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On unipotent radicals of pseudo-reductive groups

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 Added by David Stewart
 Publication date 2017
  fields
and research's language is English




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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 R_{k/k}(G) of a reductive k-group G. When G= R_{k/k}(G) we also provide some results on the orders of elements of the unipotent radical RR_u(G_{bar k}) of the extension of scalars of G to the algebraic closure bar k of k.



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Let $k/k$ be a finite purely inseparable field extension and let $G$ be a reductive $k$-group. We denote by $G=R_{k/k}(G)$ the Weil restriction of $G$ across $k/k$, a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical $mathscr{R}_{u}(G_{bar{k}})$, focusing on the case $G=GL_n$.
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