Regular Elements and Gelfand-Graev Representations for Disconnected Reductive Groups

Karine Sorlin

Bulletin de la Société Mathématique de France (2004)

  • Volume: 132, Issue: 2, page 157-199
  • ISSN: 0037-9484

Abstract

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Let G be a connected reductive group defined over 𝔽 q and let F be the corresponding Frobenius endomorphism. Let σ be a quasi-central automorphism of G , which means that σ is quasi-semi-simple (i.e. σ stabilises ( T ⊂ B ) where T is a maximal torus included in a Borel subgroup B of G ) and dim ( G σ ) > dim ( G σ ' ) for any quasi-semi-simple automorphism σ ' = σ ∘ ad ( g ) , where ad ( g ) is the conjugation by g for all g ∈ G . We suppose also that σ is rational. We define in this article Gelfand-Graev representations for the group G ˜ F = G F · 〈 σ 〉 when σ is unipotent and when it is semi-simple, which extend the σ -stable Gelfand-Graev representations for connected reductive groups. Let T be a σ -stable rational maximal torus of G included in a σ -stable rational Borel subgroup of G . Let U be the unipotent radical of B . In the connected reductive case, Gelfand-Graev representations of G F are obtained by inducing an irreducible linear character of U F which is called a regular character. We define a regular character of U F · 〈 σ 〉 as the extension of a σ -stable regular character of U F . When σ is unipotent, σ -stable Gelfand-Graev representations of G F are obtained by inducing σ -stable regular characters of U F . In this case, we define Gelfand-Graev representations of G F · 〈 σ 〉 as the representations obtained by inducing regular characters of U F · 〈 σ 〉 . When σ is semi-simple, the definition of Gelfand-Graev representations is more complicated. Gelfand-Graev representations of G F · 〈 σ 〉 have similar properties to Gelfand-Graev representations of G F . They are multiplicity free and their Harish-Chandra restrictions to a rational σ -stable Levi subgroup included in a rational σ -stable parabolic subgroup still are Gelfand-Graev representations. We say that an element of G · σ is regular if the dimension of its centralizer in  G is minimal among all elements of G · σ . The dual of any Gelfand-Graev representation of G F · σ is zero outside regular unipotent elements of  G F · σ when σ is unipotent (resp. outside regular pseudo-unipotent elements of  G F · σ ,i.e. conjugates under G of regular elements of U · σ , when σ is semi-simple). Moreover, Gelfand-Graev representations can be used to calculate the average value of irreducible characters of  G F · 〈 σ 〉 on the set of  G F -classes of regular unipotent (resp. pseudo-unipotent) elements of G F · σ if σ is unipotent (resp. semi-simple). When σ is semi-simple, the characteristic can be chosen good for ( G σ ) 0 and we can get the exact values of irreducible characters of G F · 〈 σ 〉 on G F -classes of regular pseudo-unipotent elements of G F · σ .

How to cite

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Sorlin, Karine. "Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes." Bulletin de la Société Mathématique de France 132.2 (2004): 157-199. <http://eudml.org/doc/272466>.

@article{Sorlin2004,
abstract = {Soient $G$ un groupe algébrique réductif connexe défini sur $\mathbb \{F\}_q$ et $F$ l’endomorphisme de Frobenius correspondant. Soit $\sigma $ un automorphisme rationnel quasi-central de $G$. Nous construisons ci-dessous l’équivalent des représentations de Gelfand-Graev du groupe $\widetilde\{G\}^F=G^F\!\cdot \langle \sigma \rangle $, lorsque $\sigma $ est unipotent et lorsqu’il est semi-simple. Nous montrons de plus que ces représentations vérifient des propriétés semblables à celles vérifiées par les représentations de Gelfand-Graev dans le cas connexe en particulier par rapport aux éléments réguliers.},
author = {Sorlin, Karine},
journal = {Bulletin de la Société Mathématique de France},
keywords = {finite reductive groups; disconnected algebraic groups},
language = {fre},
number = {2},
pages = {157-199},
publisher = {Société mathématique de France},
title = {Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes},
url = {http://eudml.org/doc/272466},
volume = {132},
year = {2004},
}

TY - JOUR
AU - Sorlin, Karine
TI - Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes
JO - Bulletin de la Société Mathématique de France
PY - 2004
PB - Société mathématique de France
VL - 132
IS - 2
SP - 157
EP - 199
AB - Soient $G$ un groupe algébrique réductif connexe défini sur $\mathbb {F}_q$ et $F$ l’endomorphisme de Frobenius correspondant. Soit $\sigma $ un automorphisme rationnel quasi-central de $G$. Nous construisons ci-dessous l’équivalent des représentations de Gelfand-Graev du groupe $\widetilde{G}^F=G^F\!\cdot \langle \sigma \rangle $, lorsque $\sigma $ est unipotent et lorsqu’il est semi-simple. Nous montrons de plus que ces représentations vérifient des propriétés semblables à celles vérifiées par les représentations de Gelfand-Graev dans le cas connexe en particulier par rapport aux éléments réguliers.
LA - fre
KW - finite reductive groups; disconnected algebraic groups
UR - http://eudml.org/doc/272466
ER -

References

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