Achar, Pramod N.; Hardesty, William D.; Riche, Simon

Representation Theory of Disconnected Reductive Groups

Doc. Math. 25, 2149-2177 (2020)
DOI: 10.25537/dm.2020v25.2149-2177
Communicated by Julia Pevtsova

Summary

We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic \(0\) and those in characteristic \(p\) (for groups defined over discrete valuation rings of mixed characteristic). For each of these topics, we obtain natural generalizations of the well-known results for connected reductive groups.

Mathematics Subject Classification

20G05, 17B10, 20C15

Keywords/Phrases

disconnected reductive groups, Clifford theory

References

  • 1. P. Achar, W. Hardesty, and S. Riche, Integral exotic sheaves and the modular Lusztig-Vogan bijection, preprint. arxiv 1810.06851.
  • [DG]. M. Demazure and P. Gabriel, Groupes algébriques. Tome I : Géométrie algébrique, géné\-ralités, groupes commutatifs, with an appendix Corps de classes local by M. Hazewinkel, Masson \& Cie, North-Holland Publishing Co., 1970. zbl 0203.23401; MR0302656.
  • [SGA3.1]. M. Demazure, A. Grothendieck et al., Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes. Séminaire de Géométrie Algébrique du Bois Marie 1962-64. A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J.-P. Serre. Revised and annotated edition of the 1970 French original. Edited by Philippe Gille and Patrick Polo. Documents Mathématiques 7, Société Mathématique de France, 2011. zbl 1241.14002;.
  • [SGA3.2]. M. Demazure, A. Grothendieck et al., Schémas en groupes II. Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géometrie Algébrique du Bois-Marie 1962/64 (SGA 3), Lecture Notes in Math., vol. 152, Springer-Verlag, 1970. zbl 0209.24201.
  • [SGA3.3]. M. Demazure, A. Grothendieck et al., Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs. Séminaire de Géométrie Algébrique du Bois Marie 1962-64. A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J.-P. Serre. Revised and annotated edition of the 1970 French original. Edited by Philippe Gille and Patrick Polo. Documents Mathématiques 8, Société Mathématique de France, 2011. zbl 1241.14003; MR0207705.
  • [GP]. M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, Oxford University Press, 2000. zbl 0996.20004; MR1778802.
  • [EGA4.4]. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 5-361. zbl 0153.22301; MR0238860.
  • [Ja]. J. C. Jantzen, Representations of algebraic groups, second edition, Mathematical surveys and monographs 107, Amer. Math. Soc., 2003. zbl 1034.20041; MR2015057.
  • [MR]. C. Mautner and S. Riche, On the exotic t-structure in positive characteristic, Int. Math. Res. Not. IMRN 2016, 5727-5774. DOI 10.1093/imrn/rnv330; zbl 1404.14022; MR3567258; arxiv 1412.6818; MR].
  • [Mi]. J. S. Milne, Algebraic groups: The theory of group schemes of finite type over a field, Cambridge Studies in Advanced Mathematics, no. 170, Cambridge University Press, 2017. DOI 10.1017/9781316711736; zbl 1390.14004; MR3729270.
  • [Ri]. S. Riche, Geometric Representation Theory in positive characteristic, habilitation thesis, available at https://tel.archives-ouvertes.fr/tel-01431526.
  • [Se]. J.-P. Serre, Groupes de Grothendieck des schémas en groupes réductifs déployés, Publ. Math. I.H.E.S. 34 (1968), 37-52. DOI 10.1007/BF02684589; zbl 0195.50802; MR0231831.
  • [SP]. The Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2018.

Affiliation

Achar, Pramod N.
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
Hardesty, William D.
School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia
Riche, Simon
Université Clermont Auvergne CNRS, LMBP, F-63000 Clermont-Ferrand, France

Downloads