operation of truncation is the adjoint functor of

parabolic induction In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups. If ''G'' is a reductive algebraic group and P=MAN is the Langlands decomposition of a parabol ...

Examples

*The dual of the trivial character 1 is the Steinberg character. * showed that the dual of a Deligne–Lusztig character ''R'' is ε_''G''ε_''T''''R''. *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

* * * * * * * {{DEFAULTSORT:Alvis-Curtis duality Representation theory Duality theories