## The Michigan Mathematical Journal

### On Unipotent Radicals of Pseudo-Reductive Groups

#### 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 $p^{e}$, and let $G$ denote the Weil restriction of scalars $\mathrm{R}_{k'/k}(G')$ of a reductive $k'$-group $G'$. When $G=\mathrm {R}_{k'/k}(G')$, we also provide some results on the orders of elements of the unipotent radical $\mathscr{R}_{u}(G_{\bar{k}})$ of the extension of scalars of $G$ to the algebraic closure $\bar{k}$ of $k$.

#### Article information

Source
Michigan Math. J., Volume 68, Issue 2 (2019), 277-299.

Dates
Revised: 7 September 2018
First available in Project Euclid: 18 February 2019

https://projecteuclid.org/euclid.mmj/1550480563

Digital Object Identifier
doi:10.1307/mmj/1550480563

Mathematical Reviews number (MathSciNet)
MR3961217

Zentralblatt MATH identifier
07084763

Subjects
Primary: 20G15: Linear algebraic groups over arbitrary fields

#### Citation

Bate, Michael; Martin, Benjamin; Röhrle, Gerhard; Stewart, David I. On Unipotent Radicals of Pseudo-Reductive Groups. Michigan Math. J. 68 (2019), no. 2, 277--299. doi:10.1307/mmj/1550480563. https://projecteuclid.org/euclid.mmj/1550480563

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