mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

character 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 Theoph ...

s 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 that has a finite kernel and is a ...

over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

, introduced by and studied by his student . introduced a similar duality operation for

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

s. Alvis–Curtis duality has order 2 and is an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

on generalized characters. discusses Alvis–Curtis duality in detail.

Definition

The dual ζ* of a character ζ of a

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

''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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

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 paraboli ...

Examples

*The dual of the trivial character 1 is the

Steinberg character Steinberg Media Technologies GmbH (trading as Steinberg; ) is a German musical software and hardware company based in Hamburg. It develops software for writing, recording, arranging and editing music, most notably Cubase, Nuendo, and Dorico. It ...

. * 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