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On Weil restriction of reductive groups and a theorem of Prasad

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Abstract.

Let G be a connected simple semisimple algebraic group over a local field F of arbitrary characteristic. In a previous article by the author the Zariski dense compact subgroups of G(F) were classified. In the present paper this information is used to give another proof of a theorem of Prasad [8] (also proved by Margulis [3]) which asserts that, if G is isotropic, every non-discrete closed subgroup of finite covolume contains the image of where denotes the universal covering of G. This result played a central role in Prasad’s proof of strong approximation. The present proof relies on some basic properties of Weil restrictions over possibly inseparable field extensions, which are also proved here.

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References

  1. Hiss, G.: Die adjungierten Darstellungen der Chevalley-Gruppen. Arch. Math. 42, 408–416 (1984)

    MathSciNet  Google Scholar 

  2. Hogeweij, G.M.D.: Almost Classical Lie Algebras I. Indagationes Math. 44, 441–460 (1982)

    MathSciNet  Google Scholar 

  3. Margulis, G.A.: Cobounded subgroups in algebraic groups over local fields. Funkcional. Anal. i Priložen 11 (2), 45–57 (1977); Funct. Anal. Appl. 11(2), 119–128 (1977)

    MATH  Google Scholar 

  4. Oesterlé, J.: Nombres de Tamagawa et groupes unipotents en caractéristique p. Invent. Math. 78, 13–88 (1984)

    MathSciNet  Google Scholar 

  5. Pink, R.: Compact subgroups of linear algebraic groups. J. Algebra 206, 438–504 (1998)

    Article  MathSciNet  Google Scholar 

  6. Pink, R.: Strong approximation for Zariski dense subgroups over arbitrary global fields. Comm. Math. Helv. 75, 608–643 (2000)

    Article  MathSciNet  Google Scholar 

  7. Platonov, V., Rapinchuk, A.: Algebraic Groups and Number Theory. Boston etc.: Academic Press 1994

  8. Prasad, G.: Strong approximation for semi-simple groups over function fields. Annals of Math. 105, 553–572 (1977)

    MATH  Google Scholar 

  9. Prasad, G.: Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits. Bull. Soc. math. France 110, 197–202 (1982)

    MathSciNet  Google Scholar 

  10. Prasad, G., Raghunathan, M.S.: On the Kneser-Tits problem. Comment. Math. Helv. 60(1), 107–121 (1985)

    MATH  Google Scholar 

  11. Springer, T.A.: Linear Algebraic Groups. Second Edition. Boston etc.: Birkhäuser, 1998

  12. Wang, S.P.: On density properties of S-subgroups of locally compact groups. Annals of Math. 94, 325–329 (1971)

    MATH  Google Scholar 

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Acknowledgments.

The author would like to thank Gopal Prasad and Marc Burger for interesting conversations, and especially the former for suggesting a combination of the~methods of [8] and [5] to obtain another proof of strong approximation in arbitrary characteristic.

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Authors and Affiliations

  1. Departement Mathematik, ETH Zentrum, CH, -8092, Zürich, Switzerland

    Richard Pink

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  1. Richard Pink
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Correspondence to Richard Pink.

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Mathematics Subject Classification (2000): 20G25, 14L15

in final form: 3 January 2001

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Pink, R. On Weil restriction of reductive groups and a theorem of Prasad. Math. Z. 248, 449–457 (2004). https://doi.org/10.1007/s002090100339

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  • DOI: https://doi.org/10.1007/s002090100339

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