arXiv subjects

Maike Gruchot

Publications and source records attributed to Maike Gruchot.

4 recordsLinked to original sources

arXiv2022

On the exponent of geometric unipotent radicals of pseudo-reductive groups

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 $\RR_{u}(G_{\bar{k}})$ in terms of invariants of the extension $k'/k$, starting with the case $G'=\GL_n$ and applying these results to the case where $G'$ is a simple group.

math.GR↗

arXiv2021

Complete reducibility: Variations on a theme of Serre

In this note, we unify and extend various concepts in the area of $G$-complete reducibility, where $G$ is a reductive algebraic group. By results of Serre and Bate--Martin--Röhrle, the usual notion of $G$-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of $G$. We show that other variations of this notion, such as relative complete reducibility and $σ$-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

math.GR↗

arXiv2020

Relative complete reducibility and normalised subgroups

We study a relative variant of Serre's notion of $G$-complete reducibility for a reductive algebraic group $G$. We let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ which normalise the identity component $K^{\circ}$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ}$ is completely reducible. This allows us to generalise a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$, as well as 'rational' versions over non-algebraically closed fields.

math.GR↗

arXiv2019

On relative complete reducibility

Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre's notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility which directly generalizes equivalent formulations of $G$-complete reducibility. If the ambient group $G$ is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from the aforementioned work of Bate-Martin-Roehrle-Tange.

math.GR↗