Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T04:32:10.497Z Has data issue: false hasContentIssue false
  • English
  • Français

Restrictions to G(𝔽p) and G(r) of rational G-modules

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

Published online by Cambridge University Press:  26 August 2011

Eric M. Friedlander
Eric M. Friedlander*
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We fix a prime p and consider a connected reductive algebraic group G over a perfect field k which is defined over 𝔽p. Let M be a finite-dimensional rational G-module M, a comodule for k[G]. We seek to somewhat unravel the relationship between the restriction of M to the finite Chevalley subgroup G(𝔽p)⊂G and the family of restrictions of M to Frobenius kernels G(r) ⊂G. In particular, we confront the conundrum that if M is the Frobenius twist of a rational G-module N,M=N(1), then the restrictions of M and N to G(𝔽p) are equal whereas the restriction of M to G(1) is trivial. Our analysis enables us to compare support varieties (and the finer non-maximal support varieties) for G(𝔽p) and G(r) of a rational G-module M where the choice of r depends explicitly on M.

Keywords

1-parameter subgroupsFrobenius kernelsp-nilpotent degree

MSC classification

Secondary: 20C20: Modular representations and characters 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1955 - 1978
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[BR85]Bardsey, P. and Richardson, R. W., Et́ale slices for algebraic transformation groups in characteristic p, Proc. Lond. Math. Soc. 51 (1985), 295–317.Google Scholar
[CFP07]Carlson, J., Friedlander, E. and Pevtsova, J., Modules of constant Jordan type, J. Reine Angew. Math. 614 (2007), 1–44.Google Scholar
[CLN08]Carlson, J., Lin, Z. and Nakano, D., Support varieties for modules over Chevalley groups and classical Lie algebras, Trans. Amer. Math. Soc. 360 (2008), 1870–1906.Google Scholar
[Car85]Carter, R. W., Finite groups of Lie type (Wiley-Interscience, New York, 1985).Google Scholar
[DM91]Digne, F. and Michel, J., Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21 (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[Fri10]Friedlander, E., Weil restriction and support varieties, J. Reine Angew. Math. 648 (2010), 183–200.Google Scholar
[FP]Friedlander, E. and Pevtsova, J., Constructions for infinitesimal group schemes, Trans. Amer. Math. Soc., to appear.Google Scholar
[FP05]Friedlander, E. and Pevtsova, J., Representation-theoretic support spaces for finite group schemes, Amer. J. Math. 127 (2005), 379–420.Google Scholar
[FP07]Friedlander, E. and Pevtsova, J., Π-supports for modules for finite group schemes, Duke Math. J. 139 (2007), 317–368.Google Scholar
[FP10]Friedlander, E. and Pevtsova, J., Generalized support varieties for finite group schemes, Doc. Math. (2010), 197–222 (Extra volume Suslin).Google Scholar
[FPS07]Friedlander, E., Pevtsova, J. and Suslin, A., Generic and maximal Jordan types, Invent. Math. 168 (2007), 485–522.Google Scholar
[FS97]Friedlander, E. and Suslin, A., Cohomology of a finite group scheme over a field, Invent. Math. 127 (1997), 209–270.Google Scholar
[Hum95]Humphreys, J., Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43 (American Mathematical Society, Providence, RI, 1995).Google Scholar
[Jan03]Jantzen, J. C., Representations of algebraic groups, Mathematical Surveys and Monographs, vol. 107, second edition (American Mathematical Society, Providence, RI, 2003).Google Scholar
[LN99]Lin, Z. and Nakano, D., Complexity for modules over finite Chevalley groups and classical Lie algebras, Invent. Math. 138 (1999), 85–101.Google Scholar
[NPV02]Nakano, D., Parshall, B. and Vella, D., Support varieties for algebraic groups, J. Reine Angew. Math. 547 (2002), 15–49.Google Scholar
[Pin04]Pink, R., On Weil restriction of reductive groups and a theorem of Prasad, Math. Z. 248 (2004), 449–457.Google Scholar
[Qui71]Quillen, D., The spectrum of an equivariant ring: I, II, Ann. of Math. (2) 94 (1971), 549–572, 573–602.CrossRefGoogle Scholar
[Sei00]Seitz, G., Unipotent elements, tilting modules, and saturation, Invent. Math. 141 (2000), 467–502.CrossRefGoogle Scholar
[Ser94]Serre, J.-P., Sur la semi-simplicité des produits tensoriels de repésentations de groupes, Invent. Math. 116 (1994), 513–530.Google Scholar
[Spr69]Springer, T. A., The unipotent variety of a semisimple group, Proc. Coll. Alg. Geom. (Tata Institute) (1969), 373–391.Google Scholar
[SFB97a]Suslin, A., Friedlander, E. and Bendel, C., Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc. 10 (1997), 693–728.Google Scholar
[SFB97b]Suslin, A., Friedlander, E. and Bendel, C., Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), 729–759.CrossRefGoogle Scholar
[Tes95]Testerman, D., A 1-type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups, J. Algebra 177 (1995), 34–76.Google Scholar