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Alvis–Curtis duality

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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 (1980) and studied by his student Dean Alvis (1979). Kawanaka (1981, 1982) introduced a similar duality operation for Lie algebras.

Contents

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

ζ ∗ = ∑ J ⊆ R ( − 1 ) J ζ 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–Curtis duality Wikipedia
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