Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T06:36:42.784Z Has data issue: false hasContentIssue false
  • English
  • Français

The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups

Part of: Representation theory of groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  29 November 2018

Jay Taylor
Jay Taylor*
Affiliation:
University of Arizona, Tucson, AZ 85721, USA (jaytaylor@math.arizona.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition, we obtain a parametrization of the isomorphism classes of all root data. By working at the level of root data, we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms, such embeddings were constructed by Benjamin Martin. In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. Using our investigations into root data we give new proofs of Asai's results and generalize them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.

Keywords

root datasmooth regular embeddingsunipotent supports

MSC classification

Primary: 20G07: Structure theory
Secondary: 20C33: Representations of finite groups of Lie type
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 523 - 552
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Achar, P. N. and Aubert, A.-M., Supports unipotents de faisceaux caractères, J. Inst. Math. Jussieu 6(2) (2007), 173–207.Google Scholar
2.Asai, T., Endomorphism algebras of the reductive groups over 𝔽∥ of classical type, unpublished manuscript.Google Scholar
3.Bonnafé, C., Sur les caractères des groupes réductifs finis à centre non connexe: applications aux groupes spéciaux linéaires et unitaires, Astérisque 306 (2006).Google Scholar
4.Borel, A., Linear algebraic groups, 2nd edn, Graduate Texts in Mathematics, Volume 126 (Springer-Verlag, New York, 1991).Google Scholar
5.Bourbaki, N., Lie groups and Lie algebras (Chapters 4–6), Elements of Mathematics (Springer-Verlag, Berlin, 2002), Translated from the 1968 French original by Andrew Pressley.Google Scholar
6.Conrad, B., Gabber, O. and Prasad, G., Pseudo-reductive groups, New Mathematical Monographs, Volume 17 (Cambridge University Press, Cambridge, 2010).Google Scholar
7.Deligne, P. and Lusztig, G., Representations of reductive groups over finite fields, Ann. of Math. (2) 103(1) (1976), 103–161.Google Scholar
8.Diaconis, P. W. and Graham, R. L., The graph of generating sets of an abelian group, Colloq. Math. 80(1) (1999), 31–38.Google Scholar
9.Digne, F. and Michel, J., Representations of finite groups of Lie type, London Mathematical Society Student Texts, Volume 21 (Cambridge University Press, Cambridge, 1991).Google Scholar
10.Geck, M., On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes, Doc. Math. 1(15) (1996), 293–317.Google Scholar
11.Geck, M., Hiss, G., Lübeck, F., Malle, G. and Pfeiffer, G., CHEVIE – A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175–210.Google Scholar
12.Geck, M. and Malle, G., On the existence of a unipotent support for the irreducible characters of a finite group of Lie type, Trans. Amer. Math. Soc. 352(1) (2000), 429–456.Google Scholar
13.Geck, M. and Malle, G., Reductive groups and Steinberg maps, arXiv:1608.01156 (2016).Google Scholar
14.Gille, P. and Polo, P. (eds.), Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs, Documents Mathématiques, Volume 8 (Société Mathématique de France, Paris, 2011).Google Scholar
15.Humphreys, J. E., Linear algebraic groups, Graduate Texts in Mathematics, Volume 21 (Springer-Verlag, New York, 1975).Google Scholar
16.Hungerford, T. W., Algebra, Graduate Texts in Mathematics, Volume 73 (Springer-Verlag, New York, 1980), Reprint of the 1974 original.Google Scholar
17.Jantzen, J. C., Representations of algebraic groups, 2nd edn, Mathematical Surveys and Monographs, Volume 107 (American Mathematical Society, Providence, RI, 2003).Google Scholar
18.Lübeck, F., Charaktertafeln für die gruppen CSp6(q) mit ungeradem q und Sp6(q) mit geradem q, PhD thesis, Universität Heidelberg (1993).Google Scholar
19.Lusztig, G., On the finiteness of the number of unipotent classes, Invent. Math. 34(3) (1976), 201–213.Google Scholar
20.Lusztig, G., Characters of reductive groups over a finite field, Annals of Mathematics Studies, Volume 107 (Princeton University Press, Princeton, NJ, 1984).Google Scholar
21.Lusztig, G., On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 157–166.Google Scholar
22.Lusztig, G., A unipotent support for irreducible representations, Adv. Math. 94(2) (1992), 139–179.Google Scholar
23.Martin, B. M. S., Étale slices for representation varieties in characteristic p, Indag. Math. (N.S.) 10(4) (1999), 555–564.Google Scholar
24.Michel, J., The development version of the CHEVIE package of GAP3, J. Algebra 435 (2015), 308–336.Google Scholar
25.Springer, T. A., Linear algebraic groups, Modern Birkhäuser Classics (Birkhäuser Boston Inc., Boston, MA, 2009).Google Scholar
26.Steinberg, R., The isomorphism and isogeny theorems for reductive algebraic groups, J. Algebra 216(1) (1999), 366–383.Google Scholar
27.Taylor, J., Generalized Gelfand–Graev representations in small characteristics, Nagoya Math. J. 224(1) (2016), 93–167.Google Scholar
28.Taylor, J., Action of automorphisms on irreducible characters of symplectic groups, J. Algebra 505 (2018), 211–246.Google Scholar