On Unipotent Radicals of Pseudo-Reductive Groups

Michael Bate; Benjamin Martin; Gerhard Röhrle; David I. Stewart

doi:10.1307/mmj/1550480563

On Unipotent Radicals of Pseudo-Reductive Groups

Michael Bate, Benjamin Martin, Gerhard Röhrle, David I. Stewart

Mathematical Science

Research output: Contribution to journal › Article › peer-review

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Abstract

We establish some results on the structure of the geometric unipotent radicals of
pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k^'be a purely inseparable
field extension of k of degree p^eand let G denote the Weil restriction of scalars Rk^'/k(G^') of a reductive k^'-group G^'. When G = Rk^'/k(G^') we also provide some results on the orders of elements of the unipotent radical ℜ_u(G_k¯) of the extension of scalars of G to the algebraic closure k^¯ of k.

Original language	English
Pages (from-to)	277-299
Number of pages	23
Journal	The Michigan Mathematical Journal
Volume	68
Issue number	2
Early online date	18 Feb 2019
DOIs	https://doi.org/10.1307/mmj/1550480563
Publication status	Published - 1 Jun 2019

Bibliographical note

Acknowledgements
Part of the research for this paper was carried out while the authors were
staying at the Mathematical Research Institute Oberwolfach supported by the “Research in Pairs” programme. The authors also acknowledge the financial support of EPSRC Grant EP/L005328/1 and Marsden Grant UOA1021.
We thank Brian Conrad for comments and discussion. We thank Gopal Prasad for some helpful hints to improve the results. We are also grateful to the referee for their comments.

Keywords

ALGEBRAIC-GROUPS

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Cite this

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title = "On Unipotent Radicals of Pseudo-Reductive Groups",

abstract = "We establish some results on the structure of the geometric unipotent radicals ofpseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k' be a purely inseparablefield extension of k of degree pe and let G denote the Weil restriction of scalars Rk'/k(G') of a reductive k'-group G'. When G = Rk'/k(G') we also provide some results on the orders of elements of the unipotent radical ℜu(Gk¯) of the extension of scalars of G to the algebraic closure k¯ of k.",

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note = "Acknowledgements Part of the research for this paper was carried out while the authors were staying at the Mathematical Research Institute Oberwolfach supported by the “Research in Pairs” programme. The authors also acknowledge the financial support of EPSRC Grant EP/L005328/1 and Marsden Grant UOA1021. We thank Brian Conrad for comments and discussion. We thank Gopal Prasad for some helpful hints to improve the results. We are also grateful to the referee for their comments. ",

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doi = "10.1307/mmj/1550480563",

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AU - Bate, Michael

AU - Martin, Benjamin

AU - Röhrle, Gerhard

AU - Stewart, David I.

N1 - Acknowledgements Part of the research for this paper was carried out while the authors were staying at the Mathematical Research Institute Oberwolfach supported by the “Research in Pairs” programme. The authors also acknowledge the financial support of EPSRC Grant EP/L005328/1 and Marsden Grant UOA1021. We thank Brian Conrad for comments and discussion. We thank Gopal Prasad for some helpful hints to improve the results. We are also grateful to the referee for their comments.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - We establish some results on the structure of the geometric unipotent radicals ofpseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k' be a purely inseparablefield extension of k of degree pe and let G denote the Weil restriction of scalars Rk'/k(G') of a reductive k'-group G'. When G = Rk'/k(G') we also provide some results on the orders of elements of the unipotent radical ℜu(Gk¯) of the extension of scalars of G to the algebraic closure k¯ of k.

AB - We establish some results on the structure of the geometric unipotent radicals ofpseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k' be a purely inseparablefield extension of k of degree pe and let G denote the Weil restriction of scalars Rk'/k(G') of a reductive k'-group G'. When G = Rk'/k(G') we also provide some results on the orders of elements of the unipotent radical ℜu(Gk¯) of the extension of scalars of G to the algebraic closure k¯ of k.

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EP - 299

JO - The Michigan Mathematical Journal

JF - The Michigan Mathematical Journal

IS - 2

ER -

On Unipotent Radicals of Pseudo-Reductive Groups

Abstract

Bibliographical note

Keywords

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